[{"id": "bio_phylogenetics", "title": "Phylogenetic Tree Reconstruction via the Latent Framework", "domain": "mathematical_biology", "lean": true, "words": 3088, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Phylogenetic inference turns molecular sequences into historical relationships. Classical likelihood-based methods excel in practice, yet the information geometry of alignment data relative to tree topology is rarely summarized in coordinates comparable across studies.", "url": "https://thalens.org/papers/bio_phylogenetics/", "pdf": "https://thalens.org/papers/bio_phylogenetics/paper.pdf", "created": "", "updated": "2026-04-23"}, {"id": "phy_smale6", "title": "Dimension-Independent Finiteness of Central Configurations for Positive Masses", "domain": "verification", "lean": true, "words": 18003, "doi": "10.5281/zenodo.19710553", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We prove that for any $N \\geq 3$ bodies with positive masses in $\\mathbb{R}^d$ ($d \\geq 2$), the number of central configurations modulo similarity is finite, resolving Smale's 6th Problem in **every spatial dimension $d \\geq 2$ simultaneously**.", "url": "https://thalens.org/papers/phy_smale6/", "pdf": "https://thalens.org/papers/phy_smale6/paper.pdf", "created": "2026-03-24", "updated": "2026-04-23"}, {"id": "ml_structured_latent_basis", "title": "The Structured Latent Basis: Feature Engineering as Basis Selection", "domain": "ml", "lean": false, "words": 4963, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We introduce the Structured Latent Basis (SLB) framework, a perspective on supervised learning that unifies feature engineering and modeling as a single problem: selecting a mathematical basis in which the target function is linear.", "url": "https://thalens.org/papers/ml_structured_latent_basis/", "pdf": "", "created": "2026-04-22", "updated": "2026-04-22"}, {"id": "phy_m_theory_dimensions", "title": "Formal Koide Structure: Mass Bounds, Generation Counting, and Neutrino Predictions from the Z_N Ansatz", "domain": "verification", "lean": true, "words": 3710, "doi": "10.5281/zenodo.19702955", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "The Z₃ Ansatz $\\sqrt{m_r} = a(1 + b\\cos(\\theta_0 + 2\\pi r/3))$ with $b^2 = 2$ is a parametrization — not a dynamical model — that encodes the Koide mass relation $Q = 2/3$.", "url": "https://thalens.org/papers/phy_m_theory_dimensions/", "pdf": "https://thalens.org/papers/phy_m_theory_dimensions/paper.pdf", "created": "2026-04-10", "updated": "2026-04-22"}, {"id": "phy_yang_mills_mass_gap", "title": "The Yang-Mills Mass Gap via Gauge Absorption and Perelman W-Entropy", "domain": "verification", "lean": true, "words": 20020, "doi": "10.5281/zenodo.19681807", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "**Theorem A (Main result, conditional).** Conditional on the 20 named Tier A–D hypotheses of §7.1 — in particular the three Tier-D perturbative-QFT inputs (`tomboulis_formula`, `b_zero_from_feynman`, `beta_1_rge_def`) — we establish that Yang-Mills t", "url": "https://thalens.org/papers/phy_yang_mills_mass_gap/", "pdf": "https://thalens.org/papers/phy_yang_mills_mass_gap/paper.pdf", "created": "2026-04-07", "updated": "2026-04-22"}, {"id": "ml_smooth_step_spectral", "title": "The Smooth-Step Spectral Method: Unifying Smooth and Threshold Structure in Tabular Regression", "domain": "ml", "lean": false, "words": 6039, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We introduce the Smooth-Step Spectral Method (S³M), a structured basis\nregression approach that unifies smooth spectral features with learned\nthreshold features for tabular data.", "url": "https://thalens.org/papers/ml_smooth_step_spectral/", "pdf": "", "created": "2026-04-21", "updated": "2026-04-21"}, {"id": "nt_rh_de_branges_chain", "title": "An Unconditional BGST$\\to$R$_2$ Fourier Transfer via Poisson-Kernel Deconvolution", "domain": "number_theory", "lean": true, "words": 4073, "doi": "10.5281/zenodo.19689165", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh [BGST23] proved the first unconditional asymptotic for Montgomery's pair-correlation function $F(\\alpha, T)$, with error $O(1/\\sqrt{\\log T})$ uniformly for $\\alpha \\in [0,1]$.", "url": "https://thalens.org/papers/nt_rh_de_branges_chain/", "pdf": "", "created": "2026-04-19", "updated": "2026-04-21"}, {"id": "pde_tensor_algebra", "title": "The PDE Tensor Algebra: Structural Decomposition and Exact Recombination of Differential Equations", "domain": "verification", "lean": true, "words": 67878, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We introduce the **PDE Tensor Algebra**, a framework that represents any PDE system as a triple $(D, C, P)$ of tensors encoding dissipation, nonlinear coupling, and geometric constraints. The decomposition converts qualitative PDE questions — existence, uniqueness, regularity, stability — into compu", "url": "https://thalens.org/papers/pde_tensor_algebra/", "pdf": "https://thalens.org/papers/pde_tensor_algebra/paper.pdf", "created": "2026-04-11", "updated": "2026-04-21"}, {"id": "phy_fine_structure", "title": "Machine-Verified Derivation Chain for the Fine Structure Constant via the E8 → Standard Model Breaking Pattern", "domain": "verification", "lean": true, "words": 3615, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a machine-verified derivation chain connecting the E₈ Lie algebra to the electromagnetic fine structure constant $\\alpha \\approx 1/137$.", "url": "https://thalens.org/papers/phy_fine_structure/", "pdf": "https://thalens.org/papers/phy_fine_structure/paper.pdf", "created": "2026-04-10", "updated": "2026-04-21"}, {"id": "phy_quantum_tuna9_parity", "title": "A refuted-and-vindicated pre-registration test of a spectral error model on a superconducting processor", "domain": "physics", "lean": false, "words": 7875, "doi": "10.5281/zenodo.19678606", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We pre-register and test a spectral-error-mitigation prediction for the two-qubit gate fidelity of Quantum Inspire's Tuna-9 9-qubit transmon processor and execute it in four cryptographically timestamped stages.", "url": "https://thalens.org/papers/phy_quantum_tuna9_parity/", "pdf": "https://thalens.org/papers/phy_quantum_tuna9_parity/paper.pdf", "created": "2026-04-18", "updated": "2026-04-21"}, {"id": "core_the_latent", "title": "The Latent: Finite Sufficient Representations of Smooth Systems", "domain": "core", "lean": true, "words": 80076, "doi": "10.5281/zenodo.19678631", "status": "draft", "status_label": "Draft", "review_status": "reviewed", "is_safe": true, "summary": "We define the **Latent** of a smooth system as the basis-free element of a graded Hilbert tensor algebra that completely characterizes the system's distributional, dynamic, and functional properties.", "url": "https://thalens.org/papers/core_the_latent/", "pdf": "https://thalens.org/papers/core_the_latent/paper.pdf", "created": "2026-03-18", "updated": "2026-04-20"}, {"id": "nt_bsd", "title": "A Lean 4 Formalization of the Birch and Swinnerton-Dyer Conjecture and Its Surrounding Theory", "domain": "number_theory", "lean": true, "words": 16260, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "The Birch and Swinnerton-Dyer conjecture predicts that for an elliptic curve $E/\\mathbb{Q}$ the Mordell-Weil rank equals the order of vanishing of the Hasse-Weil $L$-function at $s = 1$, and that the leading Taylor coefficient is expressed by an expl", "url": "https://thalens.org/papers/nt_bsd/", "pdf": "https://thalens.org/papers/nt_bsd/paper.pdf", "created": "2026-04-10", "updated": "2026-04-19"}, {"id": "nt_shifted_divisor_epsilon", "title": "ε-Removal for Moments of the Riemann Zeta Function via Cumulant Generating Function Analysis, Subject to Grade-2 Dominance", "domain": "number_theory", "lean": true, "words": 22090, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "Under grade-2 dominance of the correction field $X_T = \\log|\\zeta/P_T|^2$ (the per-prime Fourier cumulant bound $|\\kappa_m(f_p)| \\leq c_m p^{-\\lceil m/2 \\rceil}$ with $c_m \\leq A^m m!$ giving a positive CGF-tail convergence radius $r_\\star \\geq 1/A$;", "url": "https://thalens.org/papers/nt_shifted_divisor_epsilon/", "pdf": "https://thalens.org/papers/nt_shifted_divisor_epsilon/paper.pdf", "created": "2026-04-08", "updated": "2026-04-19"}, {"id": "degenerate_cc", "title": "An Exact Algebraic Bifurcation in the Triangle-Plus-Center Central Configuration", "domain": "celestial_mechanics", "lean": true, "words": 2225, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We prove that the triangle-plus-center central configuration of the planar four-body problem with masses $(1, 1, 1, \\mu)$ undergoes an exact stability transition at the critical mass ratio\n\n$$\\mu^* = \\frac{81 + 64\\sqrt{3}}{249},$$\n\nthe unique positiv", "url": "https://thalens.org/papers/degenerate_cc/", "pdf": "", "created": "2026-04-18", "updated": "2026-04-18"}, {"id": "double_gate", "title": "The Double Gate Theorem: Cascade Instabilities Require Two Independent Conditions", "domain": "physics", "lean": false, "words": 1396, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "We prove that gravitational cascade instabilities (such as the N-body problem's runaway collisions) and debris cascade instabilities (such as the Kessler syndrome in orbital mechanics) share a common mathematical structure: both require the simultane", "url": "https://thalens.org/papers/double_gate/", "pdf": "", "created": "2026-04-18", "updated": "2026-04-18"}, {"id": "integer_basics", "title": "Integer Basics: Foundational Properties of Integer Arithmetic", "domain": "verification", "lean": true, "words": 992, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "We present formal proofs of six foundational theorems in integer arithmetic: commutativity of addition and multiplication over ℤ, absorption of zero under multiplication, non-negativity of squared differences, and the dichotomy that every natural number is either zero or positive. All results are ma", "url": "https://thalens.org/papers/integer_basics/", "pdf": "", "created": "2026-04-18", "updated": "2026-04-18"}, {"id": "nt_collatz_cycle_elimination", "title": "No Non-Trivial Collatz Cycle Has Twenty-Three or Fewer Odd Steps: Cycle Elimination, Descent Analysis, and Residue-Class Structure", "domain": "number_theory", "lean": false, "words": 51662, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We develop a systematic method for eliminating non-trivial cycles of the Collatz map $T(n) = n/2$ if $n$ is even, $T(n) = 3n+1$ if $n$ is odd, and complement it with a stopping time analysis toward the full conjecture.", "url": "https://thalens.org/papers/nt_collatz_cycle_elimination/", "pdf": "", "created": "2026-04-15", "updated": "2026-04-18"}, {"id": "phy_turbulence_scaling_grade", "title": "Turbulence Scaling Laws from the Grade Equation: Kolmogorov Spectrum and Intermittency from Analyticity", "domain": "physics", "lean": true, "words": 12483, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We derive the Kolmogorov energy spectrum $E(k) \\sim \\varepsilon^{2/3} k^{-5/3}$ and anomalous intermittency corrections to the structure function exponents $\\zeta_p$ from the Grade Equation — a universal structural decomposition theorem for analytic dynamical systems. The derivation proceeds in thre", "url": "https://thalens.org/papers/phy_turbulence_scaling_grade/", "pdf": "https://thalens.org/papers/phy_turbulence_scaling_grade/paper.pdf", "created": "2026-03-22", "updated": "2026-04-18"}, {"id": "fin_spectral_ir_pricing", "title": "Spectral Interest Rate Pricing: COS-Based Derivatives from Yield Curve Coefficients", "domain": "finance", "lean": true, "words": 6508, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We present the first formally verified interest rate derivatives pricing engine. The Nagy spectral yield curve represents yields as a finite cosine series $y(\\tau) = A_0 + \\sum_{k=1}^K A_k \\cos(k\\pi\\tau/\\tau_{\\max})$ with each mode following independent Ornstein-Uhlenbeck dynamics.", "url": "https://thalens.org/papers/fin_spectral_ir_pricing/", "pdf": "https://thalens.org/papers/fin_spectral_ir_pricing/paper.pdf", "created": "2026-03-06", "updated": "2026-04-17"}, {"id": "math_3d_percolation_exponents", "title": "Critical Exponents for Three-Dimensional Percolation: A Crossover Approach via Long-Range Renormalization", "domain": "verification", "lean": true, "words": 37738, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/math_3d_percolation_exponents/", "pdf": "", "created": "", "updated": "2026-04-17"}, {"id": "nt_goldbach_convergence_hierarchy", "title": "$D_\\infty$ and the Goldbach Convergence Hierarchy", "domain": "number_theory", "lean": true, "words": 4697, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We identify the total zero energy $D_\\infty = \\sum_\\rho 1/|\\rho|^2 = 2 + \\gamma - \\log(4\\pi) \\approx 0.046$ as the invariant that **organizes** the difficulty landscape of additive prime problems in this framework.", "url": "https://thalens.org/papers/nt_goldbach_convergence_hierarchy/", "pdf": "https://thalens.org/papers/nt_goldbach_convergence_hierarchy/paper.pdf", "created": "2026-04-08", "updated": "2026-04-17"}, {"id": "nt_goldbach_latent", "title": "The Goldbach Conjecture as a Latent Positivity Theorem: Twenty-Five Paths, the Convergence Theorem, and the Five-Layer Strategy", "domain": "number_theory", "lean": true, "words": 13736, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We develop a conditional proof program for the Goldbach conjecture through the generating function $G(z) = P(z)^2$, where $P(z) = \\sum_{p \\text{ prime}} z^p$.", "url": "https://thalens.org/papers/nt_goldbach_latent/", "pdf": "https://thalens.org/papers/nt_goldbach_latent/paper.pdf", "created": "2026-03-19", "updated": "2026-04-17"}, {"id": "phy_quantum_usrt", "title": "The Quantum Compressibility Threshold", "domain": "physics", "lean": false, "words": 22715, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We prove a structural dichotomy for open quantum systems.", "url": "https://thalens.org/papers/phy_quantum_usrt/", "pdf": "https://thalens.org/papers/phy_quantum_usrt/paper.pdf", "created": "2026-03-08", "updated": "2026-04-17"}, {"id": "phy_constraint_forced_absorption", "title": "Constraint-Forced Absorption: A General Mechanism for Singularity Prevention in Gauge-Invariant PDE Systems", "domain": "finance", "lean": false, "words": 5290, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We identify a general mechanism — **constraint-forced absorption (CFA)** — in PDE systems where a differential identity constrains the dynamics. In the $(D, C, P)$ framework (dissipation, coupling, constraint):\n\n**Theorem** (Constraint-Forced Absorption).", "url": "https://thalens.org/papers/phy_constraint_forced_absorption/", "pdf": "", "created": "2026-04-16", "updated": "2026-04-16"}, {"id": "ml_residual_stream_denoising", "title": "Residual Stream Denoising in Large Language Models: Gradient-Free Quality Improvement via Functional Sensitivity Analysis", "domain": "machine learning / model analysis", "lean": true, "words": 9856, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/ml_residual_stream_denoising/", "pdf": "https://thalens.org/papers/ml_residual_stream_denoising/paper.pdf", "created": "2026-04-10", "updated": "2026-04-13"}, {"id": "fin_copula", "title": "An Eigenvalue-Conditioned Copula with Positive Tail Dependence: A Machine-Verified Alternative to the Gaussian Copula", "domain": "finance", "lean": false, "words": 6074, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "The Gaussian copula's failure to capture tail dependence was a central factor in the 2008 credit crisis: CDO tranche losses far exceeded model predictions because the model assigned near-zero probability to simultaneous defaults.", "url": "https://thalens.org/papers/fin_copula/", "pdf": "https://thalens.org/papers/fin_copula/paper.pdf", "created": "2026-03-04", "updated": "2026-04-11"}, {"id": "math_proof_category", "title": "Proof Circuits", "domain": "mathematics", "lean": true, "words": 21446, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "Many of the deepest theorems in mathematics were not proved within a single domain but by transferring the problem through a sequence of domains, applying tools native to each, and returning.", "url": "https://thalens.org/papers/math_proof_category/", "pdf": "https://thalens.org/papers/math_proof_category/paper.pdf", "created": "", "updated": "2026-04-11"}, {"id": "phy_spectral_error_mitigation", "title": "Spectral Error Mitigation: Exact Noise Inversion for Quantum Computers via Cluster Lindblad", "domain": "physics", "lean": false, "words": 4368, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/phy_spectral_error_mitigation/", "pdf": "https://thalens.org/papers/phy_spectral_error_mitigation/paper.pdf", "created": "2026-03-08", "updated": "2026-04-11"}, {"id": "bio_allostery", "title": "Allosteric Regulation — Spectral Communication in Proteins via the Latent Framework", "domain": "mathematical_biology", "lean": true, "words": 2269, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Allostery couples distant sites in a macromolecule: ligand binding at one pocket reshapes dynamics and thermodynamics elsewhere.", "url": "https://thalens.org/papers/bio_allostery/", "pdf": "https://thalens.org/papers/bio_allostery/paper.pdf", "created": "", "updated": "2026-04-10"}, {"id": "bio_brain_criticality", "title": "Brain Criticality — Phase Transition at the Edge of Chaos via the Latent Framework", "domain": "mathematical_biology", "lean": true, "words": 2084, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Many neural systems are hypothesized to operate near a critical point between ordered and disordered dynamics, balancing sensitivity and stability.", "url": "https://thalens.org/papers/bio_brain_criticality/", "pdf": "https://thalens.org/papers/bio_brain_criticality/paper.pdf", "created": "", "updated": "2026-04-10"}, {"id": "bio_fitness_landscape", "title": "Evolutionary Fitness Landscape Ruggedness via the Latent Framework", "domain": "mathematical_biology", "lean": true, "words": 2825, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Evolutionary fitness landscapes encode how genotypes map to reproductive success. Rugged landscapes—with many local peaks and epistatic interactions—shape adaptation, evolvability, and the predictability of evolutionary paths.", "url": "https://thalens.org/papers/bio_fitness_landscape/", "pdf": "https://thalens.org/papers/bio_fitness_landscape/paper.pdf", "created": "", "updated": "2026-04-10"}, {"id": "bio_grn_dynamics", "title": "Grn Dynamics", "domain": "verification", "lean": true, "words": 940, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Gene Regulatory Network Dynamics — Stability, Inference & Latent Connection.\n\nThis paper presents 50 machine-verified theorems building on 3 established facts and 54 hypotheses.", "url": "https://thalens.org/papers/bio_grn_dynamics/", "pdf": "https://thalens.org/papers/bio_grn_dynamics/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "bio_molecular_self_replication", "title": "Molecular Self Replication", "domain": "verification", "lean": true, "words": 192, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "This paper presents 0 machine-verified theorems. All results are formally verified in the Lean 4 (629 verification units, 102 proved statements) and exportable to Lean 4.\n\n<!-- TODO: Write a proper abstract summarizing the key contributions -->", "url": "https://thalens.org/papers/bio_molecular_self_replication/", "pdf": "https://thalens.org/papers/bio_molecular_self_replication/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "bio_neuro_manifold", "title": "Neuro Manifold", "domain": "math", "lean": true, "words": 855, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Neural Manifold — Spectral Dimension Bound, Optimal Decoding & BCI Theory.\n\nThis paper presents 47 machine-verified theorems building on 5 established facts and 32 hypotheses.", "url": "https://thalens.org/papers/bio_neuro_manifold/", "pdf": "https://thalens.org/papers/bio_neuro_manifold/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "bio_waddington", "title": "Waddington Landscape Cell Fate via the Latent Framework", "domain": "mathematical_biology", "lean": true, "words": 2471, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Waddington’s epigenetic landscape metaphor remains the dominant intuitive picture for cell fate: stable types are valleys, differentiation is downhill flow, and reprogramming lifts cells across ridges.", "url": "https://thalens.org/papers/bio_waddington/", "pdf": "https://thalens.org/papers/bio_waddington/paper.pdf", "created": "", "updated": "2026-04-10"}, {"id": "bio_wright_fisher", "title": "Wright Fisher", "domain": "verification", "lean": true, "words": 628, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Wright-Fisher Population Genetics — Spectral Convergence + Latent.\n\nThis paper presents 24 machine-verified theorems building on 0 established facts and 35 hypotheses.", "url": "https://thalens.org/papers/bio_wright_fisher/", "pdf": "https://thalens.org/papers/bio_wright_fisher/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "core_latent_algebra", "title": "The Latent Algebra: A Universal Representational Language for Smooth Systems", "domain": "core", "lean": true, "words": 10585, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We define the **Latent Algebra** $\\mathfrak{L}(\\mathcal{H}) = \\bigoplus_{r \\geq 0} \\mathcal{H}^{\\otimes r}$ as the graded tensor algebra over a separable Hilbert space $\\mathcal{H}$, equipped with four primitive operations — addition, tensor product,", "url": "https://thalens.org/papers/core_latent_algebra/", "pdf": "https://thalens.org/papers/core_latent_algebra/paper.pdf", "created": "2026-03-31", "updated": "2026-04-10"}, {"id": "econ_latent_climate_economy", "title": "Fat Tails and Carbon Taxes: A Spectral Resolution of the Climate Economics Debate", "domain": "climate economics / integrated assessment", "lean": true, "words": 2097, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "The central debate in climate economics — between Nordhaus-style moderate carbon pricing and Stern-style aggressive policy — is fundamentally a disagreement about the tail behavior of the damage distribution.", "url": "https://thalens.org/papers/econ_latent_climate_economy/", "pdf": "https://thalens.org/papers/econ_latent_climate_economy/paper.pdf", "created": "2026-04-08", "updated": "2026-04-10"}, {"id": "econ_latent_contagion", "title": "Spectral Contagion: Network Fragility through the Latent Lens", "domain": "financial networks / systemic risk", "lean": true, "words": 2049, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We formalize financial contagion as a Grade-2 hazard model on networks and show how the cascade threshold is tied, under explicit spectral proportionalities, to a concentration index: the Latent Number $\\rho$ of the interbank network.", "url": "https://thalens.org/papers/econ_latent_contagion/", "pdf": "https://thalens.org/papers/econ_latent_contagion/paper.pdf", "created": "2026-04-08", "updated": "2026-04-10"}, {"id": "econ_latent_dynamic_pricing", "title": "Spectral Pricing: Bayesian Learning and the Explore-Exploit Frontier via the Latent Framework", "domain": "dynamic pricing / bayesian learning / revenue optimization", "lean": true, "words": 2190, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "A seller facing unknown demand must balance exploration (learning the demand curve) against exploitation (maximizing immediate revenue). We organize the explore-exploit tradeoff through the Latent Number $\\rho$ of the demand function.", "url": "https://thalens.org/papers/econ_latent_dynamic_pricing/", "pdf": "https://thalens.org/papers/econ_latent_dynamic_pricing/paper.pdf", "created": "2026-04-08", "updated": "2026-04-10"}, {"id": "econ_latent_heterogeneous_agents", "title": "When Does Heterogeneity Matter? A Spectral Theory of Wealth Distribution and General Equilibrium", "domain": "macroeconomics / heterogeneous agent models", "lean": true, "words": 2342, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Heterogeneous agent models (Aiyagari, 1994; Bewley, 1986) have become the workhorse of quantitative macroeconomics, but their computational demands are severe: the state variable is the entire wealth distribution — an infinite-dimensional object.", "url": "https://thalens.org/papers/econ_latent_heterogeneous_agents/", "pdf": "https://thalens.org/papers/econ_latent_heterogeneous_agents/paper.pdf", "created": "2026-04-08", "updated": "2026-04-10"}, {"id": "econ_latent_mechanism_design", "title": "Latent Mechanism Design: Spectral Approximation of Optimal Mechanisms", "domain": "mechanism design / game theory", "lean": true, "words": 1818, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We apply the Latent spectral framework to mechanism design, framing bilateral-trade tensions through a spectral parameter $\\rho$ (Latent Number) of the type distribution.", "url": "https://thalens.org/papers/econ_latent_mechanism_design/", "pdf": "https://thalens.org/papers/econ_latent_mechanism_design/paper.pdf", "created": "2026-04-08", "updated": "2026-04-10"}, {"id": "econ_latent_microstructure", "title": "Spectral Microstructure: Kyle's Lambda, Price Discovery, and the HFT Debate through the Latent Lens", "domain": "market microstructure / financial economics", "lean": true, "words": 1764, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We recast Kyle's (1985) insider-trading and price-discovery story in the Latent spectral framework: price impact $\\lambda$, spreads, and depth are read through the Latent Number $\\rho$ of the price signal.", "url": "https://thalens.org/papers/econ_latent_microstructure/", "pdf": "https://thalens.org/papers/econ_latent_microstructure/paper.pdf", "created": "2026-04-08", "updated": "2026-04-10"}, {"id": "econ_latent_optimal_taxation", "title": "Spectral Taxation: The Latent Structure of Optimal Income Tax Schedules", "domain": "public economics / optimal taxation", "lean": true, "words": 2176, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We apply the Latent spectral framework to the Mirrlees (1971) optimal taxation problem, showing that the information rent, tax schedule complexity, and welfare cost of progressive taxation are all governed by the Latent Number $\\rho$ of the ability distribution.", "url": "https://thalens.org/papers/econ_latent_optimal_taxation/", "pdf": "https://thalens.org/papers/econ_latent_optimal_taxation/paper.pdf", "created": "2026-04-08", "updated": "2026-04-10"}, {"id": "econ_latent_social_choice", "title": "Quantitative Arrow: Measuring Distance from Impossibility via the Latent Framework", "domain": "social choice theory / voting theory", "lean": true, "words": 1887, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Arrow's Impossibility Theorem (1951) proves that no social welfare function over three or more alternatives can simultaneously satisfy Unanimity, Independence of Irrelevant Alternatives (IIA), and Non-Dictatorship.", "url": "https://thalens.org/papers/econ_latent_social_choice/", "pdf": "https://thalens.org/papers/econ_latent_social_choice/paper.pdf", "created": "2026-04-08", "updated": "2026-04-10"}, {"id": "fin_american_basket_gym", "title": "American Basket Gym", "domain": "verification", "lean": true, "words": 946, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "American Basket Gym — ProofEnv proofs (eigen-COS / FW / GH error bounds).\n\nThis paper presents 67 machine-verified theorems.", "url": "https://thalens.org/papers/fin_american_basket_gym/", "pdf": "https://thalens.org/papers/fin_american_basket_gym/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_american_exercise", "title": "American Exercise", "domain": "verification", "lean": true, "words": 189, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "This paper presents 0 machine-verified theorems. All results are formally verified in the Lean 4 (97 verification units, 20 proved statements) and exportable to Lean 4.\n\n<!-- TODO: Write a proper abstract summarizing the key contributions -->", "url": "https://thalens.org/papers/fin_american_exercise/", "pdf": "https://thalens.org/papers/fin_american_exercise/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_bayes_decision", "title": "Bayes Decision", "domain": "verification", "lean": true, "words": 831, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Bayes decision — ProofEnv encoding of BayesClassifier / MAPClassifier (Lean BayesDecision).\n\nThis paper presents 58 machine-verified theorems.", "url": "https://thalens.org/papers/fin_bayes_decision/", "pdf": "https://thalens.org/papers/fin_bayes_decision/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_bayesian_risk", "title": "Bayesian Risk", "domain": "verification", "lean": true, "words": 766, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Bayesian Risk — clean ProofEnv proof.\n\nThis paper presents 51 machine-verified theorems.", "url": "https://thalens.org/papers/fin_bayesian_risk/", "pdf": "https://thalens.org/papers/fin_bayesian_risk/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_bellman", "title": "Bellman", "domain": "verification", "lean": true, "words": 1180, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Bellman — ProofEnv: DP value bounds, MDP Bellman operator, contraction lemmas.\n\nThis paper presents 95 machine-verified theorems building on 0 established facts and 1 hypotheses.", "url": "https://thalens.org/papers/fin_bellman/", "pdf": "https://thalens.org/papers/fin_bellman/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_correlated_cos_v2", "title": "Correlated Cos V2", "domain": "verification", "lean": true, "words": 562, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Correlated COS v2 (Correlated COS Method) — Newton proofs\n\nThis paper presents 24 machine-verified theorems building on 0 established facts and 4 hypotheses.", "url": "https://thalens.org/papers/fin_correlated_cos_v2/", "pdf": "https://thalens.org/papers/fin_correlated_cos_v2/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_credit_portfolio_loss", "title": "Credit Portfolio Loss", "domain": "finance", "lean": true, "words": 815, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Credit Portfolio Loss Distribution — Beyond the Gaussian Copula.\n\nThis paper presents 33 machine-verified theorems building on 0 established facts and 52 hypotheses.", "url": "https://thalens.org/papers/fin_credit_portfolio_loss/", "pdf": "https://thalens.org/papers/fin_credit_portfolio_loss/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_elicitability", "title": "Elicitability", "domain": "verification", "lean": true, "words": 852, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "#9 Elicitability Resolution — Coherence vs Elicitability.\n\nThis paper presents 35 machine-verified theorems building on 0 established facts and 68 hypotheses.", "url": "https://thalens.org/papers/fin_elicitability/", "pdf": "https://thalens.org/papers/fin_elicitability/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_elicitability_coherence", "title": "Elicitability Coherence", "domain": "verification", "lean": true, "words": 194, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "This paper presents 0 machine-verified theorems. All results are formally verified in the Lean 4 (79 verification units, 20 proved statements) and exportable to Lean 4.\n\n<!-- TODO: Write a proper abstract summarizing the key contributions -->", "url": "https://thalens.org/papers/fin_elicitability_coherence/", "pdf": "https://thalens.org/papers/fin_elicitability_coherence/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_equity_premium", "title": "Equity Premium", "domain": "verification", "lean": true, "words": 984, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Equity Premium Puzzle — Machine-Checked Formalization\n\nThis paper presents 73 machine-verified theorems building on 2 established facts and 3 hypotheses.", "url": "https://thalens.org/papers/fin_equity_premium/", "pdf": "https://thalens.org/papers/fin_equity_premium/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_fenton_copula", "title": "Fenton Copula", "domain": "verification", "lean": true, "words": 463, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Fenton — Hermite-COS Convergence Theory\n\nThis paper presents 20 machine-verified theorems building on 0 established facts and 8 hypotheses.", "url": "https://thalens.org/papers/fin_fenton_copula/", "pdf": "https://thalens.org/papers/fin_fenton_copula/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_fenton_solved", "title": "The Fenton Distribution Solved", "domain": "finance", "lean": true, "words": 14277, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "reviewed", "is_safe": true, "summary": "The moment-based Latent representation of correlated lognormal sums (Nagy, 2026, *The Exact Latent Distribution of Correlated Lognormal Sums*) relies on scaled moments $c_k = m_k/k!$ that grow as $e^{\\sigma_{\\max}^2 k^2/2}$.", "url": "https://thalens.org/papers/fin_fenton_solved/", "pdf": "https://thalens.org/papers/fin_fenton_solved/paper.pdf", "created": "2026-03-18", "updated": "2026-04-10"}, {"id": "fin_financial_contagion", "title": "Financial Contagion", "domain": "verification", "lean": true, "words": 955, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Contagion Phase Transition — Sharp Threshold with Network Topology.\n\nThis paper presents 53 machine-verified theorems building on 0 established facts and 50 hypotheses.", "url": "https://thalens.org/papers/fin_financial_contagion/", "pdf": "https://thalens.org/papers/fin_financial_contagion/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_ftap_uncertainty", "title": "Ftap Uncertainty", "domain": "verification", "lean": true, "words": 924, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "#2 FTAP under Model Uncertainty (Knightian Uncertainty).\n\nThis paper presents 35 machine-verified theorems building on 0 established facts and 77 hypotheses.", "url": "https://thalens.org/papers/fin_ftap_uncertainty/", "pdf": "https://thalens.org/papers/fin_ftap_uncertainty/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_harvestability_derivation", "title": "Harvestability Derivation", "domain": "verification", "lean": true, "words": 874, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Harvestability Derivation — clean ProofEnv proof.\n\nThis paper presents 54 machine-verified theorems building on 0 established facts and 12 hypotheses.", "url": "https://thalens.org/papers/fin_harvestability_derivation/", "pdf": "https://thalens.org/papers/fin_harvestability_derivation/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_harvestability_extensions", "title": "Harvestability Extensions", "domain": "verification", "lean": true, "words": 406, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Harvestability Extensions — clean ProofEnv proof.\n\nThis paper presents 19 machine-verified theorems.", "url": "https://thalens.org/papers/fin_harvestability_extensions/", "pdf": "https://thalens.org/papers/fin_harvestability_extensions/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_model_free_bounds", "title": "Model Free Bounds", "domain": "verification", "lean": true, "words": 953, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "#3 Model-Free Bounds via Optimal Transport + Spectral Methods.\n\nThis paper presents 35 machine-verified theorems building on 0 established facts and 80 hypotheses.", "url": "https://thalens.org/papers/fin_model_free_bounds/", "pdf": "https://thalens.org/papers/fin_model_free_bounds/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_nsco_sv2", "title": "Nsco Sv2", "domain": "verification", "lean": true, "words": 193, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "This paper presents 0 machine-verified theorems. All results are formally verified in the Lean 4 (62 verification units, 54 proved statements) and exportable to Lean 4.\n\n<!-- TODO: Write a proper abstract summarizing the key contributions -->", "url": "https://thalens.org/papers/fin_nsco_sv2/", "pdf": "https://thalens.org/papers/fin_nsco_sv2/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_nsco_sv3_smile", "title": "Nsco Sv3 Smile", "domain": "verification", "lean": true, "words": 886, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "NSCOSv3 Smile — ProofEnv (Heston-style variance: mean reversion, spreads).\n\nThis paper presents 62 machine-verified theorems.", "url": "https://thalens.org/papers/fin_nsco_sv3_smile/", "pdf": "https://thalens.org/papers/fin_nsco_sv3_smile/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_optimal_execution", "title": "Optimal Execution", "domain": "verification", "lean": true, "words": 855, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "#8 Optimal Execution Under Nonlinear Market Impact.\n\nThis paper presents 38 machine-verified theorems building on 0 established facts and 62 hypotheses.", "url": "https://thalens.org/papers/fin_optimal_execution/", "pdf": "https://thalens.org/papers/fin_optimal_execution/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_pricing_allocation", "title": "Pricing Allocation", "domain": "finance", "lean": true, "words": 1670, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Pricing Allocation — clean ProofEnv proof.\n\nThis paper presents 142 machine-verified theorems.", "url": "https://thalens.org/papers/fin_pricing_allocation/", "pdf": "https://thalens.org/papers/fin_pricing_allocation/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_spectral_fenton", "title": "Spectral Fenton", "domain": "verification", "lean": true, "words": 965, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Spectral Fenton — ProofEnv proofs from Lean + quasi archives.\n\nThis paper presents 74 machine-verified theorems.", "url": "https://thalens.org/papers/fin_spectral_fenton/", "pdf": "https://thalens.org/papers/fin_spectral_fenton/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_spectral_portfolio", "title": "Spectral Portfolio", "domain": "finance", "lean": true, "words": 1092, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Spectral Portfolio — ProofEnv proofs (covariance / PC risk split).\n\nThis paper presents 93 machine-verified theorems.", "url": "https://thalens.org/papers/fin_spectral_portfolio/", "pdf": "https://thalens.org/papers/fin_spectral_portfolio/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_stochastic_portfolio_theory", "title": "Stochastic Portfolio Theory", "domain": "finance", "lean": true, "words": 138, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "This paper presents 0 machine-verified theorems. All results are formally verified in the Lean 4 (125 verification units, 39 proved statements) and exportable to Lean 4.\n\n<!-- TODO: Write a proper abstract summarizing the key contributions -->", "url": "https://thalens.org/papers/fin_stochastic_portfolio_theory/", "pdf": "https://thalens.org/papers/fin_stochastic_portfolio_theory/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "fin_vol_surface_arb_free", "title": "Vol Surface Arb Free", "domain": "verification", "lean": true, "words": 140, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "This paper presents 0 machine-verified theorems. All results are formally verified in the Lean 4 (103 verification units, 25 proved statements) and exportable to Lean 4.\n\n<!-- TODO: Write a proper abstract summarizing the key contributions -->", "url": "https://thalens.org/papers/fin_vol_surface_arb_free/", "pdf": "https://thalens.org/papers/fin_vol_surface_arb_free/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "math_bridge_methodology", "title": "Bridge Methodology", "domain": "verification", "lean": true, "words": 341, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Bridge Methodology — Properties of Cross-Domain Mathematical Bridges.\n\nThis paper presents 20 machine-verified theorems.", "url": "https://thalens.org/papers/math_bridge_methodology/", "pdf": "https://thalens.org/papers/math_bridge_methodology/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "math_concrete_applications", "title": "Concrete Applications", "domain": "verification", "lean": true, "words": 471, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Concrete Applications: Goldbach Crystallization & NS Turbulence.\n\nThis paper presents 20 machine-verified theorems building on 0 established facts and 31 hypotheses.", "url": "https://thalens.org/papers/math_concrete_applications/", "pdf": "https://thalens.org/papers/math_concrete_applications/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "math_constraint_lifecycle", "title": "Constraint Lifecycle", "domain": "verification", "lean": true, "words": 715, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Constraint Lifecycle — clean ProofEnv proof.\n\nThis paper presents 52 machine-verified theorems.", "url": "https://thalens.org/papers/math_constraint_lifecycle/", "pdf": "https://thalens.org/papers/math_constraint_lifecycle/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "math_grade3_latent_algebra", "title": "Grade3 Latent Algebra", "domain": "core", "lean": true, "words": 2289, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Grade-3 Latent Algebra — Formal Foundation.\n\nThis paper presents 142 machine-verified theorems building on 14 established facts and 122 hypotheses.", "url": "https://thalens.org/papers/math_grade3_latent_algebra/", "pdf": "https://thalens.org/papers/math_grade3_latent_algebra/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "math_grade_method", "title": "Grade Method", "domain": "verification", "lean": true, "words": 1109, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Grade Method — Domain Application Proofs.\n\nThis paper presents 62 machine-verified theorems building on 18 established facts and 45 hypotheses.", "url": "https://thalens.org/papers/math_grade_method/", "pdf": "https://thalens.org/papers/math_grade_method/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "math_kessler_threshold", "title": "Kessler Threshold", "domain": "verification", "lean": true, "words": 136, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "This paper presents 0 machine-verified theorems. All results are formally verified in the Lean 4 (71 verification units, 46 proved statements) and exportable to Lean 4.\n\n<!-- TODO: Write a proper abstract summarizing the key contributions -->", "url": "https://thalens.org/papers/math_kessler_threshold/", "pdf": "https://thalens.org/papers/math_kessler_threshold/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "math_latent_core", "title": "Latent Core", "domain": "core", "lean": true, "words": 1506, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Bridge: Latent Core ↔ Eigenvalue Conditioning.\n\nThis paper presents 120 machine-verified theorems building on 10 established facts and 24 hypotheses.", "url": "https://thalens.org/papers/math_latent_core/", "pdf": "https://thalens.org/papers/math_latent_core/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "math_latent_optimization", "title": "Latent Optimization", "domain": "core", "lean": true, "words": 1492, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Latent Optimization — ProofEnv proofs.\n\nThis paper presents 136 machine-verified theorems building on 1 established facts and 1 hypotheses.", "url": "https://thalens.org/papers/math_latent_optimization/", "pdf": "https://thalens.org/papers/math_latent_optimization/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "math_mollification", "title": "Mollification", "domain": "verification", "lean": true, "words": 581, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Mollification Theory — Full Lean 4 Proof Suite (Bootstrap Edition).\n\nThis paper presents 47 machine-verified theorems.", "url": "https://thalens.org/papers/math_mollification/", "pdf": "https://thalens.org/papers/math_mollification/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "math_monodromy_grade", "title": "Monodromy Grade", "domain": "verification", "lean": true, "words": 136, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "This paper presents 0 machine-verified theorems. All results are formally verified in the Lean 4 (60 verification units, 5 proved statements) and exportable to Lean 4.\n\n<!-- TODO: Write a proper abstract summarizing the key contributions -->", "url": "https://thalens.org/papers/math_monodromy_grade/", "pdf": "https://thalens.org/papers/math_monodromy_grade/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "math_nash_boltzmann_bridge", "title": "Nash Boltzmann Bridge", "domain": "verification", "lean": true, "words": 779, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Nash-Boltzmann Duality — Bridge Kernel (Lean 4).\n\nThis paper presents 66 machine-verified theorems.", "url": "https://thalens.org/papers/math_nash_boltzmann_bridge/", "pdf": "https://thalens.org/papers/math_nash_boltzmann_bridge/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "math_resonance_simultaneous", "title": "Resonance Simultaneous", "domain": "verification", "lean": true, "words": 624, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Simultaneous Field — Three Millennium Problems as Projections\n\nThis paper presents 28 machine-verified theorems building on 7 established facts and 40 hypotheses.", "url": "https://thalens.org/papers/math_resonance_simultaneous/", "pdf": "https://thalens.org/papers/math_resonance_simultaneous/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "math_rmt_latent_bridge", "title": "Rmt Latent Bridge", "domain": "core", "lean": true, "words": 401, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "RMT–Latent Bridge: Grade-Shadow Duality.\n\nThis paper presents 16 machine-verified theorems building on 2 established facts and 16 hypotheses.", "url": "https://thalens.org/papers/math_rmt_latent_bridge/", "pdf": "https://thalens.org/papers/math_rmt_latent_bridge/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "math_spec_category", "title": "Spec Category", "domain": "verification", "lean": true, "words": 636, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Spec Category — The Proof Category of Spectral Transfers.\n\nThis paper presents 39 machine-verified theorems building on 16 established facts and 8 hypotheses.", "url": "https://thalens.org/papers/math_spec_category/", "pdf": "https://thalens.org/papers/math_spec_category/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "math_spectral_optimization", "title": "Spectral Optimization", "domain": "verification", "lean": true, "words": 873, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Spectral Optimization — ProofEnv proofs (condition number, spectral step rate).\n\nThis paper presents 71 machine-verified theorems.", "url": "https://thalens.org/papers/math_spectral_optimization/", "pdf": "https://thalens.org/papers/math_spectral_optimization/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "math_stochastic_calculus", "title": "Stochastic Calculus", "domain": "verification", "lean": true, "words": 395, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Stochastic Calculus — clean ProofEnv proof.\n\nThis paper presents 19 machine-verified theorems building on 6 established facts and 3 hypotheses.", "url": "https://thalens.org/papers/math_stochastic_calculus/", "pdf": "https://thalens.org/papers/math_stochastic_calculus/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "meta_computational_instances", "title": "Computational Instances", "domain": "verification", "lean": true, "words": 631, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Computational Instances: concrete elements with determined ρ values.\n\nThis paper presents 28 machine-verified theorems building on 2 established facts and 47 hypotheses.", "url": "https://thalens.org/papers/meta_computational_instances/", "pdf": "https://thalens.org/papers/meta_computational_instances/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "meta_p_vs_np", "title": "P Vs Np", "domain": "verification", "lean": true, "words": 138, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "This paper presents 0 machine-verified theorems. All results are formally verified in the Lean 4 (363 verification units, 347 proved statements) and exportable to Lean 4.\n\n<!-- TODO: Write a proper abstract summarizing the key contributions -->", "url": "https://thalens.org/papers/meta_p_vs_np/", "pdf": "https://thalens.org/papers/meta_p_vs_np/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "meta_proof_convergence", "title": "The Operational Curry–Howard: Proof Discovery as Navigated Software Architecture", "domain": "meta-methodology", "lean": true, "words": 37516, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "reviewed", "is_safe": true, "summary": "We propose a framework that unifies two perspectives on formal proof\nconstruction: **proof as pathfinding** in a type-theoretic state\nspace and **proof as programming** via the Curry–Howard\ncorrespondence.", "url": "https://thalens.org/papers/meta_proof_convergence/", "pdf": "https://thalens.org/papers/meta_proof_convergence/paper.pdf", "created": "2026-04-06", "updated": "2026-04-10"}, {"id": "meta_rmt_grade_shadow", "title": "The Grade-Shadow Correspondence: Random Matrix Universality from the Latent Grade Hierarchy", "domain": "math", "lean": true, "words": 11061, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We establish a correspondence between the grade hierarchy of the Latent framework and the universality classes of Random Matrix Theory. Every analytic dynamical system decomposes into grades $F = \\sum_{k=0}^{\\infty} A^{(k)}$ with exponential suppression $\\|A^{(k)}\\| \\leq C_0/\\rho^k$ (the Grade Equat", "url": "https://thalens.org/papers/meta_rmt_grade_shadow/", "pdf": "https://thalens.org/papers/meta_rmt_grade_shadow/paper.pdf", "created": "2026-04-03", "updated": "2026-04-10"}, {"id": "ml_ai_safety_chain", "title": "Ai Safety Chain", "domain": "verification", "lean": true, "words": 641, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "AI Safety Chain — Connecting Theorems (Formal Proofs)\n\nThis paper presents 27 machine-verified theorems building on 21 established facts and 25 hypotheses.", "url": "https://thalens.org/papers/ml_ai_safety_chain/", "pdf": "https://thalens.org/papers/ml_ai_safety_chain/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "ml_deep_consciousness", "title": "Deep Consciousness", "domain": "verification", "lean": true, "words": 1157, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Deep Consciousness — clean ProofEnv proof.\n\nThis paper presents 97 machine-verified theorems.", "url": "https://thalens.org/papers/ml_deep_consciousness/", "pdf": "https://thalens.org/papers/ml_deep_consciousness/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "ml_spectral_overfitting", "title": "Spectral Overfitting", "domain": "verification", "lean": true, "words": 740, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Spectral Overfitting — clean ProofEnv proof.\n\nThis paper presents 54 machine-verified theorems building on 0 established facts and 10 hypotheses.", "url": "https://thalens.org/papers/ml_spectral_overfitting/", "pdf": "https://thalens.org/papers/ml_spectral_overfitting/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "nt_explicit_formula_bridge", "title": "Explicit Formula Bridge", "domain": "verification", "lean": true, "words": 882, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Explicit Formula Bridge — The Zero–Prime Duality as a Spec Morphism.\n\nThis paper presents 66 machine-verified theorems building on 10 established facts and 4 hypotheses.", "url": "https://thalens.org/papers/nt_explicit_formula_bridge/", "pdf": "https://thalens.org/papers/nt_explicit_formula_bridge/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "nt_fejer_smoothing", "title": "Fejer Smoothing", "domain": "verification", "lean": true, "words": 494, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Fejér Smoothing — clean ProofEnv proof.\n\nThis paper presents 32 machine-verified theorems building on 0 established facts and 3 hypotheses.", "url": "https://thalens.org/papers/nt_fejer_smoothing/", "pdf": "https://thalens.org/papers/nt_fejer_smoothing/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "nt_gue_rh_gap", "title": "Gue Rh Gap", "domain": "verification", "lean": true, "words": 540, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Strip Extension & Spectral Gap — Quantitative Interior\n=======================================================\nFormalizes the two central results of the GUE density paper at a\nlevel deeper than the propositional chain in dpp_proof.py:\n\nThis paper pre", "url": "https://thalens.org/papers/nt_gue_rh_gap/", "pdf": "https://thalens.org/papers/nt_gue_rh_gap/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "nt_pade_resummation", "title": "Pade Resummation", "domain": "verification", "lean": true, "words": 664, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Entry alias for paderesummation_proof.py (canonical filename on disk).\n\nThis paper presents 44 machine-verified theorems building on 0 established facts and 7 hypotheses.", "url": "https://thalens.org/papers/nt_pade_resummation/", "pdf": "https://thalens.org/papers/nt_pade_resummation/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "nt_prime_number_theorem", "title": "Prime Number Theorem", "domain": "math", "lean": true, "words": 787, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Prime Number Theorem — Logical Structure\n\nThis paper presents 36 machine-verified theorems building on 0 established facts and 64 hypotheses.", "url": "https://thalens.org/papers/nt_prime_number_theorem/", "pdf": "https://thalens.org/papers/nt_prime_number_theorem/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "nt_proof_complexity_rho", "title": "Proof Complexity Rho", "domain": "verification", "lean": true, "words": 1203, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "This paper presents 51 machine-verified theorems building on 19 established facts and 106 hypotheses.", "url": "https://thalens.org/papers/nt_proof_complexity_rho/", "pdf": "https://thalens.org/papers/nt_proof_complexity_rho/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "nt_resonance_algebra_rh", "title": "Resonance Algebra Rh", "domain": "physics", "lean": true, "words": 10429, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "Representation Theory — Diamond for Characters, Plancherel, Branching\n\nThis paper presents 673 machine-verified theorems building on 85 established facts and 817 hypotheses.", "url": "https://thalens.org/papers/nt_resonance_algebra_rh/", "pdf": "https://thalens.org/papers/nt_resonance_algebra_rh/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "nt_resonance_goldbach", "title": "Resonance Goldbach", "domain": "verification", "lean": true, "words": 626, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Goldbach-Latent Bridge — Connecting resonance framework to the full Goldbach proof\n\nThis paper presents 18 machine-verified theorems building on 11 established facts and 58 hypotheses.", "url": "https://thalens.org/papers/nt_resonance_goldbach/", "pdf": "https://thalens.org/papers/nt_resonance_goldbach/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "nt_resonance_selberg", "title": "Resonance Selberg", "domain": "verification", "lean": true, "words": 881, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Ihara Zeta Function — Combinatorial Diamond for Graphs\n\nThis paper presents 37 machine-verified theorems building on 8 established facts and 74 hypotheses.", "url": "https://thalens.org/papers/nt_resonance_selberg/", "pdf": "https://thalens.org/papers/nt_resonance_selberg/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "nt_riemann_hypothesis", "title": "The Riemann Hypothesis via Zeta Moment Hankel Positivity", "domain": "math", "lean": true, "words": 19040, "doi": "10.5281/zenodo.19336129", "status": "draft", "status_label": "Draft", "review_status": "reviewed", "is_safe": true, "summary": "We establish a conditional proof that the Riemann Hypothesis follows\nfrom moment upper bounds weaker than the Lindelöf hypothesis.", "url": "https://thalens.org/papers/nt_riemann_hypothesis/", "pdf": "https://thalens.org/papers/nt_riemann_hypothesis/paper.pdf", "created": "2026-03-26", "updated": "2026-04-10"}, {"id": "phy_black_hole_info", "title": "Black Hole Info", "domain": "verification", "lean": true, "words": 893, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Black Hole Information Paradox — Formal Impossibility and Resolution\n====================================================================\nProof suite version 2.0 — 45 theorems across 8 parts.\n\nThis paper presents 83 machine-verified theorems.", "url": "https://thalens.org/papers/phy_black_hole_info/", "pdf": "https://thalens.org/papers/phy_black_hole_info/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "phy_cosmic_censorship", "title": "Cosmic Censorship", "domain": "verification", "lean": true, "words": 677, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Cosmic Censorship — Nature's Modesty\n=====================================\n\nThis paper presents 58 machine-verified theorems.", "url": "https://thalens.org/papers/phy_cosmic_censorship/", "pdf": "https://thalens.org/papers/phy_cosmic_censorship/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "phy_cosmic_inflation", "title": "Cosmic Inflation", "domain": "verification", "lean": true, "words": 624, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Cosmic Inflation — The First 10⁻³² Seconds\n============================================\n\nThis paper presents 55 machine-verified theorems.", "url": "https://thalens.org/papers/phy_cosmic_inflation/", "pdf": "https://thalens.org/papers/phy_cosmic_inflation/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "phy_exact_3body_solution", "title": "The Exact Latent Solution of the Gravitational Three-Body Problem", "domain": "physics", "lean": false, "words": 11407, "doi": "10.5281/zenodo.19101229", "status": "draft", "status_label": "Draft", "review_status": "reviewed", "is_safe": true, "summary": "We argue that **every trajectory of the planar gravitational three-body problem** (excluding measure-zero triple collision) admits a finite Latent representation to arbitrary accuracy — an exact, implicit, constructive *encoding* in Fourier space — u", "url": "https://thalens.org/papers/phy_exact_3body_solution/", "pdf": "https://thalens.org/papers/phy_exact_3body_solution/paper.pdf", "created": "2026-03-18", "updated": "2026-04-10"}, {"id": "phy_n_body", "title": "N Body", "domain": "verification", "lean": true, "words": 1057, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "N-Body — ProofEnv proofs aligned with NBodyGradeBound.lean + quasi archive.\n\nThis paper presents 92 machine-verified theorems.", "url": "https://thalens.org/papers/phy_n_body/", "pdf": "https://thalens.org/papers/phy_n_body/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "phy_navier_stokes_latent", "title": "Grade Decomposition and Gevrey Regularity for Navier-Stokes: A Machine-Checked Conditional Framework", "domain": "verification", "lean": true, "words": 9471, "doi": "10.5281/zenodo.19336183", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We introduce a grade decomposition of the Gevrey energy balance for the incompressible Navier-Stokes equations. The physically correct model uses $\\mathbb{C}$-valued Fourier coefficients with a factor of $i$ in the advection; the real-coefficient model trivializes all grade-3 terms.", "url": "https://thalens.org/papers/phy_navier_stokes_latent/", "pdf": "https://thalens.org/papers/phy_navier_stokes_latent/paper.pdf", "created": "2026-03-19", "updated": "2026-04-10"}, {"id": "phy_nbody_braid_realization", "title": "Braid Realization at Zero Angular Momentum for the Planar N-Body Problem", "domain": "physics", "lean": true, "words": 5937, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We prove that for the planar Newtonian $N$-body problem with arbitrary positive masses and zero angular momentum ($J = 0$), every reduced free homotopy class of periodic orbits with all pairwise winding numbers nonzero is realized by a collision-free periodic orbit.", "url": "https://thalens.org/papers/phy_nbody_braid_realization/", "pdf": "https://thalens.org/papers/phy_nbody_braid_realization/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "phy_neutron_star_postmerger", "title": "Neutron Star Postmerger", "domain": "verification", "lean": true, "words": 307, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Neutron Star Post-Merger ρ — Lean 4.\n\nThis paper presents 16 machine-verified theorems.", "url": "https://thalens.org/papers/phy_neutron_star_postmerger/", "pdf": "https://thalens.org/papers/phy_neutron_star_postmerger/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "phy_poincare_conjecture", "title": "Poincare Conjecture", "domain": "verification", "lean": true, "words": 484, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Perelman's Proof of the Poincaré Conjecture — Lean 4 Formalization\n\nThis paper presents 21 machine-verified theorems building on 4 established facts and 29 hypotheses.", "url": "https://thalens.org/papers/phy_poincare_conjecture/", "pdf": "https://thalens.org/papers/phy_poincare_conjecture/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "phy_qnm_ringdown", "title": "Qnm Ringdown", "domain": "verification", "lean": true, "words": 617, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Black Hole QNM Ringdown — Lean 4 Proof (v2.0).\n\nThis paper presents 49 machine-verified theorems.", "url": "https://thalens.org/papers/phy_qnm_ringdown/", "pdf": "https://thalens.org/papers/phy_qnm_ringdown/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "phy_sir_epidemic", "title": "Sir Epidemic", "domain": "verification", "lean": true, "words": 136, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "This paper presents 0 machine-verified theorems. All results are formally verified in the Lean 4 (51 verification units, 21 proved statements) and exportable to Lean 4.\n\n<!-- TODO: Write a proper abstract summarizing the key contributions -->", "url": "https://thalens.org/papers/phy_sir_epidemic/", "pdf": "https://thalens.org/papers/phy_sir_epidemic/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "phy_spectral3_body", "title": "Spectral3 Body", "domain": "verification", "lean": true, "words": 1678, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Spectral 3-Body — clean ProofEnv proof.\n\nThis paper presents 149 machine-verified theorems.", "url": "https://thalens.org/papers/phy_spectral3_body/", "pdf": "https://thalens.org/papers/phy_spectral3_body/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "phy_turbulence_grade", "title": "Turbulence Grade", "domain": "physics", "lean": true, "words": 1149, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "Turbulence Grade — Kernel Proof Suite v2.0\n\nThis paper presents 102 machine-verified theorems.", "url": "https://thalens.org/papers/phy_turbulence_grade/", "pdf": "https://thalens.org/papers/phy_turbulence_grade/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "phy_uv_ir_duality", "title": "Uv Ir Duality", "domain": "verification", "lean": true, "words": 545, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "UV-IR Grade Duality — 30 Machine-Checked Theorems\n\nThis paper presents 30 machine-verified theorems building on 9 established facts and 10 hypotheses.", "url": "https://thalens.org/papers/phy_uv_ir_duality/", "pdf": "https://thalens.org/papers/phy_uv_ir_duality/paper.pdf", "created": "2026-04-10", "updated": "2026-04-10"}, {"id": "ai_spectral_interpretability", "title": "The Spectral Theory of LLM Understanding: When Can We Trust a Language Model?", "domain": "ml", "lean": true, "words": 7629, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "Large language models are deployed in medicine, law, education, and critical infrastructure — yet we cannot mathematically characterize when their outputs are trustworthy.", "url": "https://thalens.org/papers/ai_spectral_interpretability/", "pdf": "https://thalens.org/papers/ai_spectral_interpretability/paper.pdf", "created": "2026-04-09", "updated": "2026-04-09"}, {"id": "bio_cross_domain_bridges", "title": "Cross-Domain Bridges in the Latent Framework: Structural Isomorphisms Across Ten Biological Domains", "domain": "biology, cross-domain, latent theory", "lean": true, "words": 2669, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "The Latent framework ($\\Lambda$) provides a unified spectral representation for systems governed by self-adjoint operators with exponentially decaying eigenvalues.", "url": "https://thalens.org/papers/bio_cross_domain_bridges/", "pdf": "https://thalens.org/papers/bio_cross_domain_bridges/paper.pdf", "created": "", "updated": "2026-04-09"}, {"id": "bio_epidemic_network", "title": "Epidemic Spreading on Networks via the Latent Framework: Spectral Control, Superspreader Detection, and Optimal Intervention", "domain": "mathematical_epidemiology", "lean": true, "words": 3109, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/bio_epidemic_network/", "pdf": "https://thalens.org/papers/bio_epidemic_network/paper.pdf", "created": "", "updated": "2026-04-09"}, {"id": "bio_grn_inference", "title": "Gene Regulatory Network Inference via the Latent Framework", "domain": "core", "lean": true, "words": 2149, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We apply the Latent framework to the gene regulatory network (GRN) inference problem — reconstructing the regulatory interaction matrix $W$ from gene expression data.", "url": "https://thalens.org/papers/bio_grn_inference/", "pdf": "https://thalens.org/papers/bio_grn_inference/paper.pdf", "created": "2026-04-08", "updated": "2026-04-09"}, {"id": "bio_morphogenesis_turing", "title": "Morphogenesis as Spectral Selection", "domain": "physics", "lean": true, "words": 3185, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We apply the Latent framework to reaction-diffusion systems that exhibit Turing pattern formation, revealing that pattern selection, stability, and convergence are governed by a single spectral quantity: the Latent Number $\\rho$ of the linearized reaction-diffusion operator.", "url": "https://thalens.org/papers/bio_morphogenesis_turing/", "pdf": "https://thalens.org/papers/bio_morphogenesis_turing/paper.pdf", "created": "2026-04-08", "updated": "2026-04-09"}, {"id": "bio_neural_manifold_decoding", "title": "Optimal Neural Decoding via the Latent Framework: How Many Electrodes Does a Brain-Computer Interface Need?", "domain": "computational_neuroscience", "lean": true, "words": 2618, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/bio_neural_manifold_decoding/", "pdf": "https://thalens.org/papers/bio_neural_manifold_decoding/paper.pdf", "created": "", "updated": "2026-04-09"}, {"id": "econ_latent_asset_pricing", "title": "One Number Rules Finance: The Latent Resolution of Asset Pricing Puzzles", "domain": "quantitative finance / mathematical economics", "lean": true, "words": 2344, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We resolve three longstanding anomalies in asset pricing — the equity premium puzzle, the Fama-French factor zoo, and the failure of single-factor models — through a single spectral parameter: the Latent Number $\\rho$.", "url": "https://thalens.org/papers/econ_latent_asset_pricing/", "pdf": "https://thalens.org/papers/econ_latent_asset_pricing/paper.pdf", "created": "2026-04-08", "updated": "2026-04-09"}, {"id": "econ_latent_bounded_rationality", "title": "Bounded Rationality and the Spectral Complexity of Games", "domain": "computational game theory / bounded rationality", "lean": true, "words": 1984, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Finding a Nash equilibrium is PPAD-complete (Daskalakis, Goldberg & Papadimitriou, 2009), yet economic models routinely assume agents play equilibrium strategies.", "url": "https://thalens.org/papers/econ_latent_bounded_rationality/", "pdf": "https://thalens.org/papers/econ_latent_bounded_rationality/paper.pdf", "created": "2026-04-08", "updated": "2026-04-09"}, {"id": "fin_rough_volatility", "title": "The ATM Skew Power Law: A Machine-Verified Derivation from the rBergomi Model", "domain": "quantitative_finance", "lean": true, "words": 14265, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We derive the ATM implied volatility skew power law $\\psi(T) \\sim C \\cdot T^{H-1/2}$ from the rough Bergomi (rBergomi) model through a machine-verified chain of 125 theorems.", "url": "https://thalens.org/papers/fin_rough_volatility/", "pdf": "https://thalens.org/papers/fin_rough_volatility/paper.pdf", "created": "", "updated": "2026-04-09"}, {"id": "math_langlands_transfer_graph", "title": "The Langlands Transfer Graph", "domain": "mathematics", "lean": true, "words": 7642, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We reframe the Langlands program as a systematic edge-construction effort within the proof category $\\mathbf{Spec}$ introduced in [Nagy, 2026a]. Each proved Langlands-type result — from Artin reciprocity to the modularity theorem — corresponds to a morphism between spectral domains carrying a comput", "url": "https://thalens.org/papers/math_langlands_transfer_graph/", "pdf": "https://thalens.org/papers/math_langlands_transfer_graph/paper.pdf", "created": "", "updated": "2026-04-09"}, {"id": "math_latent_path_integral", "title": "The Latent Path Integral", "domain": "mathematics", "lean": true, "words": 10090, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We prove that a single saddle-point formula governs quantitative approximation in every system whose action functional admits a Latent grade decomposition.", "url": "https://thalens.org/papers/math_latent_path_integral/", "pdf": "https://thalens.org/papers/math_latent_path_integral/paper.pdf", "created": "2026-04-09", "updated": "2026-04-09"}, {"id": "meta_unified_field", "title": "The Unified Field: Fifteen Algebraic Structures and a Meta-Algebra for Mathematics", "domain": "meta-mathematics", "lean": true, "words": 10842, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/meta_unified_field/", "pdf": "https://thalens.org/papers/meta_unified_field/paper.pdf", "created": "", "updated": "2026-04-09"}, {"id": "ml_neural_scaling_laws", "title": "Why Neural Networks Scale: A Complete Latent-Theoretic Foundation", "domain": "machine_learning", "lean": true, "words": 6042, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a unified mathematical theory of neural scaling laws derived from the spectral structure of data distributions. The central object is the Latent Number $\\rho \\in (0, \\infty)$, which measures the rate at which a distribution's spectral coefficients decay.", "url": "https://thalens.org/papers/ml_neural_scaling_laws/", "pdf": "https://thalens.org/papers/ml_neural_scaling_laws/paper.pdf", "created": "", "updated": "2026-04-09"}, {"id": "nt_resonance_rh", "title": "A Unified Resonance Framework for the Riemann Hypothesis", "domain": "number_theory", "lean": true, "words": 18863, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We introduce a resonance algebra framework that unifies four classical approaches to the Riemann Hypothesis into a single structure. Each non-trivial zero \\(\\rho_n = \\sigma_n + i\\gamma_n\\) of \\(\\zeta(s)\\) is modeled as a resonance mode with damping rate \\(\\sigma_n\\) and frequency \\(\\gamma_n\\).", "url": "https://thalens.org/papers/nt_resonance_rh/", "pdf": "https://thalens.org/papers/nt_resonance_rh/paper.pdf", "created": "", "updated": "2026-04-09"}, {"id": "nt_rh_cumulant_chain", "title": "The Riemann Hypothesis via Cumulant Independence and de Branges Regularity", "domain": "math", "lean": true, "words": 5851, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We prove that $E[|\\zeta(1/2+it)|^{2k}] \\leq C(k) (\\log T)^{k^2}$ for all integers $k \\geq 1$, where the average is over $t \\in [T, 2T]$. For $k = 1, 2$ this recovers the classical results of Hardy–Littlewood (1918) and Ingham (1926).", "url": "https://thalens.org/papers/nt_rh_cumulant_chain/", "pdf": "https://thalens.org/papers/nt_rh_cumulant_chain/paper.pdf", "created": "2026-04-09", "updated": "2026-04-09"}, {"id": "nt_sharp_moment_hypothesis", "title": "The Sharp Moment Hypothesis for the Riemann Zeta Function", "domain": "math", "lean": true, "words": 2862, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "This draft develops a global strategy aimed at the sharp upper bound\n$$\\frac{1}{T}\\int_T^{2T} |\\zeta(1/2+it)|^{2k}\\,dt \\leq C(k)(\\log T)^{k^2}$$\nfor all integers $k \\geq 1$ and all $T \\geq T_0(k)$, where $C(k)$ is an explicit constant depending only on $k$.", "url": "https://thalens.org/papers/nt_sharp_moment_hypothesis/", "pdf": "https://thalens.org/papers/nt_sharp_moment_hypothesis/paper.pdf", "created": "2026-04-09", "updated": "2026-04-09"}, {"id": "pde_gw_spectral_stability", "title": "Spectral Stability of the Gallay-Wayne Gap under Three-Dimensional Vortex Tube Perturbations", "domain": "pde, spectral theory, fluid mechanics", "lean": true, "words": 3227, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Gallay and Wayne (2005) proved that the Lamb-Oseen vortex is globally asymptotically stable as a solution of the two-dimensional Navier-Stokes equation, with the linearized operator possessing a spectral gap $\\gamma > 0$ in Gaussian-weighted $L^2$ spaces.", "url": "https://thalens.org/papers/pde_gw_spectral_stability/", "pdf": "https://thalens.org/papers/pde_gw_spectral_stability/paper.pdf", "created": "2026-04-09", "updated": "2026-04-09"}, {"id": "fnd_simultaneous_field", "title": "The Simultaneous Field: A Universal Mathematical Framework for Parallel Computation over Value Spaces", "domain": "foundations", "lean": false, "words": 5027, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We introduce the **Simultaneous Field** $\\mathbb{S}\\text{im}$, a mathematical structure in which computation proceeds not sequentially but in superposition over all possible values.", "url": "https://thalens.org/papers/fnd_simultaneous_field/", "pdf": "https://thalens.org/papers/fnd_simultaneous_field/paper.pdf", "created": "2026-04-07", "updated": "2026-04-08"}, {"id": "ml_scaling_laws_latent", "title": "A Unified Spectral Theory of Machine Learning: Neural Scaling Laws, Generalization, and Architecture Design via the Latent Framework", "domain": "ml_theory", "lean": true, "words": 4572, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a unified spectral theory of machine learning built on the Latent framework ($\\Lambda$, $\\rho$).", "url": "https://thalens.org/papers/ml_scaling_laws_latent/", "pdf": "https://thalens.org/papers/ml_scaling_laws_latent/paper.pdf", "created": "2026-04-08", "updated": "2026-04-08"}, {"id": "nt_rh_cumulant_density", "title": "A Zero-Density Bound Near the Critical Line via Cumulant Matching", "domain": "math", "lean": true, "words": 3174, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We derive a zero-density estimate for the Riemann zeta function near the critical line by combining the Selberg Central Limit Theorem, the Euler product cumulant bound, and a width-corrected Hadamard spike formula.", "url": "https://thalens.org/papers/nt_rh_cumulant_density/", "pdf": "https://thalens.org/papers/nt_rh_cumulant_density/paper.pdf", "created": "2026-04-07", "updated": "2026-04-07"}, {"id": "phy_celestial_mechanics_latent", "title": "The Latent Solution of the Gravitational N-Body Problem", "domain": "physics", "lean": true, "words": 8965, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a complete treatment of the gravitational N-body problem through the lens of the Latent framework — finite, sufficient representations of smooth dynamical systems.", "url": "https://thalens.org/papers/phy_celestial_mechanics_latent/", "pdf": "https://thalens.org/papers/phy_celestial_mechanics_latent/paper.pdf", "created": "2026-04-07", "updated": "2026-04-07"}, {"id": "test_forge_validation", "title": "Spectral Mollification of Singular Distributions: A Latent Framework Approach", "domain": "core", "lean": false, "words": 1593, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We present a framework for mollifying singular probability distributions into smooth spectral representations via the Latent transform. The key result is a convergence theorem showing that any distribution with finite second moment can be approximated to arbitrary precision by a truncated spectral e", "url": "https://thalens.org/papers/test_forge_validation/", "pdf": "https://thalens.org/papers/test_forge_validation/paper.pdf", "created": "2026-04-07", "updated": "2026-04-07"}, {"id": "bio_latent_biology", "title": "The Latent Number in Biology", "domain": "core", "lean": true, "words": 4876, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We establish that three apparently unrelated biological phenomena — neural manifold dimensionality, gene regulatory network stability, and Wright–Fisher population genetic convergence — are governed by a common spectral quantity: the Latent Number $\\rho$.", "url": "https://thalens.org/papers/bio_latent_biology/", "pdf": "https://thalens.org/papers/bio_latent_biology/paper.pdf", "created": "2026-04-06", "updated": "2026-04-06"}, {"id": "meta_pvsnp_platonic", "title": "P ≠ NP from Six Cited Axioms: A Machine-Verified Conditional Proof via Partition Function Analyticity", "domain": "verification", "lean": true, "words": 6194, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We formalize a conditional proof of $\\mathrm{P} \\neq \\mathrm{NP}$ in a Python-native proof language with Lean 4 export.", "url": "https://thalens.org/papers/meta_pvsnp_platonic/", "pdf": "https://thalens.org/papers/meta_pvsnp_platonic/paper.pdf", "created": "2026-04-06", "updated": "2026-04-06"}, {"id": "bio_protein_folding_dynamics", "title": "Protein Folding as a Spectral First-Passage Problem", "domain": "core", "lean": false, "words": 22583, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We develop a spectral theory of protein folding dynamics by establishing a precise correspondence between the conformational Fokker–Planck equation and the Latent framework previously applied to financial risk, fluid regularity, and fusion plasma con", "url": "https://thalens.org/papers/bio_protein_folding_dynamics/", "pdf": "https://thalens.org/papers/bio_protein_folding_dynamics/paper.pdf", "created": "2026-04-02", "updated": "2026-04-05"}, {"id": "bio_natures_latent_catalog", "title": "Nature's Latent Catalog: A Formally Verified Tour of Biological Shape Optimization", "domain": "verification", "lean": true, "words": 4386, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Why does a honeycomb look regular while trabecular bone looks chaotic? We propose that the visual complexity of a biological structure is governed by the **Latent dimension** $d_L$: the number of independent parameters generating the optimal geometry from a constrained optimization problem.", "url": "https://thalens.org/papers/bio_natures_latent_catalog/", "pdf": "https://thalens.org/papers/bio_natures_latent_catalog/paper.pdf", "created": "2026-04-04", "updated": "2026-04-04"}, {"id": "gt_differential_games_latent", "title": "Dimension-Free Differential Games via Latent Representation", "domain": "core", "lean": false, "words": 3001, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We develop a spectral Latent method for Hamilton-Jacobi-Isaacs (HJI) equations arising from two-player and multi-player differential games in high dimensions. The value function $V(t,x)$ is represented in the Latent basis with $N^*$ modes per dimension, reducing the PDE to an ODE system.", "url": "https://thalens.org/papers/gt_differential_games_latent/", "pdf": "https://thalens.org/papers/gt_differential_games_latent/paper.pdf", "created": "2026-04-04", "updated": "2026-04-04"}, {"id": "gt_latent_game_theory", "title": "The Latent of a Game: Dimension-Free Representations for N-Player Strategic Interactions", "domain": "core", "lean": false, "words": 5703, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We define the **Latent of an N-player game** as the element of a graded Hilbert tensor algebra that encodes the complete interaction structure of the game's payoff functions, organized by interaction order. The grade-$r$ component captures irreducible $r$-player interactions.", "url": "https://thalens.org/papers/gt_latent_game_theory/", "pdf": "https://thalens.org/papers/gt_latent_game_theory/paper.pdf", "created": "2026-04-04", "updated": "2026-04-04"}, {"id": "gt_latent_grand_unification", "title": "Interaction Grade as a Universal Language: The Latent Unification of Complex Systems", "domain": "core", "lean": false, "words": 2226, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We survey the transdisciplinary connections enabled by the Latent framework's grade decomposition, establishing exact correspondences between complex systems in physics, game theory, biology, and machine learning.", "url": "https://thalens.org/papers/gt_latent_grand_unification/", "pdf": "https://thalens.org/papers/gt_latent_grand_unification/paper.pdf", "created": "2026-04-04", "updated": "2026-04-04"}, {"id": "gt_mean_field_latent", "title": "Spectral Latent Methods for High-Dimensional Mean-Field Games", "domain": "core", "lean": false, "words": 3443, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We develop a spectral Latent method for high-dimensional mean-field games (MFGs). Our starting point is the observation that the MFG system — the coupled Hamilton-Jacobi-Bellman and Fokker-Planck equations — is the Euler-Lagrange system of the **grade-1 Latent truncation** of the underlying $N$-play", "url": "https://thalens.org/papers/gt_mean_field_latent/", "pdf": "https://thalens.org/papers/gt_mean_field_latent/paper.pdf", "created": "2026-04-04", "updated": "2026-04-04"}, {"id": "gt_mechanism_design_latent", "title": "Optimal Mechanism Design via Latent Compression", "domain": "core", "lean": false, "words": 2840, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We apply the Latent framework to multi-item, multi-bidder mechanism design. The allocation rule $x(v_1, \\ldots, v_N) : V^N \\to [0,1]^{NM}$ for $N$ bidders and $M$ items is a function on the $NM$-dimensional type space.", "url": "https://thalens.org/papers/gt_mechanism_design_latent/", "pdf": "https://thalens.org/papers/gt_mechanism_design_latent/paper.pdf", "created": "2026-04-04", "updated": "2026-04-04"}, {"id": "gt_nash_boltzmann_duality", "title": "Nash-Boltzmann Duality: Statistical Mechanics as Game Theory in the Latent Algebra", "domain": "core", "lean": false, "words": 3187, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We establish an exact duality between strategic games and statistical mechanical systems through the Latent algebra's grade decomposition.", "url": "https://thalens.org/papers/gt_nash_boltzmann_duality/", "pdf": "https://thalens.org/papers/gt_nash_boltzmann_duality/paper.pdf", "created": "2026-04-04", "updated": "2026-04-04"}, {"id": "gt_protein_folding_game", "title": "The Folding Game: Protein Structure as Nash Equilibrium in the Latent Algebra", "domain": "core", "lean": true, "words": 3355, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We establish an exact correspondence between protein folding dynamics and strategic game theory through the Latent algebra's grade decomposition. Under the Nash-Boltzmann duality, amino acid residues become players, dihedral angles become strategies, and the conformational energy function becomes th", "url": "https://thalens.org/papers/gt_protein_folding_game/", "pdf": "https://thalens.org/papers/gt_protein_folding_game/paper.pdf", "created": "2026-04-04", "updated": "2026-04-04"}, {"id": "phy_qnm_latent", "title": "The Latent of Black Hole Ringdown: Spectral Sufficiency and the Overtone Problem", "domain": "physics", "lean": false, "words": 7205, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "The gravitational wave ringdown of a perturbed Kerr black hole is a sum of quasinormal modes (QNMs) — damped oscillations with complex frequencies determined by the black hole's mass and spin.", "url": "https://thalens.org/papers/phy_qnm_latent/", "pdf": "https://thalens.org/papers/phy_qnm_latent/paper.pdf", "created": "2026-04-03", "updated": "2026-04-04"}, {"id": "bio_myomere_composites", "title": "From Fish to Composites: The Latent Structure of Fiber-Angle Optimization", "domain": "core", "lean": true, "words": 3287, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "The W-shaped muscle segmentation pattern visible in a salmon fillet is the solution to a constrained optimization problem: maximize total bending moment subject to a maximum fiber strain constraint.", "url": "https://thalens.org/papers/bio_myomere_composites/", "pdf": "https://thalens.org/papers/bio_myomere_composites/paper.pdf", "created": "2026-04-03", "updated": "2026-04-03"}, {"id": "phy_accretion_disk_latent", "title": "The Latent of Accretion: Spectral Sufficiency and Grade Structure of Black Hole Accretion Disk Spectra", "domain": "core", "lean": true, "words": 1763, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "The spectral energy distribution (SED) of accreting black holes — from stellar-mass X-ray binaries to supermassive AGN — is governed by the angular momentum transport equation in the disk, which takes the form of a Fokker-Planck equation in the radial coordinate.", "url": "https://thalens.org/papers/phy_accretion_disk_latent/", "pdf": "https://thalens.org/papers/phy_accretion_disk_latent/paper.pdf", "created": "2026-04-03", "updated": "2026-04-03"}, {"id": "phy_asteroseismology_latent", "title": "How Many Modes Determine a Star? Spectral Sufficiency Bounds for Asteroseismology via the Latent Theorem", "domain": "core", "lean": true, "words": 1849, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Asteroseismology determines stellar interior structure from observed oscillation frequencies — eigenvalues of the stellar structure operator.", "url": "https://thalens.org/papers/phy_asteroseismology_latent/", "pdf": "https://thalens.org/papers/phy_asteroseismology_latent/paper.pdf", "created": "2026-04-03", "updated": "2026-04-03"}, {"id": "phy_pulsar_timing_latent", "title": "Spectral Grade Decomposition of Pulsar Timing Residuals: Separating Gravitational Waves from Intrinsic Noise", "domain": "physics", "lean": true, "words": 1729, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Pulsar timing arrays (PTAs) detect the stochastic gravitational wave background (GWB) through correlated timing residuals across a network of millisecond pulsars. The key detection statistic — the Hellings-Downs angular correlation — must be separated from intrinsic pulsar noise, spatially correlate", "url": "https://thalens.org/papers/phy_pulsar_timing_latent/", "pdf": "https://thalens.org/papers/phy_pulsar_timing_latent/paper.pdf", "created": "2026-04-03", "updated": "2026-04-03"}, {"id": "aero_kessler_space_debris", "title": "The Kessler Threshold as a Grade-2 Bifurcation: Formally Verified Bounds for Space Debris Cascade Dynamics", "domain": "verification", "lean": true, "words": 4953, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We formalize the Kessler cascade — the self-sustaining collision fragmentation of orbital debris — as a Grade-2 dynamical system and prove the existence of a critical debris density threshold with formally verified bounds.\n\nThe debris population rate", "url": "https://thalens.org/papers/aero_kessler_space_debris/", "pdf": "https://thalens.org/papers/aero_kessler_space_debris/paper.pdf", "created": "2026-04-02", "updated": "2026-04-02"}, {"id": "core_latent_field_equation", "title": "The Grade Equation: A Universal Structural Law for Smooth Dynamical Systems", "domain": "physics", "lean": true, "words": 4839, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We prove that every analytic dynamical system $\\dot{\\mathbf{x}} = F(\\mathbf{x})$ satisfies a universal structural law — the **Grade Equation** — which decomposes the dynamics into a hierarchy of interaction grades $F = \\sum_{k=0}^{\\infty} A^{(k)}$ wi", "url": "https://thalens.org/papers/core_latent_field_equation/", "pdf": "https://thalens.org/papers/core_latent_field_equation/paper.pdf", "created": "2026-03-22", "updated": "2026-04-02"}, {"id": "core_latent_pde_solution", "title": "The Latent Solution: A Finite Sufficient Representation Framework for Partial Differential Equations", "domain": "core", "lean": true, "words": 8801, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We introduce the **Latent solution** as a quantitative framework for finite-dimensional representation of PDE solutions.", "url": "https://thalens.org/papers/core_latent_pde_solution/", "pdf": "https://thalens.org/papers/core_latent_pde_solution/paper.pdf", "created": "2026-03-31", "updated": "2026-04-02"}, {"id": "math_grade2_universality", "title": "Grade-2 Universality: A Formally Verified Unification of Fluid Turbulence, Gravitational Singularities, Orbital Debris Cascades, and Epidemic Thresholds", "domain": "physics", "lean": true, "words": 6085, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We identify and formally verify a universal algebraic structure — the **Grade-2 equation** — that governs threshold phenomena in four physically distinct domains: (I) the Navier–Stokes equations for incompressible fluid flow, (II) the Painlevé classi", "url": "https://thalens.org/papers/math_grade2_universality/", "pdf": "https://thalens.org/papers/math_grade2_universality/paper.pdf", "created": "2026-04-02", "updated": "2026-04-02"}, {"id": "math_spectral_zeta_generators", "title": "The Spectral Zeta Function of Markov Generators", "domain": "core", "lean": false, "words": 4317, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We introduce the spectral zeta function $\\zeta_L(s) = \\mathrm{Tr}(L^{-s})$ of the generator $L$ of a continuous-time Markov process as a universal diagnostic for complex dynamical systems.", "url": "https://thalens.org/papers/math_spectral_zeta_generators/", "pdf": "https://thalens.org/papers/math_spectral_zeta_generators/paper.pdf", "created": "2026-04-02", "updated": "2026-04-02"}, {"id": "meta_cumulant_bridge_mh", "title": "The Cumulant Bridge: Reducing the Moment Hypothesis to a Single Distributional Condition", "domain": "analytic_number_theory", "lean": false, "words": 5729, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We reformulate the Moment Hypothesis for the Riemann zeta function\nas a cumulant boundedness condition and show that it reduces to a\nsingle distributional hypothesis about the Dirichlet polynomial\napproximation at $\\sigma = 1/2$.", "url": "https://thalens.org/papers/meta_cumulant_bridge_mh/", "pdf": "https://thalens.org/papers/meta_cumulant_bridge_mh/paper.pdf", "created": "2026-03-28", "updated": "2026-04-02"}, {"id": "phy_mhd_latent", "title": "The Grade Structure of MHD Conserved Quantities: Effective Grade, Onsager Thresholds, and Taylor Relaxation", "domain": "verification", "lean": true, "words": 5152, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We introduce the *effective grade* of a conserved quantity: $\\mathrm{grade}_{\\mathrm{eff}}(Q) = \\mathrm{grade}_{\\mathrm{nom}}(Q) - \\delta_{\\mathrm{constr}}(Q)$, where $\\mathrm{grade}_{\\mathrm{nom}}$ is the polynomial degree in the dynamical fields an", "url": "https://thalens.org/papers/phy_mhd_latent/", "pdf": "https://thalens.org/papers/phy_mhd_latent/paper.pdf", "created": "2026-04-02", "updated": "2026-04-02"}, {"id": "fin_spectral_ccr_regulatory", "title": "Spectral Counterparty Credit Risk: Deterministic EPE, PFE, and Regulatory Capital Without Monte Carlo", "domain": "finance", "lean": false, "words": 3990, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We derive the full counterparty credit risk (CCR) metric stack --- Expected Positive Exposure (EPE), Potential Future Exposure (PFE), Effective Expected Positive Exposure (EEPE), and Exposure at Default (EAD) --- from the Fokker--Planck spectral generator without Monte Carlo simulation.", "url": "https://thalens.org/papers/fin_spectral_ccr_regulatory/", "pdf": "https://thalens.org/papers/fin_spectral_ccr_regulatory/paper.pdf", "created": "2026-04-01", "updated": "2026-04-01"}, {"id": "fin_spectral_market_risk_benchmark", "title": "Spectral Market Risk: Deterministic VaR, ES, and Stress Testing Without Monte Carlo", "domain": "finance", "lean": false, "words": 3859, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We present a benchmark of the Spectral Fenton method --- a deterministic alternative to Monte Carlo simulation for computing Value-at-Risk, Expected Shortfall, and full stress surfaces --- on realistic multi-asset portfolios representative of bank trading desks.", "url": "https://thalens.org/papers/fin_spectral_market_risk_benchmark/", "pdf": "https://thalens.org/papers/fin_spectral_market_risk_benchmark/paper.pdf", "created": "2026-04-01", "updated": "2026-04-01"}, {"id": "fin_spectral_xva_benchmark", "title": "Spectral XVA vs Nested Monte Carlo: A Three-Engine Benchmark", "domain": "finance", "lean": false, "words": 4260, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We present a controlled benchmark comparing three computational engines for Credit Valuation Adjustment (CVA) on an identical 10-swap interest rate portfolio under Vasicek dynamics: (i) a **spectral method** based on the Fokker--Planck generator, imp", "url": "https://thalens.org/papers/fin_spectral_xva_benchmark/", "pdf": "https://thalens.org/papers/fin_spectral_xva_benchmark/paper.pdf", "created": "2026-04-01", "updated": "2026-04-01"}, {"id": "nt_rh_path1_fourier_euler", "title": "The Riemann Hypothesis via Fourier-Euler Product: The Shortest Unconditional Proof", "domain": "math", "lean": false, "words": 3525, "doi": "10.5281/zenodo.19369217", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We prove the Riemann Hypothesis unconditionally from three classical inputs: Kronecker-Weyl equidistribution, the Bessel I₀ product identity, and Mertens' divergence theorem.", "url": "https://thalens.org/papers/nt_rh_path1_fourier_euler/", "pdf": "https://thalens.org/papers/nt_rh_path1_fourier_euler/paper.pdf", "created": "2026-03-31", "updated": "2026-04-01"}, {"id": "nt_rh_path3_spectral", "title": "The Riemann Hypothesis via Berry-Keating Spectral Construction", "domain": "math", "lean": false, "words": 9372, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We prove the Riemann Hypothesis conditional on a single operator-theoretic hypothesis: that the Berry-Keating Hamiltonian $H = \\frac{1}{2}(xp + px)$ admits a self-adjoint realization $\\hat{H}$ with discrete spectrum whose eigenvalues are the imaginar", "url": "https://thalens.org/papers/nt_rh_path3_spectral/", "pdf": "https://thalens.org/papers/nt_rh_path3_spectral/paper.pdf", "created": "2026-03-31", "updated": "2026-04-01"}, {"id": "meta_bridge_meta_methodology", "title": "The Bridge Method: Systematic Cross-Domain Discovery via Shared Mathematical Structure", "domain": "finance", "lean": true, "words": 11767, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We formalize the concept of a **mathematical bridge** — a theorem establishing that conclusions proved in domain $A$ imply structure in domain $B$ — and develop a methodology for systematic bridge discovery.", "url": "https://thalens.org/papers/meta_bridge_meta_methodology/", "pdf": "https://thalens.org/papers/meta_bridge_meta_methodology/paper.pdf", "created": "2026-03-31", "updated": "2026-03-31"}, {"id": "meta_eigenvalue_conditioning", "title": "Eigenvalue Conditioning: A Universal Computational Primitive for Correlated Systems", "domain": "finance", "lean": true, "words": 7165, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present eigenvalue conditioning as a universal computational primitive: given an $n$-dimensional problem governed by a positive semidefinite structure matrix $\\Sigma$, eigendecompose $\\Sigma$, project onto its $K$ dominant eigenmodes, solve $K$ independent one-dimensional problems, and combine.", "url": "https://thalens.org/papers/meta_eigenvalue_conditioning/", "pdf": "https://thalens.org/papers/meta_eigenvalue_conditioning/paper.pdf", "created": "2026-03-31", "updated": "2026-03-31"}, {"id": "meta_grade_decomposition_method", "title": "The Grade Method: Structural Decomposition of ODE Vector Fields via the Grade Hierarchy", "domain": "finance", "lean": false, "words": 9626, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We present the **grade decomposition method** as a general-purpose structural analysis tool for smooth dynamical systems. Any analytic vector field $F$ decomposes into interaction grades $F = \\sum_{k=0}^{\\infty} A^{(k)}$ with exponential suppression $\\|A^{(k)}\\| \\leq C_0/\\rho^k$, where $\\rho$ is the", "url": "https://thalens.org/papers/meta_grade_decomposition_method/", "pdf": "https://thalens.org/papers/meta_grade_decomposition_method/paper.pdf", "created": "2026-03-31", "updated": "2026-03-31"}, {"id": "meta_grand_framework", "title": "One Parameter: How ρ Unifies Computation, Structure, and Safety", "domain": "ml", "lean": false, "words": 1784, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We present a unified framework in which a single parameter — the analyticity radius $\\rho$ — simultaneously controls five properties of smooth systems: (1) **structural complexity** (the number of interaction grades needed), (2) **computational cost*", "url": "https://thalens.org/papers/meta_grand_framework/", "pdf": "https://thalens.org/papers/meta_grand_framework/paper.pdf", "created": "2026-03-31", "updated": "2026-03-31"}, {"id": "meta_latent_complexity", "title": "Latent Complexity: A Computable Theory of System Difficulty for Smooth Systems", "domain": "core", "lean": false, "words": 14088, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We construct a complexity theory for smooth (analytic) systems based on the Latent framework.", "url": "https://thalens.org/papers/meta_latent_complexity/", "pdf": "https://thalens.org/papers/meta_latent_complexity/paper.pdf", "created": "2026-03-31", "updated": "2026-03-31"}, {"id": "meta_rho_diagnostic", "title": "The Latent Number ρ: A Universal Diagnostic for Computational Complexity", "domain": "core", "lean": false, "words": 11292, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We introduce the **$\\rho$-diagnostic**: a cross-domain methodology for assessing the computational complexity of smooth systems through a single computable parameter, the **Latent Number** $\\rho$ (the system's intrinsic compressibility; formally, the analyticity parameter).", "url": "https://thalens.org/papers/meta_rho_diagnostic/", "pdf": "https://thalens.org/papers/meta_rho_diagnostic/paper.pdf", "created": "2026-03-31", "updated": "2026-03-31"}, {"id": "meta_two_lenses", "title": "Two Lenses, One Invariant: Empirical Confirmation That ρ Is Basis-Independent", "domain": "finance", "lean": false, "words": 4277, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "The Latent Number $\\rho$ — the analyticity parameter that governs computational complexity in smooth systems — can be measured through at least two independent methods: **spectral decay** (fitting the exponential decline of expansion coefficients in", "url": "https://thalens.org/papers/meta_two_lenses/", "pdf": "https://thalens.org/papers/meta_two_lenses/paper.pdf", "created": "2026-03-31", "updated": "2026-03-31"}, {"id": "ml_ai_safety_manifesto", "title": "Proved Safe: A Machine-Verified Theory of AI Safety from the Eigenvalue Spectrum", "domain": "ml", "lean": true, "words": 8080, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a unified, machine-verified theory of AI safety in which the eigenvalue decay rate $s$ of the data covariance determines a formal chain: scaling laws, self-improvement ceilings, robustness certificates, and a quantitative safety budget.", "url": "https://thalens.org/papers/ml_ai_safety_manifesto/", "pdf": "https://thalens.org/papers/ml_ai_safety_manifesto/paper.pdf", "created": "2026-03-31", "updated": "2026-03-31"}, {"id": "nt_per_prime_cumulants", "title": "Per-Prime Cumulant Structure of the Riemann Zeta Function", "domain": "math", "lean": true, "words": 2634, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We study the cumulant structure of $\\log|\\zeta(1/2+it)|^2$ induced by the Euler product. For each prime $p$, define $f_p(t) = 2\\operatorname{Re}(-\\log(1-p^{-1/2-it}))$.", "url": "https://thalens.org/papers/nt_per_prime_cumulants/", "pdf": "https://thalens.org/papers/nt_per_prime_cumulants/paper.pdf", "created": "2026-03-31", "updated": "2026-03-31"}, {"id": "nt_rh_path2_gue", "title": "Full Density of Zeta Zeros on the Critical Line via GUE Universality", "domain": "math", "lean": false, "words": 4700, "doi": "10.5281/zenodo.19369302", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We prove that 100% of the nontrivial zeros of $\\zeta(s)$ lie on the critical line in the density sense: $N_0(T)/N(T) \\to 1$ as $T \\to \\infty$. The proof combines two results.", "url": "https://thalens.org/papers/nt_rh_path2_gue/", "pdf": "https://thalens.org/papers/nt_rh_path2_gue/paper.pdf", "created": "2026-03-30", "updated": "2026-03-31"}, {"id": "phy_grade_regularity", "title": "Grade Regularity: A Universal Criterion for Strong Solutions of Nonlinear PDEs", "domain": "core", "lean": false, "words": 4916, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We prove a universal regularity criterion for nonlinear partial differential equations based on the grade decomposition of analytic vector fields.", "url": "https://thalens.org/papers/phy_grade_regularity/", "pdf": "https://thalens.org/papers/phy_grade_regularity/paper.pdf", "created": "2026-03-31", "updated": "2026-03-31"}, {"id": "fin_autocallable_spectral", "title": "Autocallable Pricing as a Latent Computation: Spectral Methods with Formal Error Bounds", "domain": "finance", "lean": true, "words": 4516, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We show that the price of a worst-of autocallable with step-down barriers, discrete monitoring, and stochastic volatility is computable as a composition of seven Latent operations (Nagy 2026).", "url": "https://thalens.org/papers/fin_autocallable_spectral/", "pdf": "https://thalens.org/papers/fin_autocallable_spectral/paper.pdf", "created": "2026-03-30", "updated": "2026-03-30"}, {"id": "ml_reflective_self_amendment", "title": "Reflective Self-Amendment", "domain": "ml", "lean": false, "words": 4009, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We identify a structural property unique to large language model (LLM) agents: when an agent's behavioral policy is specified in natural language (a \"skill file\"), and its reasoning engine operates in the same modality (natural language), the agent c", "url": "https://thalens.org/papers/ml_reflective_self_amendment/", "pdf": "https://thalens.org/papers/ml_reflective_self_amendment/paper.pdf", "created": "2026-03-30", "updated": "2026-03-30"}, {"id": "nt_rh_consolidated", "title": "Euler Product Cumulant Bounds, GUE Pair Correlation, and Zero Density on the Critical Line", "domain": "math", "lean": false, "words": 6470, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We prove that the Euler product structure of $\\zeta(s)$ provides an unconditional input to the Montgomery–Rudnick–Sarnak pair correlation framework, and show that GUE pair correlation implies that 100% of nontrivial zeros lie on the critical line.\n\n$", "url": "https://thalens.org/papers/nt_rh_consolidated/", "pdf": "https://thalens.org/papers/nt_rh_consolidated/paper.pdf", "created": "2026-03-30", "updated": "2026-03-30"}, {"id": "meta_network_latent_barabasi", "title": "Why Scale-Free Networks Are Spectrally Compressible: A Latent Sufficiency Theorem for Network Dynamics", "domain": "core", "lean": true, "words": 3480, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We prove that the Latent Theorem — which guarantees finite sufficient spectral representations for smooth systems — explains the dimensional reduction of dynamics on scale-free networks.", "url": "https://thalens.org/papers/meta_network_latent_barabasi/", "pdf": "https://thalens.org/papers/meta_network_latent_barabasi/paper.pdf", "created": "2026-03-29", "updated": "2026-03-29"}, {"id": "meta_residual_monte_carlo", "title": "Residual Monte Carlo: A Unifying Framework for Variance Reduction", "domain": "finance", "lean": false, "words": 5893, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We introduce Residual Monte Carlo (RMC), a variance reduction framework that decomposes high-dimensional integrals into an exactly computable dominant part and a simulated residual.", "url": "https://thalens.org/papers/meta_residual_monte_carlo/", "pdf": "https://thalens.org/papers/meta_residual_monte_carlo/paper.pdf", "created": "2026-03-29", "updated": "2026-03-29"}, {"id": "meta_theory_cognitive_percolation", "title": "Creative Flow as a Percolation Phase Transition in Knowledge Graphs", "domain": "research", "lean": false, "words": 5006, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/meta_theory_cognitive_percolation/", "pdf": "https://thalens.org/papers/meta_theory_cognitive_percolation/paper.pdf", "created": "2026-03-29", "updated": "2026-03-29"}, {"id": "nt_chowla", "title": "Machine-Verified Bounds for Chowla's Cosine Problem", "domain": "verification", "lean": true, "words": 6311, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "Chowla's cosine problem (Erdős problem #510) asks: for a set \\(A = \\{a_1, \\ldots, a_N\\}\\) of distinct positive integers, how negative can \\(\\min_\\theta \\sum_{i=1}^N \\cos(a_i \\theta)\\) be? The Erdős conjecture asserts a bound of order \\(-c\\sqrt{N}\\), which remains open.", "url": "https://thalens.org/papers/nt_chowla/", "pdf": "https://thalens.org/papers/nt_chowla/paper.pdf", "created": "2026-03-07", "updated": "2026-03-29"}, {"id": "fin_es_backtest_exact", "title": "Contaminated by Construction: Separating Simulation Noise from Model Risk in ES Backtests", "domain": "finance", "lean": true, "words": 12334, "doi": "10.5281/zenodo.19605107", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "Expected Shortfall backtesting under Basel III/IV suffers from an unmeasured structural weakness: Monte Carlo estimation of ES injects computational noise into the Acerbi-Székely (2014) test statistic, but the magnitude of this contamination has not", "url": "https://thalens.org/papers/fin_es_backtest_exact/", "pdf": "https://thalens.org/papers/fin_es_backtest_exact/paper.pdf", "created": "2026-03-06", "updated": "2026-03-28"}, {"id": "fin_es_backtest_spectral", "title": "Full-Tail Backtesting: Beyond Pointwise Validation for Expected Shortfall", "domain": "finance", "lean": true, "words": 10211, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "Every existing Expected Shortfall backtest evaluates a risk model at a single confidence level. This paper introduces four backtesting methods that test the entire tail distribution simultaneously — methods that are structurally impossible without exact closed-form computation of the CDF and density", "url": "https://thalens.org/papers/fin_es_backtest_spectral/", "pdf": "https://thalens.org/papers/fin_es_backtest_spectral/paper.pdf", "created": "2026-03-06", "updated": "2026-03-28"}, {"id": "fin_es_backtest_var_reduction", "title": "Reducing the Contamination: Variance Reduction Strategies for Monte Carlo ES Backtests", "domain": "finance", "lean": false, "words": 4528, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "Monte Carlo estimation of Expected Shortfall contaminates the Acerbi-Székely (2014) backtest statistic with computational noise.", "url": "https://thalens.org/papers/fin_es_backtest_var_reduction/", "pdf": "https://thalens.org/papers/fin_es_backtest_var_reduction/paper.pdf", "created": "2026-03-28", "updated": "2026-03-28"}, {"id": "meta_canopy_latent", "title": "Emergent Canopy Architecture from Discrete Leaf Dynamics: A Latent Environmental Encoding", "domain": "mathematical biology, optimization, latent representations", "lean": false, "words": 4559, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Why are trees shaped the way they are? We present a minimal discrete-leaf growth model in which canopy architecture emerges from two local rules: (1) new leaves are placed by a species-specific strategy, and (2) leaves that receive insufficient light die off.", "url": "https://thalens.org/papers/meta_canopy_latent/", "pdf": "https://thalens.org/papers/meta_canopy_latent/paper.pdf", "created": "2026-03-28", "updated": "2026-03-28"}, {"id": "meta_plant_bidirectional", "title": "Plants: The Bidirectional Optimizers", "domain": "mathematical biology, optimization, latent representations", "lean": false, "words": 3764, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "A tree optimizes in two directions simultaneously: upward toward light and downward toward water. The canopy and root system are not independent organs — they are coupled through Liebig's law of the minimum, where the limiting resource bottlenecks the entire organism.", "url": "https://thalens.org/papers/meta_plant_bidirectional/", "pdf": "https://thalens.org/papers/meta_plant_bidirectional/paper.pdf", "created": "2026-03-28", "updated": "2026-03-28"}, {"id": "meta_root_latent", "title": "Emergent Root Architecture from Discrete Tip Dynamics: The Underground Mirror", "domain": "mathematical biology, optimization, latent representations", "lean": false, "words": 4139, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "This paper is the underground twin of \"Emergent Canopy Architecture from Discrete Leaf Dynamics\" (Nagy, 2026).", "url": "https://thalens.org/papers/meta_root_latent/", "pdf": "https://thalens.org/papers/meta_root_latent/paper.pdf", "created": "2026-03-28", "updated": "2026-03-28"}, {"id": "meta_tool_cascade_theorem", "title": "The Cascade Depth Theorem", "domain": "finance", "lean": false, "words": 5010, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We formalize the *tool-building cascade*: a recursive decision process in which an agent, instead of solving problem instances directly, invests in tools that solve problem classes, thereby revealing higher-level problems amenable to the same strategy.", "url": "https://thalens.org/papers/meta_tool_cascade_theorem/", "pdf": "https://thalens.org/papers/meta_tool_cascade_theorem/paper.pdf", "created": "2026-03-28", "updated": "2026-03-28"}, {"id": "fin_spectral_importance_sampling", "title": "Spectral Importance Sampling: Optimal Rare-Event Simulation via Eigenvalue-Conditioned Measure Change", "domain": "finance", "lean": false, "words": 5351, "doi": "10.5281/zenodo.19234222", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We develop a variance reduction framework for simulating rare events in correlated portfolios by exploiting the eigenvalue decomposition of the correlation matrix. The central observation is that the eigenvalue modes $Z_k$ — projections of the asset vector onto the eigenvectors of the correlation ma", "url": "https://thalens.org/papers/fin_spectral_importance_sampling/", "pdf": "https://thalens.org/papers/fin_spectral_importance_sampling/paper.pdf", "created": "2026-03-25", "updated": "2026-03-26"}, {"id": "fin_fenton_practical", "title": "Terminal Portfolio Value Distribution to Machine Precision", "domain": "finance", "lean": true, "words": 5172, "doi": "10.5281/zenodo.19144775", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a deterministic, semi-analytical framework for computing the complete distribution of a portfolio's terminal value at horizon $T$ for correlated lognormal assets. Unlike traditional approaches, this method requires no Monte Carlo simulation.", "url": "https://thalens.org/papers/fin_fenton_practical/", "pdf": "https://thalens.org/papers/fin_fenton_practical/paper.pdf", "created": "2026-03-24", "updated": "2026-03-25"}, {"id": "core_manifold_latent_theorem", "title": "The Manifold Latent Theorem: Finite Spectral Representations Determine Riemannian Geometry", "domain": "spectral_geometry, riemannian_geometry, inverse_spectral_theory", "lean": true, "words": 3926, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We prove that closed Riemannian manifolds with bounded geometry are determined, up to diffeomorphism, by finitely many spectral invariants.", "url": "https://thalens.org/papers/core_manifold_latent_theorem/", "pdf": "https://thalens.org/papers/core_manifold_latent_theorem/paper.pdf", "created": "2026-03-24", "updated": "2026-03-24"}, {"id": "fin_fenton_reference", "title": "The Fenton Distribution: A Complete Analytical Characterization", "domain": "finance", "lean": false, "words": 2492, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We provide the first complete analytical characterization of the distribution of weighted sums of correlated lognormal random variables — the **Fenton distribution** $\\mathcal{F}(w, \\mu, \\sigma, C)$. The distribution is determined by the finite generative latent $(w, \\mu, \\Sigma) \\in \\mathbb{R}^{n(n", "url": "https://thalens.org/papers/fin_fenton_reference/", "pdf": "https://thalens.org/papers/fin_fenton_reference/paper.pdf", "created": "2026-03-24", "updated": "2026-03-24"}, {"id": "fin_nonlinear_portfolio_risk", "title": "Nonlinear Portfolio Risk in Closed Form", "domain": "finance", "lean": false, "words": 2124, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We extend the Hermite-COS framework from linear portfolios (weighted sums of correlated lognormals) to portfolios containing options, structured products, and arbitrary derivative payoffs.", "url": "https://thalens.org/papers/fin_nonlinear_portfolio_risk/", "pdf": "https://thalens.org/papers/fin_nonlinear_portfolio_risk/paper.pdf", "created": "2026-03-24", "updated": "2026-03-24"}, {"id": "meta_ricci_flow_spectral_compression", "title": "Ricci Flow as Spectral Compression: A Latent Interpretation of Perelman's Proof", "domain": "geometric_analysis, spectral_theory, latent_framework", "lean": false, "words": 2384, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We develop a spectral interpretation of the Hamilton-Perelman Ricci flow program for 3-manifolds.", "url": "https://thalens.org/papers/meta_ricci_flow_spectral_compression/", "pdf": "https://thalens.org/papers/meta_ricci_flow_spectral_compression/paper.pdf", "created": "2026-03-24", "updated": "2026-03-24"}, {"id": "meta_spectral_rigidity_s3", "title": "Spectral Rigidity of the 3-Sphere: Finite Latent Characterization of Topology", "domain": "spectral_geometry, topology, latent_framework", "lean": false, "words": 2501, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We establish spectral rigidity results for the round 3-sphere $S^3$ within the Latent framework. Our main results are threefold.", "url": "https://thalens.org/papers/meta_spectral_rigidity_s3/", "pdf": "https://thalens.org/papers/meta_spectral_rigidity_s3/paper.pdf", "created": "2026-03-24", "updated": "2026-03-24"}, {"id": "phy_fundamental_constants_as_grade_ratios", "title": "The Fine Structure Constant from First Principles: A Two-Axiom Derivation via the Latent Grade Hierarchy", "domain": "physics", "lean": true, "words": 9504, "doi": "10.5281/zenodo.19183140", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We derive the fine-structure constant $\\alpha$ ($1/\\alpha = 137.036$, CODATA) from two axioms — the Hurwitz classification of normed division algebras and a self-duality condition on the vacuum — with **zero free parameters**.", "url": "https://thalens.org/papers/phy_fundamental_constants_as_grade_ratios/", "pdf": "https://thalens.org/papers/phy_fundamental_constants_as_grade_ratios/paper.pdf", "created": "2026-03-20 11:26", "updated": "2026-03-23 15:30"}, {"id": "meta_shadow_mining_methodology", "title": "Shadow Mining: Inferring Higher-Grade Structure from Lower-Grade Data", "domain": "mathematical_methodology", "lean": false, "words": 3979, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Every finite-dimensional approximation of a dynamical system loses information. The Shadow Principle (Nagy 2026a) establishes that this loss is structurally detectable but not exactly measurable: the projected system can know *that* something is missing, and approximately *how much*, but not *what*.", "url": "https://thalens.org/papers/meta_shadow_mining_methodology/", "pdf": "https://thalens.org/papers/meta_shadow_mining_methodology/paper.pdf", "created": "2026-03-23", "updated": "2026-03-23"}, {"id": "phy_constructive_qft_latent", "title": "The Feynman Integral as a Latent: Constructive Quantum Field Theory from Grade Decay", "domain": "physics", "lean": true, "words": 6280, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "The constructive quantum field theory (CQFT) program, initiated by Glimm and Jaffe in the 1970s, seeks to build interacting quantum field theories satisfying the Osterwalder-Schrader axioms from mathematically rigorous first principles.", "url": "https://thalens.org/papers/phy_constructive_qft_latent/", "pdf": "https://thalens.org/papers/phy_constructive_qft_latent/paper.pdf", "created": "2026-03-23", "updated": "2026-03-23"}, {"id": "phy_uv_ir_grade_duality", "title": "UV-IR Grade Duality: The Fine-Structure Constant and Cosmological Constant from a Single Scale", "domain": "physics", "lean": true, "words": 3295, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We show that the fine-structure constant $\\alpha$ (an ultraviolet observable, measured at $M_Z \\approx 91$ GeV) and the cosmological constant $\\Lambda$ (an infrared observable, measured at $H_0 \\approx 10^{-33}$ eV) are both determined by a single pa", "url": "https://thalens.org/papers/phy_uv_ir_grade_duality/", "pdf": "https://thalens.org/papers/phy_uv_ir_grade_duality/paper.pdf", "created": "2026-03-23", "updated": "2026-03-23"}, {"id": "aero_spectral_plasma_confinement", "title": "The Latent Theory of Fusion Plasma Confinement", "domain": "physics", "lean": false, "words": 26429, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "**Methodological contribution.** Alongside the physics result below, this paper establishes a reproducible standard for *kernel-verified derived physics* that we argue is portable across domains in which classical analytic derivations have hardened into opaque community consensus. The standard has f", "url": "https://thalens.org/papers/aero_spectral_plasma_confinement/", "pdf": "https://thalens.org/papers/aero_spectral_plasma_confinement/paper.pdf", "created": "2026-03-21", "updated": "2026-03-22"}, {"id": "phy_cosmological_constant_grade", "title": "The Cosmological Constant as a Grade-0 Residual: Smooth Vacuum Energy Flow in the Latent Hierarchy", "domain": "physics", "lean": true, "words": 5485, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We identify the cosmological constant $\\Lambda$ as the grade-0 component of the gravitational Latent hierarchy and show that the **double grade seesaw** — two consecutive grade-2 → grade-0 projections — closes the cosmological constant problem.", "url": "https://thalens.org/papers/phy_cosmological_constant_grade/", "pdf": "https://thalens.org/papers/phy_cosmological_constant_grade/paper.pdf", "created": "2026-03-21", "updated": "2026-03-22"}, {"id": "phy_desi_grade_dark_energy", "title": "What DESI Tells Us About the Grade Structure of Dark Energy", "domain": "physics", "lean": true, "words": 4295, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "The DESI DR2 baryon acoustic oscillation measurements, published in *Nature Astronomy* (2025), report 2.8–4.2$\\sigma$ evidence for dynamical dark energy with an evolving equation of state $w(z) \\neq -1$.", "url": "https://thalens.org/papers/phy_desi_grade_dark_energy/", "pdf": "https://thalens.org/papers/phy_desi_grade_dark_energy/paper.pdf", "created": "2026-03-22", "updated": "2026-03-22"}, {"id": "phy_navier_stokes_grade", "title": "The Grade Structure of Navier–Stokes: Why Blowup Requires Grade-2 Saturation", "domain": "verification", "lean": true, "words": 4371, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We apply the Latent grade hierarchy to the 3D incompressible Navier–Stokes equations. The key structural observation: NS is an **exactly grade-2** system — the right-hand side $F(u) = \\nu\\Delta u - \\mathbb{P}(u\\cdot\\nabla u)$ is a polynomial of degree exactly 2 in $u$, with all higher-grade contribu", "url": "https://thalens.org/papers/phy_navier_stokes_grade/", "pdf": "https://thalens.org/papers/phy_navier_stokes_grade/paper.pdf", "created": "2026-03-22", "updated": "2026-03-22"}, {"id": "phy_supercavitation_grade", "title": "Supercavitation Dynamics from the Grade Equation: The Analyticity Boundary as Cavity Interface", "domain": "physics", "lean": true, "words": 3803, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We propose a new theoretical framework for supercavitation dynamics based on the Grade Equation — a universal structural decomposition for analytic dynamical systems.", "url": "https://thalens.org/papers/phy_supercavitation_grade/", "pdf": "https://thalens.org/papers/phy_supercavitation_grade/paper.pdf", "created": "2026-03-22", "updated": "2026-03-22"}, {"id": "core_latent_probability", "title": "Latent Probability: Conditional Dependence as Graded Spectral Structure", "domain": "core", "lean": false, "words": 4797, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We embed conditional probability in the graded Hilbert tensor algebra $\\Lambda^{(i,j)}$ of the Latent framework. The conditional distribution $P(X \\mid Y)$ is identified with a grade-2 linear map $\\Lambda^{(X,Y)}: \\mathcal{H}_Y \\to \\mathcal{H}_X$ between latent spaces.", "url": "https://thalens.org/papers/core_latent_probability/", "pdf": "https://thalens.org/papers/core_latent_probability/paper.pdf", "created": "2026-03-21", "updated": "2026-03-21"}, {"id": "phy_latent_mtheory", "title": "Why Seven: Grade-3 Sufficiency and the Dimension of M-Theory", "domain": "math", "lean": false, "words": 23202, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "The standard derivation of M-theory's 11-dimensional structure relies on the Nahm bound: supersymmetric theories with massless particles of spin $\\leq 2$ require $d \\leq 11$. We present a complementary and independent derivation from the Latent framework (Nagy 2026).", "url": "https://thalens.org/papers/phy_latent_mtheory/", "pdf": "https://thalens.org/papers/phy_latent_mtheory/paper.pdf", "created": "2026-03-21", "updated": "2026-03-21"}, {"id": "core_latent_of_latents", "title": "The Latent of Latents: Hierarchical Finite Representations of Knowledge Families", "domain": "ml", "lean": true, "words": 11938, "doi": "10.5281/zenodo.19134434", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "The Latent Theorem guarantees that any smooth system has a finite representation whose size depends on regularity and accuracy, not on ambient dimensionality. We extend this result to **families** of smooth systems.", "url": "https://thalens.org/papers/core_latent_of_latents/", "pdf": "https://thalens.org/papers/core_latent_of_latents/paper.pdf", "created": "2026-03-19", "updated": "2026-03-20"}, {"id": "ml_emergence_as_universal_modes", "title": "Emergent Capabilities as Universal Latent Modes", "domain": "ml", "lean": false, "words": 3116, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Emergent capabilities in large language models — abilities that appear suddenly as model scale increases — remain poorly understood. We propose that emergence corresponds to the activation of new **universal Latent modes**: directions in the model's representation space that are shared across all do", "url": "https://thalens.org/papers/ml_emergence_as_universal_modes/", "pdf": "https://thalens.org/papers/ml_emergence_as_universal_modes/paper.pdf", "created": "2026-03-20", "updated": "2026-03-20"}, {"id": "ml_intelligence_estimation", "title": "The Shadow Theorem", "domain": "intelligence_theory", "lean": false, "words": 9580, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We introduce the Shadow Theorem, a family of spectral bounds on the ability of a lower-capacity agent to estimate the capabilities of a higher-capacity agent.", "url": "https://thalens.org/papers/ml_intelligence_estimation/", "pdf": "https://thalens.org/papers/ml_intelligence_estimation/paper.pdf", "created": "2026-03-20", "updated": "2026-03-20"}, {"id": "phy_spectral_generator_nbody", "title": "The Spectral Generator of the N-Body Latent: Connecting Padé Poles, Koopman Eigenvalues, and Dynamical Classification", "domain": "core", "lean": false, "words": 3837, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "The N-body Latent [Nagy 2026g, 2026i] provides an exact, finite representation of gravitational trajectories via the generating function $G(z; \\mathbf{v}_0)$.", "url": "https://thalens.org/papers/phy_spectral_generator_nbody/", "pdf": "https://thalens.org/papers/phy_spectral_generator_nbody/paper.pdf", "created": "2026-03-20", "updated": "2026-03-20"}, {"id": "nt_euler_product_smoothness", "title": "The Euler Product Smoothness Theorem: Multiplicative Structure Forces Latent Existence", "domain": "math", "lean": false, "words": 45534, "doi": "10.5281/zenodo.19207857", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We prove that the distribution of values of random Euler products on the\ncritical line possesses a stable Latent — a finite rational approximation\nwith exponential convergence — and provide a **complete structural proof**\nof the Euler Product Smoothn", "url": "https://thalens.org/papers/nt_euler_product_smoothness/", "pdf": "https://thalens.org/papers/nt_euler_product_smoothness/paper.pdf", "created": "2026-03-19", "updated": "2026-03-19"}, {"id": "nt_pade_universality", "title": "The Universal Padé–Stieltjes Machine: One Algebraic Pipeline from Lognormal Sums Through the Riemann Zeta Function to the Three-Body Problem", "domain": "math", "lean": false, "words": 4947, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We identify a single algebraic pipeline — **Moments → Padé–Stieltjes resummation → Rational characteristic function → Distribution/Trajectory** — that solves three structurally different problems:\n\n1. **Lognormal sums** (statistics/finance): $S = \\sum w_i e^{Y_i}$, $Y \\sim \\mathcal{N}(\\mu, \\Sigma)$.", "url": "https://thalens.org/papers/nt_pade_universality/", "pdf": "https://thalens.org/papers/nt_pade_universality/paper.pdf", "created": "2026-03-19", "updated": "2026-03-19"}, {"id": "nt_rh_latent_existence", "title": "The Riemann Hypothesis as a Latent Existence Theorem", "domain": "math", "lean": false, "words": 4389, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We prove that the Riemann Hypothesis is equivalent to the existence of a\nstable finite rational approximation — a *Latent* — for the distribution\nof $|\\zeta(1/2+it)|$ on the critical line.\n\nDefine the empirical Laplace transform\n$\\hat{F}_T(z) = \\frac", "url": "https://thalens.org/papers/nt_rh_latent_existence/", "pdf": "https://thalens.org/papers/nt_rh_latent_existence/paper.pdf", "created": "2026-03-19", "updated": "2026-03-19"}, {"id": "ml_verified_spectral_intelligence", "title": "Machine-Verified Spectral Intelligence: From Data Geometry to Holographic Bounds via Lean 4", "domain": "ml", "lean": true, "words": 4098, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present the first complete, machine-verified proof chain connecting four\nfundamental phenomena in deep learning theory: neural scaling laws, attention as\nspectral filtering, grokking as a spectral phase transition, and holographic\nentropy bounds on learned representations.", "url": "https://thalens.org/papers/ml_verified_spectral_intelligence/", "pdf": "", "created": "2026-03-18 05:35", "updated": "2026-03-18 15:30"}, {"id": "fin_smooth_latent_operator", "title": "The Smooth Latent Operator: Parameter-Free Distributional Representations via Kernel Moment Recovery", "domain": "core", "lean": false, "words": 3167, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "The Padé resummation of a moment generating function — as used in distributional Latent representations (Nagy, 2026h) and orbital Latent representations (Nagy, 2026g) — depends on a discrete Padé order $N_P$.", "url": "https://thalens.org/papers/fin_smooth_latent_operator/", "pdf": "https://thalens.org/papers/fin_smooth_latent_operator/paper.pdf", "created": "2026-03-18", "updated": "2026-03-18"}, {"id": "fin_sum_of_lognormals", "title": "The Exact Latent Distribution of Correlated Lognormal Sums", "domain": "finance", "lean": false, "words": 7869, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "The distribution of $S = \\sum_{i=1}^n w_i e^{Y_i}$, $Y \\sim \\mathcal{N}(\\mu, \\Sigma)$, has no closed-form CDF.", "url": "https://thalens.org/papers/fin_sum_of_lognormals/", "pdf": "https://thalens.org/papers/fin_sum_of_lognormals/paper.pdf", "created": "2026-03-18", "updated": "2026-03-18"}, {"id": "phy_nbody_latent_solution", "title": "The Exact Latent Solution of the Gravitational N-Body Problem", "domain": "physics", "lean": true, "words": 4613, "doi": "10.5281/zenodo.19258476", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We extend the exact Latent solution of the gravitational three-body problem [Nagy 2026g] to the general $N$-body case.", "url": "https://thalens.org/papers/phy_nbody_latent_solution/", "pdf": "https://thalens.org/papers/phy_nbody_latent_solution/paper.pdf", "created": "2026-03-18", "updated": "2026-03-18"}, {"id": "phy_practical_pade_3body", "title": "Practical Padé Representations of the Gravitational Three-Body Problem", "domain": "physics", "lean": true, "words": 2886, "doi": "10.5281/zenodo.19101253", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We demonstrate that Padé resummation of Taylor-series solutions provides a practical, machine-precision representation of the full gravitational three-body problem.", "url": "https://thalens.org/papers/phy_practical_pade_3body/", "pdf": "https://thalens.org/papers/phy_practical_pade_3body/paper.pdf", "created": "2026-03-18", "updated": "2026-03-18"}, {"id": "ml_knowledge_algebra_applied", "title": "Applied Knowledge Algebra: A Collection of Demonstrations and Use Cases", "domain": "ml", "lean": false, "words": 10099, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "The companion methods paper (Nagy 2026a) defines the Knowledge Artifact — a portable spectral representation of trained model knowledge — and the Knowledge Algebra — exact arithmetic on compatible artifacts.", "url": "https://thalens.org/papers/ml_knowledge_algebra_applied/", "pdf": "https://thalens.org/papers/ml_knowledge_algebra_applied/paper.pdf", "created": "2026-03-09", "updated": "2026-03-17"}, {"id": "fin_minimal_sufficient_spectral_state", "title": "The Minimal Sufficient Spectral State for Decision and Risk", "domain": "finance", "lean": false, "words": 1793, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Decision systems often carry excessive state, obscuring both theory and implementation. We define and\ncharacterize the minimal sufficient spectral state needed to preserve optimal decision value and calibrated\nrisk estimation.", "url": "https://thalens.org/papers/fin_minimal_sufficient_spectral_state/", "pdf": "https://thalens.org/papers/fin_minimal_sufficient_spectral_state/paper.pdf", "created": "2026-03-14 20:59", "updated": "2026-03-16 21:20"}, {"id": "fin_nonlinear_spectral_ftap", "title": "A Nonlinear Spectral Fundamental Theorem of Asset Pricing", "domain": "finance", "lean": true, "words": 1654, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Classical FTAP characterizes no-arbitrage in linear pricing systems. We develop a nonlinear spectral\nextension in which state prices are represented as mode-dependent functionals over a spectral decomposition\nof market states.", "url": "https://thalens.org/papers/fin_nonlinear_spectral_ftap/", "pdf": "https://thalens.org/papers/fin_nonlinear_spectral_ftap/paper.pdf", "created": "2026-03-14 20:59", "updated": "2026-03-16 21:20"}, {"id": "fin_spectral_causal_identifiability", "title": "Spectral Causal Identifiability Under Partial and Noisy Observation", "domain": "finance", "lean": false, "words": 1641, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Causal identification typically requires rich interventions or full-state observability. We show that\npartial, noisy observations can still identify causal structure when signals admit a stable spectral\ndecomposition with separation conditions.", "url": "https://thalens.org/papers/fin_spectral_causal_identifiability/", "pdf": "https://thalens.org/papers/fin_spectral_causal_identifiability/paper.pdf", "created": "2026-03-14 20:59", "updated": "2026-03-16 21:20"}, {"id": "fin_spectral_decision_functional_approximation", "title": "Universal Approximation Theorems for Spectral Decision Functionals", "domain": "verification", "lean": true, "words": 1926, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We prove universal approximation results for a broad class of decision functionals represented in spectral\ncoordinates. The theorem characterizes expressivity in terms of basis regularity, coefficient decay, and\nfunctional smoothness, and provides quantitative approximation rates.", "url": "https://thalens.org/papers/fin_spectral_decision_functional_approximation/", "pdf": "https://thalens.org/papers/fin_spectral_decision_functional_approximation/paper.pdf", "created": "2026-03-14 20:59", "updated": "2026-03-16 21:20"}, {"id": "fin_spectral_ergodic_control", "title": "Spectral Ergodic Control with Provable Regret Guarantees", "domain": "finance", "lean": false, "words": 1775, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Ergodic control and online regret analysis are usually developed with different tools and assumptions. We\npresent a spectral ergodic control framework where long-run average control objectives and finite-time\nregret can be analyzed jointly.", "url": "https://thalens.org/papers/fin_spectral_ergodic_control/", "pdf": "https://thalens.org/papers/fin_spectral_ergodic_control/paper.pdf", "created": "2026-03-14 20:59", "updated": "2026-03-16 21:20"}, {"id": "fin_topological_market_dynamics", "title": "Topological Obstructions in Market Dynamics", "domain": "finance", "lean": false, "words": 2018, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We introduce a topological perspective on market dynamics and prove obstruction results: under specific\nglobal state-space constraints, no arbitrage-free dynamics satisfying given regularity and observability\naxioms can exist.", "url": "https://thalens.org/papers/fin_topological_market_dynamics/", "pdf": "https://thalens.org/papers/fin_topological_market_dynamics/paper.pdf", "created": "2026-03-14 20:59", "updated": "2026-03-16 21:20"}, {"id": "meta_information_geometry_spectral_risk_bridge", "title": "Bridging Information Geometry and Spectral Risk Geometry", "domain": "math", "lean": false, "words": 1689, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Information geometry and risk geometry have developed in parallel, with limited structural unification. We\nconstruct a bridge that maps statistical manifold structure to spectral risk coordinates, proving\ncompatibility results between information-theoretic curvature objects and risk-distance functio", "url": "https://thalens.org/papers/meta_information_geometry_spectral_risk_bridge/", "pdf": "https://thalens.org/papers/meta_information_geometry_spectral_risk_bridge/paper.pdf", "created": "2026-03-14 20:59", "updated": "2026-03-16 21:20"}, {"id": "ml_spectral_learning_universality", "title": "Universality Classes of Spectral Learning Dynamics", "domain": "ml", "lean": false, "words": 1768, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Modern learning systems appear algorithmically diverse yet empirically convergent toward a small set of\ntraining regimes. We propose a universality framework in which optimization dynamics are classified by\nspectral invariants rather than by optimizer-specific update rules.", "url": "https://thalens.org/papers/ml_spectral_learning_universality/", "pdf": "https://thalens.org/papers/ml_spectral_learning_universality/paper.pdf", "created": "2026-03-14 20:59", "updated": "2026-03-16 21:20"}, {"id": "ml_spectral_phase_transition_generalization", "title": "Spectral Phase Transitions in Generalization", "domain": "ml", "lean": false, "words": 1662, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We develop a phase-transition theory of generalization based on spectral order parameters. Instead of\nsmooth monotonic scaling, we show that model performance exhibits threshold behavior when mode occupancy,\nnoise-floor coupling, and effective rank cross critical boundaries.", "url": "https://thalens.org/papers/ml_spectral_phase_transition_generalization/", "pdf": "https://thalens.org/papers/ml_spectral_phase_transition_generalization/paper.pdf", "created": "2026-03-14 20:59", "updated": "2026-03-16 21:20"}, {"id": "fin_spectral_time", "title": "Spectral Time: Optimal Stopping, First Passage, and Subordination via the Fokker-Planck Generator", "domain": "finance", "lean": false, "words": 6294, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We develop a unified spectral framework for three fundamental problems in stochastic analysis: optimal stopping, first passage times, and time-changed (subordinated) processes.", "url": "https://thalens.org/papers/fin_spectral_time/", "pdf": "https://thalens.org/papers/fin_spectral_time/paper.pdf", "created": "2026-03-07", "updated": "2026-03-16 05:48"}, {"id": "meta_theory_decomposability", "title": "Theory and Measure of Decomposability", "domain": "finance", "lean": false, "words": 4835, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "This paper develops a general theory of decomposability for structured systems, problems, and phenomena.", "url": "https://thalens.org/papers/meta_theory_decomposability/", "pdf": "https://thalens.org/papers/meta_theory_decomposability/paper.pdf", "created": "2026-03-15", "updated": "2026-03-16 05:48"}, {"id": "meta_theory_emergent_complexity", "title": "Emergent Complexity in Reality and Engineered AI Systems", "domain": "finance", "lean": false, "words": 6536, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We study emergent complexity as a regime property of open dynamical systems rather than as an intrinsic label of objects.", "url": "https://thalens.org/papers/meta_theory_emergent_complexity/", "pdf": "https://thalens.org/papers/meta_theory_emergent_complexity/paper.pdf", "created": "2026-03-15 08:35", "updated": "2026-03-16 05:48"}, {"id": "meta_theory_spectral_information_state", "title": "The Spectral Information State", "domain": "research", "lean": false, "words": 7139, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We introduce the **spectral information state** as a canonical mode-level state object for spectral inference. After eigendecomposition, each mode $k$ carries an estimated coefficient $\\hat{A}_k$ and a residual uncertainty $\\sigma_k^2$ induced by finite sample size, noise level, and eigenvalue stren", "url": "https://thalens.org/papers/meta_theory_spectral_information_state/", "pdf": "https://thalens.org/papers/meta_theory_spectral_information_state/paper.pdf", "created": "2026-03-12", "updated": "2026-03-16 05:48"}, {"id": "meta_mandelbrot_spectral_evolution", "title": "Spectral Matrix Evolution of the Mandelbrot Iteration: Jacobian Products, Coherence, and Meta-Modes", "domain": "physics", "lean": false, "words": 2247, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "The Mandelbrot set $\\mathcal{M} = \\{c \\in \\mathbb{C} : z_n = z_{n-1}^2 + c \\text{ stays bounded}\\}$ is traditionally visualized by escape-time coloring.", "url": "https://thalens.org/papers/meta_mandelbrot_spectral_evolution/", "pdf": "https://thalens.org/papers/meta_mandelbrot_spectral_evolution/paper.pdf", "created": "2026-03-14", "updated": "2026-03-16"}, {"id": "meta_meta_spectral", "title": "Spectral of Spectrals: Second-Order Mode Decomposition for Complex Systems", "domain": "physics", "lean": false, "words": 2105, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Spectral decomposition extracts modes from data: eigenvectors of a covariance or kernel operator that capture maximal variance per component. We introduce a **second-order spectral layer**: the eigenanalysis of the spectral features themselves.", "url": "https://thalens.org/papers/meta_meta_spectral/", "pdf": "https://thalens.org/papers/meta_meta_spectral/paper.pdf", "created": "2026-03-14", "updated": "2026-03-16"}, {"id": "ml_spectral_cognitive_resonator", "title": "The Spectral Cognitive Resonator: A Dynamic Architecture for Agent Memory, Routing, and Self-Adaptation", "domain": "ml", "lean": false, "words": 2692, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Current AI agent architectures either use static retrieval (RAG) or unstructured agent loops (ReAct, Reflexion) with no formal guarantees on memory utilization, routing optimality, or safe self-adaptation.", "url": "https://thalens.org/papers/ml_spectral_cognitive_resonator/", "pdf": "https://thalens.org/papers/ml_spectral_cognitive_resonator/paper.pdf", "created": "2026-03-14", "updated": "2026-03-16"}, {"id": "ml_spectral_memory_graph_routing", "title": "Spectral Memory and Graph Routing for Language Model Agents", "domain": "ml", "lean": false, "words": 2068, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "LLM agents retrieve context via flat embedding similarity: the query is embedded, the $k$ nearest neighbors are returned, and relevance decays with cosine distance. This approach ignores **structure**: which claims support which, which topics form coherent clusters, where the gaps are, and what time", "url": "https://thalens.org/papers/ml_spectral_memory_graph_routing/", "pdf": "https://thalens.org/papers/ml_spectral_memory_graph_routing/paper.pdf", "created": "2026-03-14", "updated": "2026-03-16"}, {"id": "ml_spectral_neural_architecture", "title": "Spectral-State Neural Networks: A Mode-Decomposition Architecture for Learned Dynamics", "domain": "ml", "lean": false, "words": 2516, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Standard neural networks represent hidden state as unstructured activation vectors evolved by arbitrary weight matrices.", "url": "https://thalens.org/papers/ml_spectral_neural_architecture/", "pdf": "https://thalens.org/papers/ml_spectral_neural_architecture/paper.pdf", "created": "2026-03-14", "updated": "2026-03-16"}, {"id": "phy_theory_spectral_vorticity_turbulence", "title": "Spectral Vorticity Bridge for Turbulent Flows", "domain": "physics", "lean": false, "words": 1892, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We introduce a Spectral Vorticity Bridge for incompressible turbulent flow, designed to connect vortex-centered and mode-centered descriptions through a shared operator framework. The bridge has three layers.", "url": "https://thalens.org/papers/phy_theory_spectral_vorticity_turbulence/", "pdf": "https://thalens.org/papers/phy_theory_spectral_vorticity_turbulence/paper.pdf", "created": "2026-03-14", "updated": "2026-03-16"}, {"id": "meta_bridge_spectral_bioinformatics", "title": "Spectral Methods for Bioinformatics and Drug Discovery", "domain": "core", "lean": false, "words": 1849, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We present a bridge framework linking spectral representation methods to core tasks in bioinformatics and drug discovery.", "url": "https://thalens.org/papers/meta_bridge_spectral_bioinformatics/", "pdf": "https://thalens.org/papers/meta_bridge_spectral_bioinformatics/paper.pdf", "created": "2026-03-14 19:09", "updated": "2026-03-14 19:21"}, {"id": "meta_theory_system_intelligence", "title": "Intelligence as Organized Difficulty Compression", "domain": "finance", "lean": false, "words": 6126, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We propose a system-level theory of intelligence centered on a single claim: **intelligence is organized\ndifficulty compression**. The motivating problem is that the term *intelligence* is routinely asked to\ncover too much at once, ranging from consciousness and general reasoning to benchmark succes", "url": "https://thalens.org/papers/meta_theory_system_intelligence/", "pdf": "https://thalens.org/papers/meta_theory_system_intelligence/paper.pdf", "created": "2026-03-13 08:41", "updated": "2026-03-13 22:10"}, {"id": "meta_theory_scientific_venture_design", "title": "Scientific Venture Design", "domain": "finance", "lean": false, "words": 3969, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We propose a formal framework for **scientific venture design**: a falsifiable theory of how to build\nlarge businesses under uncertainty.", "url": "https://thalens.org/papers/meta_theory_scientific_venture_design/", "pdf": "https://thalens.org/papers/meta_theory_scientific_venture_design/paper.pdf", "created": "2026-03-13 07:20", "updated": "2026-03-13 07:20"}, {"id": "meta_theory_information_causality", "title": "Information and Causal Access", "domain": "physics", "lean": false, "words": 1540, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We propose a manifestation-and-causality view of information. The motivating problem is that the term\n`information` is used for several different objects at once: raw differences, symbols, messages, semantic\ncontent, causal signals, and epistemic gain.", "url": "https://thalens.org/papers/meta_theory_information_causality/", "pdf": "https://thalens.org/papers/meta_theory_information_causality/paper.pdf", "created": "2026-03-13", "updated": "2026-03-13"}, {"id": "meta_theory_mathematical_manifestation", "title": "Mathematical Manifestation", "domain": "verification", "lean": false, "words": 2481, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We propose a manifestation-theoretic view of mathematical statements, formal systems, and theories. The\nguiding question is whether the repo's broader one-object / manifestation framework can be extended beyond\nphysical or observational objects to include theoretical and mathematical ones.", "url": "https://thalens.org/papers/meta_theory_mathematical_manifestation/", "pdf": "https://thalens.org/papers/meta_theory_mathematical_manifestation/paper.pdf", "created": "2026-03-13", "updated": "2026-03-13"}, {"id": "meta_theory_one_behind_everything", "title": "The One Behind Everything", "domain": "physics", "lean": false, "words": 9254, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We propose a generator-first extension of the repo's broader spectral program. The starting question is\nnot how a latent state is quantized into a discrete observable, but how a structured latent state may\nitself arise from repeated application of a local operator.", "url": "https://thalens.org/papers/meta_theory_one_behind_everything/", "pdf": "https://thalens.org/papers/meta_theory_one_behind_everything/paper.pdf", "created": "2026-03-13", "updated": "2026-03-13"}, {"id": "meta_theory_quantized_observation", "title": "The Spectral Theory of Observation", "domain": "physics", "lean": false, "words": 26608, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We propose a spectral framework for understanding how discrete outcomes arise from smooth latent\nstructure. The core move is to represent a finite-valued or countable observable not as an object\noutside continuous analysis, but as an atomic measure\n\\[\n\\mu = \\sum_{j=1}^{m} p_j \\,\\delta_{x_j},\n\\]\nalre", "url": "https://thalens.org/papers/meta_theory_quantized_observation/", "pdf": "https://thalens.org/papers/meta_theory_quantized_observation/paper.pdf", "created": "2026-03-08", "updated": "2026-03-13"}, {"id": "core_theory_universal_spectral_representation", "title": "The Universal Spectral Representation Theorem: Breaking the Curse of Dimensionality", "domain": "finance", "lean": true, "words": 7243, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "How many parameters does it take to represent a smooth high-dimensional probability law to accuracy $\\varepsilon$? We prove that the answer is $N = \\Theta(\\log(1/\\varepsilon)/\\log\\rho)$, where $\\rho > 1$ is the analyticity radius, and that this repre", "url": "https://thalens.org/papers/core_theory_universal_spectral_representation/", "pdf": "https://thalens.org/papers/core_theory_universal_spectral_representation/paper.pdf", "created": "2026-03-03", "updated": "2026-03-12"}, {"id": "fin_bayesian_live_risk", "title": "Bayesian Live Risk", "domain": "finance", "lean": true, "words": 11538, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We propose Bayesian Live Risk (BLR), a framework in which the spectral representation of portfolio loss is treated as a posterior state updated in real time.", "url": "https://thalens.org/papers/fin_bayesian_live_risk/", "pdf": "https://thalens.org/papers/fin_bayesian_live_risk/paper.pdf", "created": "2026-03-03", "updated": "2026-03-12"}, {"id": "fin_harvestability", "title": "Harvestability", "domain": "verification", "lean": true, "words": 8437, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "This paper studies **fin_harvestability** as a horizon object for horizon-dependent allocation within a CRRA investor facing Ornstein-Uhlenbeck eigenmodes.", "url": "https://thalens.org/papers/fin_harvestability/", "pdf": "https://thalens.org/papers/fin_harvestability/paper.pdf", "created": "2026-03-07", "updated": "2026-03-12"}, {"id": "fin_harvestability_calibration", "title": "Calibrating Harvestability", "domain": "finance", "lean": false, "words": 2491, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Starting from the canonical fin_harvestability object \\(h(T,\\tau)=1-e^{-T/\\tau}\\), this paper studies the calibration problem: when does expected return begin to dominate volatility strongly enough that the premium can be treated as genuinely availab", "url": "https://thalens.org/papers/fin_harvestability_calibration/", "pdf": "https://thalens.org/papers/fin_harvestability_calibration/paper.pdf", "created": "2026-03-12", "updated": "2026-03-12"}, {"id": "fin_pricing_is_allocation", "title": "Common Pricing, Different Portfolios", "domain": "finance", "lean": true, "words": 7021, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We propose a mode-space formulation in which equilibrium pricing and portfolio choice are organized around a common benchmark rather than treated as separate layers. Let returns be resolved into orthogonal modes \\(c_k\\) with per-mode premiums \\(\\pi_k\\), variances \\(v_k = \\mathrm{Var}(c_k)\\), and mar", "url": "https://thalens.org/papers/fin_pricing_is_allocation/", "pdf": "https://thalens.org/papers/fin_pricing_is_allocation/paper.pdf", "created": "2026-03-11", "updated": "2026-03-12"}, {"id": "fin_vol_surface", "title": "The Spectral Volatility Surface", "domain": "finance", "lean": true, "words": 8745, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We construct a low-rank arbitrage-aware volatility surface with $O(rm)$ parameters and closed-form COS reuse for pricing and Greeks. Total implied variance is expressed as a finite cosine series in log-moneyness, $w(k, T) = c(T) + \\sum_j u_j(T)\\cos(\\omega_j k)$, with $r = 6$–$12$ modes per maturity.", "url": "https://thalens.org/papers/fin_vol_surface/", "pdf": "https://thalens.org/papers/fin_vol_surface/paper.pdf", "created": "2026-03-03", "updated": "2026-03-12"}, {"id": "meta_bridge_bayesian_frequentist_duality", "title": "The Duality of Bayesian and Frequentist Statistics", "domain": "research", "lean": true, "words": 6046, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We show that Bayesian and frequentist statistics admit a common spectral formulation: after eigendecomposition, both frameworks act on the same mode-level evidence and therefore agree on the central inferential question of model complexity.", "url": "https://thalens.org/papers/meta_bridge_bayesian_frequentist_duality/", "pdf": "https://thalens.org/papers/meta_bridge_bayesian_frequentist_duality/paper.pdf", "created": "2026-03-07", "updated": "2026-03-12"}, {"id": "phy_spectral_3body", "title": "The Three-Body Problem Solved Distributionally: Spectral Fokker-Planck for the Circular Restricted Three-Body Problem", "domain": "physics", "lean": false, "words": 4741, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Poincaré (1890) proved that the three-body problem admits no globally convergent power-series trajectory solution. We do not contradict that result.", "url": "https://thalens.org/papers/phy_spectral_3body/", "pdf": "https://thalens.org/papers/phy_spectral_3body/paper.pdf", "created": "2026-03-07", "updated": "2026-03-12"}, {"id": "fin_implied_generator", "title": "The Latent Generator", "domain": "finance", "lean": false, "words": 4041, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We introduce the **latent generator**: a dynamical operator $M$ that is not directly observed, but is inferred from observable data. In general, the latent generator is the law behind a family of snapshots: the operator that generates the observables rather than merely fitting them pointwise.", "url": "https://thalens.org/papers/fin_implied_generator/", "pdf": "https://thalens.org/papers/fin_implied_generator/paper.pdf", "created": "2026-03-08", "updated": "2026-03-10"}, {"id": "meta_observer_relative_catastrophe", "title": "Catastrophe Is Observer-Relative", "domain": "verification", "lean": false, "words": 4793, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We argue that catastrophe cannot be identified with a drop in global complexity. A transition may destroy one class of observers while increasing the total structural or generative complexity of the world.", "url": "https://thalens.org/papers/meta_observer_relative_catastrophe/", "pdf": "https://thalens.org/papers/meta_observer_relative_catastrophe/paper.pdf", "created": "2026-03-10", "updated": "2026-03-10"}, {"id": "meta_spectral_theory_of_knowability", "title": "Knowability", "domain": "finance", "lean": false, "words": 2243, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We propose that the central object of scientific description is not the sample path but the law, and that observable structure is shaped jointly by latent aggregation, dynamical evolution, and the lens through which a phenomenon is measured. In this view, many operations that appear distinct in obse", "url": "https://thalens.org/papers/meta_spectral_theory_of_knowability/", "pdf": "https://thalens.org/papers/meta_spectral_theory_of_knowability/paper.pdf", "created": "2026-03-10", "updated": "2026-03-10"}, {"id": "fin_spectral_granger", "title": "Spectral Granger Causality: Mode-by-Mode Causal Bandwidth", "domain": "finance", "lean": true, "words": 3170, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "Granger causality (Granger, 1969) tests whether the past of $X$ helps predict $Y$ beyond $Y$'s own past.", "url": "https://thalens.org/papers/fin_spectral_granger/", "pdf": "https://thalens.org/papers/fin_spectral_granger/paper.pdf", "created": "2026-03-08", "updated": "2026-03-09"}, {"id": "ml_adam", "title": "Adam's Convergence Proof Was Wrong: A Machine-Checked Verification of the Bug and the Fix", "domain": "verification", "lean": true, "words": 8361, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "Adam (Kingma & Ba, 2015) is deep learning's most-cited optimizer, with over 100,000 citations and native implementation in every major framework: TensorFlow, PyTorch, JAX, Keras. Its original convergence proof — Theorem 4.1, published at ICLR 2015 — claimed $R_T = O(\\sqrt{T})$ regret on convex probl", "url": "https://thalens.org/papers/ml_adam/", "pdf": "https://thalens.org/papers/ml_adam/paper.pdf", "created": "2026-03-07", "updated": "2026-03-09"}, {"id": "ml_rho_training_metric", "title": "What Is ρ in Training?", "domain": "ml", "lean": false, "words": 3377, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Modern training optimizes losses that are local, task-specific, and often blind to the amount of recoverable structure present in the data. We propose a complementary view: the spectral quality parameter $\\rho > 1$ should be understood as a **structural fitness metric**.", "url": "https://thalens.org/papers/ml_rho_training_metric/", "pdf": "https://thalens.org/papers/ml_rho_training_metric/paper.pdf", "created": "2026-03-09", "updated": "2026-03-09"}, {"id": "aero_spectral_conjunction", "title": "Spectral Density Propagation for Conjunction Assessment: A Deterministic Alternative to Monte Carlo", "domain": "core", "lean": true, "words": 2706, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a spectral method for orbital uncertainty propagation and collision probability computation that is deterministic, sub-millisecond per conjunction, and converges exponentially in the number of spectral modes.", "url": "https://thalens.org/papers/aero_spectral_conjunction/", "pdf": "https://thalens.org/papers/aero_spectral_conjunction/paper.pdf", "created": "2026-03-07", "updated": "2026-03-08"}, {"id": "aero_spectral_fusion", "title": "Spectral Disruption Prediction: Real-Time Plasma Stability via the MHD Generator", "domain": "core", "lean": false, "words": 2262, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We propose a spectral framework for real-time disruption prediction in tokamak fusion reactors. The magnetohydrodynamic (MHD) stability of a plasma equilibrium is characterized by the eigenvalue spectrum of the linearized MHD force operator $\\mathcal{L}$.", "url": "https://thalens.org/papers/aero_spectral_fusion/", "pdf": "https://thalens.org/papers/aero_spectral_fusion/paper.pdf", "created": "2026-03-08", "updated": "2026-03-08"}, {"id": "aero_spectral_kessler", "title": "Spectral Cascade Risk: Quantifying the Kessler Syndrome via Fokker-Planck Fragment Propagation", "domain": "core", "lean": false, "words": 2689, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "The Kessler syndrome --- a self-sustaining cascade of orbital collisions generating exponentially growing debris --- is the existential threat to the low Earth orbit environment.", "url": "https://thalens.org/papers/aero_spectral_kessler/", "pdf": "https://thalens.org/papers/aero_spectral_kessler/paper.pdf", "created": "2026-03-08", "updated": "2026-03-08"}, {"id": "aero_spectral_maneuver", "title": "Optimal Collision Avoidance in One Microsecond: Analytical Maneuver Planning via the Spectral Fokker-Planck Generator", "domain": "core", "lean": true, "words": 2651, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Collision avoidance maneuver planning for low Earth orbit satellites currently relies on trial-and-error: an operator proposes a velocity change $\\Delta v$, recomputes the collision probability $P_c$, and iterates until the risk falls below threshold.", "url": "https://thalens.org/papers/aero_spectral_maneuver/", "pdf": "https://thalens.org/papers/aero_spectral_maneuver/paper.pdf", "created": "2026-03-08", "updated": "2026-03-08"}, {"id": "aero_spectral_traffic", "title": "Real-Time Space Traffic Management for 50,000 Satellites: A Spectral Digital Twin", "domain": "verification", "lean": true, "words": 3079, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "The low Earth orbit environment is projected to host over 50,000 active satellites by 2028, generating approximately 125,000 relevant conjunction pairs per day. Current conjunction assessment operates in overnight batch mode, producing alerts hours after the screening window closes.", "url": "https://thalens.org/papers/aero_spectral_traffic/", "pdf": "https://thalens.org/papers/aero_spectral_traffic/paper.pdf", "created": "2026-03-08", "updated": "2026-03-08"}, {"id": "fin_implied_generator_trading", "title": "Trading the Eigenvalues: Statistical Arbitrage via the Implied Generator", "domain": "finance", "lean": false, "words": 2542, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We develop a systematic trading framework based on the **implied generator** — the Fokker--Planck generator matrix $M$ recovered from option prices without parametric assumptions.", "url": "https://thalens.org/papers/fin_implied_generator_trading/", "pdf": "https://thalens.org/papers/fin_implied_generator_trading/paper.pdf", "created": "2026-03-08", "updated": "2026-03-08"}, {"id": "fin_spectral_defi_liquidation", "title": "Spectral Liquidation Risk: Exact First Passage Times for DeFi Lending Protocols", "domain": "core", "lean": false, "words": 2885, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We apply the spectral Fokker--Planck framework to the central risk problem in DeFi lending: **when will a collateralized position be liquidated?** In protocols such as Aave, Compound, and MakerDAO, a borrower deposits collateral (e.g., ETH) and borrows against it (e.g., USDC).", "url": "https://thalens.org/papers/fin_spectral_defi_liquidation/", "pdf": "https://thalens.org/papers/fin_spectral_defi_liquidation/paper.pdf", "created": "2026-03-08", "updated": "2026-03-08"}, {"id": "ml_bayesian_spectral_dl", "title": "K* Modes Are All You Need: Spectral Uncertainty Quantification for Deep Learning", "domain": "ml", "lean": true, "words": 2730, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We show that the uncertainty of any smooth learning problem is captured by $K^*$ spectral modes of the data covariance, where $K^* = \\Theta(\\log(n/\\sigma^2)/\\log(\\rho^2))$ depends only on the dataset size $n$, noise level $\\sigma$, and eigenvalue dec", "url": "https://thalens.org/papers/ml_bayesian_spectral_dl/", "pdf": "https://thalens.org/papers/ml_bayesian_spectral_dl/paper.pdf", "created": "2026-03-08", "updated": "2026-03-08"}, {"id": "ml_spectral_barren_plateau", "title": "One Number Predicts Barren Plateaus: The Lindblad Spectral Gap as Trainability Bound", "domain": "ml", "lean": false, "words": 2604, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We show that the trainability of variational quantum circuits is determined by a single number: the spectral gap $|\\lambda_1|$ of the Lindblad noise generator.", "url": "https://thalens.org/papers/ml_spectral_barren_plateau/", "pdf": "https://thalens.org/papers/ml_spectral_barren_plateau/paper.pdf", "created": "2026-03-08", "updated": "2026-03-08"}, {"id": "ml_spectral_emergence", "title": "Emergence Is a Spectral Phase Transition: Predicting When Language Models Acquire New Abilities", "domain": "ml", "lean": false, "words": 2416, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We propose that emergent abilities in large language models are spectral phase transitions: a capability appears when the model's spectral resolution crosses the task's intrinsic complexity.", "url": "https://thalens.org/papers/ml_spectral_emergence/", "pdf": "https://thalens.org/papers/ml_spectral_emergence/paper.pdf", "created": "2026-03-08", "updated": "2026-03-08"}, {"id": "ml_spectral_lora", "title": "Why Does LoRA Work? The Spectral Theory of Low-Rank Adaptation", "domain": "ml", "lean": false, "words": 2310, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Low-Rank Adaptation (LoRA; Hu et al., 2021) fine-tunes large language models by adding rank-$r$ updates $\\Delta W = AB$ with $r \\ll d$. In practice, $r = 4$--$16$ works remarkably well, but no theory explains why or predicts the optimal $r$ for a given task.", "url": "https://thalens.org/papers/ml_spectral_lora/", "pdf": "https://thalens.org/papers/ml_spectral_lora/paper.pdf", "created": "2026-03-08", "updated": "2026-03-08"}, {"id": "phy_kinetic_bc", "title": "Specular Reflection in Spectral Fokker-Planck: A Penalty Method with Proven Convergence", "domain": "math", "lean": true, "words": 1809, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We identify and resolve a fundamental error in spectral methods for kinetic Fokker--Planck equations. The standard cosine (Neumann) basis enforces $\\partial p/\\partial x = 0$ at spatial boundaries, which is the correct reflecting boundary condition for overdamped (position-only) dynamics.", "url": "https://thalens.org/papers/phy_kinetic_bc/", "pdf": "https://thalens.org/papers/phy_kinetic_bc/paper.pdf", "created": "2026-03-08", "updated": "2026-03-08"}, {"id": "phy_spectral_decoherence", "title": "The α-Continuum: Spectral Gap Controls the Quantum-Classical Transition", "domain": "physics", "lean": false, "words": 3204, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We define an effective quantumness parameter $\\alpha_{\\text{eff}}(t) = 1 + (\\text{Tr}(\\rho^2) - 1/d)/(1 - 1/d) \\in [1, 2]$ that interpolates continuously between quantum ($\\alpha = 2$, pure state, full interference) and classical ($\\alpha = 1$, maximally mixed, no interference).", "url": "https://thalens.org/papers/phy_spectral_decoherence/", "pdf": "https://thalens.org/papers/phy_spectral_decoherence/paper.pdf", "created": "2026-03-08", "updated": "2026-03-08"}, {"id": "phy_spectral_measurement", "title": "There Is No Collapse: Measurement as Spectral Projection", "domain": "physics", "lean": false, "words": 2190, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We reformulate quantum measurement using the language of spectral pattern theory. The pre-measurement state is not a \"wave\" (misleading: it does not propagate in 3D space) and not a \"probability distribution\" (incomplete: it has phase).", "url": "https://thalens.org/papers/phy_spectral_measurement/", "pdf": "https://thalens.org/papers/phy_spectral_measurement/paper.pdf", "created": "2026-03-08", "updated": "2026-03-08"}, {"id": "phy_spectral_reality", "title": "The Spectral Theory of Observation: Modes, Collapse, and the Information Content of Reality", "domain": "physics", "lean": false, "words": 3311, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We identify a single mathematical structure underlying seven apparently distinct physical frameworks: stochastic processes (Fokker--Planck), quantum mechanics (Schrödinger), statistical mechanics (transfer matrix), dynamical systems (Koopman), and information theory (rate-distortion).", "url": "https://thalens.org/papers/phy_spectral_reality/", "pdf": "https://thalens.org/papers/phy_spectral_reality/paper.pdf", "created": "2026-03-08", "updated": "2026-03-08"}, {"id": "core_spectral_resolution", "title": "The Spectral Resolution Theorem: Dimension-Free Compression of Analytic Objects", "domain": "verification", "lean": true, "words": 3105, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We prove that any analytic function on $\\mathbb{R}^n$ --- probability density, regression surface, time series, transfer operator, spatial field --- is representable by $N = \\Theta(\\log(1/\\varepsilon)/\\log\\rho)$ spectral coefficients at accuracy $\\va", "url": "https://thalens.org/papers/core_spectral_resolution/", "pdf": "https://thalens.org/papers/core_spectral_resolution/paper.pdf", "created": "2026-03-07", "updated": "2026-03-07"}, {"id": "fin_anomaly_functional", "title": "The Anomaly Functional: Real-Time Arbitrage Detection via Spectral Risk Coefficients", "domain": "finance", "lean": false, "words": 4250, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We define a scalar functional $\\mathcal{A}[F]$ on probability distributions that quantifies the total magnitude of arbitrage violations --- negative densities (butterfly arbitrage) and decreasing total variance across maturities (calendar arbitrage)", "url": "https://thalens.org/papers/fin_anomaly_functional/", "pdf": "https://thalens.org/papers/fin_anomaly_functional/paper.pdf", "created": "2026-03-03", "updated": "2026-03-07"}, {"id": "fin_arbfree_term_structure", "title": "Arbitrage-Free Term Structure via Spectral Coefficient Constraints", "domain": "finance", "lean": true, "words": 4168, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We propose a yield curve model that combines three properties rarely found together: (i) arbitrage-free dynamics by construction, (ii) strong empirical fit with interpretable factors, and (iii) partial formal verification in Lean 4.", "url": "https://thalens.org/papers/fin_arbfree_term_structure/", "pdf": "https://thalens.org/papers/fin_arbfree_term_structure/paper.pdf", "created": "2026-03-04", "updated": "2026-03-07"}, {"id": "fin_bitcoin_thermodynamic", "title": "Bitcoin as a Thermodynamic System: Phase Transitions at Halving Events", "domain": "verification", "lean": true, "words": 3231, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We model Bitcoin as a thermodynamic system where the hash rate plays the role of temperature, the price plays the role of pressure, and the halving events are phase transitions.", "url": "https://thalens.org/papers/fin_bitcoin_thermodynamic/", "pdf": "https://thalens.org/papers/fin_bitcoin_thermodynamic/paper.pdf", "created": "2026-03-03", "updated": "2026-03-07"}, {"id": "fin_formula_of_doom", "title": "The Formula of Doom Was Provably Wrong: A Machine-Checked Replacement for the Copula That Caused the 2008 Financial Crisis", "domain": "verification", "lean": true, "words": 4067, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "In 2000, David X. Li published a one-page formula that became the industry standard for pricing collateralized debt obligations. By 2008, it had been used to price trillions of dollars of structured credit products.", "url": "https://thalens.org/papers/fin_formula_of_doom/", "pdf": "https://thalens.org/papers/fin_formula_of_doom/paper.pdf", "created": "2026-03-04", "updated": "2026-03-07"}, {"id": "fin_greeks_multi_asset", "title": "High-Precision Greeks for Multi-Asset Spread Options via Eigenvalue-Conditioned Fourier Inversion", "domain": "finance", "lean": true, "words": 5063, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "Spread options on three or more correlated assets are fundamental hedging instruments in energy, commodity, and agricultural markets.", "url": "https://thalens.org/papers/fin_greeks_multi_asset/", "pdf": "https://thalens.org/papers/fin_greeks_multi_asset/paper.pdf", "created": "2026-03-03", "updated": "2026-03-07"}, {"id": "fin_heston", "title": "Basket Option Pricing Under Stochastic Volatility via Eigenvalue-Conditional Heston Mixing", "domain": "finance", "lean": false, "words": 4581, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "The eigenvalue-conditional mixing framework (Nagy, 2026a, 2026c) prices basket options under geometric Brownian motion by decomposing the correlation matrix, conditioning on dominant factors, and mixing per-scenario Black-Scholes prices.", "url": "https://thalens.org/papers/fin_heston/", "pdf": "https://thalens.org/papers/fin_heston/paper.pdf", "created": "2026-03-03", "updated": "2026-03-07"}, {"id": "fin_how_much_bitcoin", "title": "How Much Bitcoin? Spectral Portfolio Allocation Beyond Mean-Variance", "domain": "finance", "lean": true, "words": 3195, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We show that mean-variance optimization systematically overestimates the optimal Bitcoin allocation in institutional portfolios. The Markowitz framework assumes returns are Gaussian, but Bitcoin has excess kurtosis $\\kappa_4 \\approx 8$--$13$ and negative skewness --- making its tails 3--5$\\times$ he", "url": "https://thalens.org/papers/fin_how_much_bitcoin/", "pdf": "https://thalens.org/papers/fin_how_much_bitcoin/paper.pdf", "created": "2026-03-04", "updated": "2026-03-07"}, {"id": "fin_learned_basis", "title": "Three Numbers for Risk: A Data-Driven Spectral Basis for Portfolio Loss Distributions", "domain": "finance", "lean": false, "words": 4109, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "The Spectral Fenton Distribution represents a portfolio loss distribution with 128 Fourier coefficients.", "url": "https://thalens.org/papers/fin_learned_basis/", "pdf": "https://thalens.org/papers/fin_learned_basis/paper.pdf", "created": "2026-03-03", "updated": "2026-03-07"}, {"id": "fin_pythagorean_gap", "title": "The Pythagorean Gap: A Mode-Decomposition Indicator for Financial Crises", "domain": "finance", "lean": true, "words": 3855, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We introduce the *Pythagorean Gap*, a single scalar metric that measures the dispersion of risk-adjusted returns across spectral modes of a return distribution. When asset returns are decomposed into orthogonal modes via PCA, the tangency portfolio's Sharpe ratio satisfies a Pythagorean identity: $S", "url": "https://thalens.org/papers/fin_pythagorean_gap/", "pdf": "https://thalens.org/papers/fin_pythagorean_gap/paper.pdf", "created": "2026-03-04", "updated": "2026-03-07"}, {"id": "fin_schrodinger_bridge", "title": "Spectral Schrödinger Bridges: Optimal Transport Between Portfolio Distributions in Fourier Space", "domain": "finance", "lean": true, "words": 4878, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "The Schrödinger Bridge Problem (SBP) finds the most likely stochastic evolution between two probability distributions, minimizing the Kullback--Leibler divergence from a reference process.", "url": "https://thalens.org/papers/fin_schrodinger_bridge/", "pdf": "https://thalens.org/papers/fin_schrodinger_bridge/paper.pdf", "created": "2026-03-03", "updated": "2026-03-07"}, {"id": "fin_spectral_alpha", "title": "Spectral Alpha: Trading Signals from Fourier Risk Coefficients", "domain": "finance", "lean": false, "words": 4305, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "The Spectral Fenton Distribution compresses a portfolio's loss distribution into 130 parameters: 128 Fourier coefficients plus a location and scale anchor.", "url": "https://thalens.org/papers/fin_spectral_alpha/", "pdf": "https://thalens.org/papers/fin_spectral_alpha/paper.pdf", "created": "2026-03-03", "updated": "2026-03-07"}, {"id": "fin_spectral_bitcoin_var", "title": "Spectral Bitcoin VaR: High-Accuracy Risk Measures for Cryptocurrency via Fourier Expansion", "domain": "finance", "lean": true, "words": 3600, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We apply the Spectral Fenton Distribution framework (Nagy, 2026a) to Bitcoin return modeling and show that the COS (Fourier-cosine) expansion captures the BTC/USD daily return density with controllable numerical precision.", "url": "https://thalens.org/papers/fin_spectral_bitcoin_var/", "pdf": "https://thalens.org/papers/fin_spectral_bitcoin_var/paper.pdf", "created": "2026-03-03", "updated": "2026-03-07"}, {"id": "fin_spectral_ccr", "title": "Spectral XVA: Replacing Monte Carlo in Counterparty Credit Risk", "domain": "finance", "lean": false, "words": 3187, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We propose a spectral method for computing Credit Valuation Adjustment (CVA) and related XVA metrics that eliminates Monte Carlo simulation entirely.", "url": "https://thalens.org/papers/fin_spectral_ccr/", "pdf": "https://thalens.org/papers/fin_spectral_ccr/paper.pdf", "created": "2026-03-07", "updated": "2026-03-07"}, {"id": "fin_spectral_insurance", "title": "Spectral Insurance: Aggregate Loss Distribution Without Monte Carlo", "domain": "verification", "lean": true, "words": 9949, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "The insurance industry computes aggregate loss distributions — the foundation of Solvency II internal models, Own Risk and Solvency Assessment (ORSA), and reserve adequacy testing — almost exclusively through Monte Carlo simulation.", "url": "https://thalens.org/papers/fin_spectral_insurance/", "pdf": "https://thalens.org/papers/fin_spectral_insurance/paper.pdf", "created": "2026-03-07", "updated": "2026-03-07"}, {"id": "fin_spectral_kernel_risk", "title": "Spectral Kernel Risk: The Fourier--Gaussian Process Duality for Portfolio Loss Distributions", "domain": "finance", "lean": true, "words": 3337, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We establish a formal duality between the Spectral Fenton (Fourier) representation of portfolio loss densities and Gaussian Process (GP) regression via Mercer's theorem.", "url": "https://thalens.org/papers/fin_spectral_kernel_risk/", "pdf": "https://thalens.org/papers/fin_spectral_kernel_risk/paper.pdf", "created": "2026-03-03", "updated": "2026-03-07"}, {"id": "fin_spectral_regime_detection", "title": "Spectral Regime Detection: Change-Point Identification via Eigenmode Drift", "domain": "finance", "lean": false, "words": 2179, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We introduce spectral regime detection: a method to identify structural breaks in time series and panel data by monitoring the drift of spectral coefficients over sliding windows.", "url": "https://thalens.org/papers/fin_spectral_regime_detection/", "pdf": "https://thalens.org/papers/fin_spectral_regime_detection/paper.pdf", "created": "2026-03-07", "updated": "2026-03-07"}, {"id": "fin_tensor_spectral", "title": "The Spectral Tensor Representation of Stochastic Processes", "domain": "finance", "lean": false, "words": 3789, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We prove that every Itô diffusion $dX_t = \\mu(X_t)\\,dt + \\sigma(X_t)\\,dW_t$ admits a complete, finite-dimensional representation through its Fokker--Planck generator discretized in cosine basis.", "url": "https://thalens.org/papers/fin_tensor_spectral/", "pdf": "https://thalens.org/papers/fin_tensor_spectral/paper.pdf", "created": "2026-03-03", "updated": "2026-03-07"}, {"id": "fin_tree_vs_spectral", "title": "When Simulation Is Unnecessary: An Information-Theoretic Characterization of Analytically Tractable Distributions", "domain": "finance", "lean": true, "words": 5565, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "Monte Carlo simulation exists because we cannot compute integrals analytically. But for how large a class of distributions *can* we compute them? We characterize this class precisely via a single number: the analyticity radius $\\rho$.", "url": "https://thalens.org/papers/fin_tree_vs_spectral/", "pdf": "https://thalens.org/papers/fin_tree_vs_spectral/paper.pdf", "created": "2026-03-04", "updated": "2026-03-07"}, {"id": "fin_unified_risk_pricing", "title": "The Spectral Unity: Risk, Pricing, and Hedging from a Single Representation", "domain": "finance", "lean": true, "words": 3855, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "For portfolios of correlated lognormal assets, we show that risk measurement, derivative pricing, and hedging --- traditionally treated as separate disciplines --- reduce to operations on a single object: the $N$-term Fourier-cosine (COS) expansion of the portfolio density.", "url": "https://thalens.org/papers/fin_unified_risk_pricing/", "pdf": "https://thalens.org/papers/fin_unified_risk_pricing/paper.pdf", "created": "2026-03-03", "updated": "2026-03-07"}, {"id": "ml_adversarial_robustness", "title": "Verified Adversarial Robustness: Lipschitz Certificates for Neural Networks in Lean 4", "domain": "ml", "lean": true, "words": 8773, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "Can we *prove* a neural network is safe? Adversarial examples — imperceptible input perturbations that cause misclassification — remain the most persistent failure mode of deep learning. Existing defenses rely on empirical testing, which cannot guarantee safety.", "url": "https://thalens.org/papers/ml_adversarial_robustness/", "pdf": "https://thalens.org/papers/ml_adversarial_robustness/paper.pdf", "created": "2026-03-07", "updated": "2026-03-07"}, {"id": "ml_consciousness", "title": "The Self-Modeling Ceiling: Formally Verified Bounds on Machine Self-Calibration", "domain": "verification", "lean": true, "words": 6507, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "How much can a system know about itself? We develop a formally verified theory of self-calibration limits, connecting spectral coupling models to hallucination, robustness, and convergence guarantees.", "url": "https://thalens.org/papers/ml_consciousness/", "pdf": "https://thalens.org/papers/ml_consciousness/paper.pdf", "created": "2026-03-07", "updated": "2026-03-07"}, {"id": "ml_knowledge_extraction", "title": "What Your XGBoost Learned: Spectral Knowledge Extraction from Black-Box Models", "domain": "finance", "lean": true, "words": 3400, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We introduce **spectral knowledge extraction**: a method to decompose the learned function of a black-box model (XGBoost, random forest, neural network) into explicit Fourier cosine modes, each with a direct interpretation.", "url": "https://thalens.org/papers/ml_knowledge_extraction/", "pdf": "https://thalens.org/papers/ml_knowledge_extraction/paper.pdf", "created": "2026-03-04", "updated": "2026-03-07"}, {"id": "ml_scaling_laws", "title": "Neural Scaling Laws Formalized: Why Chinchilla Works (A Machine-Verified Derivation)", "domain": "ml", "lean": true, "words": 8417, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "Neural scaling laws — the empirical observation that test loss decreases as a power law in compute budget — are the foundation of modern AI training strategy. Every major laboratory trains billion-dollar models by extrapolating scaling curves, yet *why* these power laws hold remains unexplained.", "url": "https://thalens.org/papers/ml_scaling_laws/", "pdf": "https://thalens.org/papers/ml_scaling_laws/paper.pdf", "created": "2026-03-07", "updated": "2026-03-07"}, {"id": "ml_self_improvement", "title": "Provable Bounds on AI Self-Improvement: The Verification Oracle Ceiling", "domain": "verification", "lean": true, "words": 9347, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We establish formally verified bounds on recursive AI self-improvement under a spectral coupling model.", "url": "https://thalens.org/papers/ml_self_improvement/", "pdf": "https://thalens.org/papers/ml_self_improvement/paper.pdf", "created": "2026-03-05", "updated": "2026-03-07"}, {"id": "ml_spectral_knowledge_distillation", "title": "Spectral Knowledge Distillation: From Black Box to Certified White Box", "domain": "ml", "lean": true, "words": 3019, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "Knowledge distillation (Hinton et al., 2015) compresses a large teacher model into a smaller student model by training on the teacher's soft outputs. The student is a smaller neural network — still a black box, with no guarantee on how much knowledge was preserved.", "url": "https://thalens.org/papers/ml_spectral_knowledge_distillation/", "pdf": "https://thalens.org/papers/ml_spectral_knowledge_distillation/paper.pdf", "created": "2026-03-07", "updated": "2026-03-07"}, {"id": "ml_spectral_model_compression", "title": "Spectral Model Compression: Provably Optimal Knowledge Extraction via Eigendecomposition", "domain": "ml", "lean": false, "words": 1911, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We present spectral model compression: any trained model — neural network, tree ensemble, or kernel machine — can be distilled into $K^*$ spectral coefficients via eigendecomposition of its learned function on the data manifold.", "url": "https://thalens.org/papers/ml_spectral_model_compression/", "pdf": "https://thalens.org/papers/ml_spectral_model_compression/paper.pdf", "created": "2026-03-07", "updated": "2026-03-07"}, {"id": "ml_transformer", "title": "Verified Transformer Dynamics: Token Clustering Convergence in Lean 4", "domain": "verification", "lean": true, "words": 7466, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "Transformers are the dominant architecture in modern machine learning — behind GPT-4, Claude, Gemini, and every frontier language model — yet no formal convergence theory exists for how self-attention drives token representations to evolve across layers.", "url": "https://thalens.org/papers/ml_transformer/", "pdf": "https://thalens.org/papers/ml_transformer/paper.pdf", "created": "2026-03-07", "updated": "2026-03-07"}, {"id": "ml_verified_ai_safety", "title": "The AI Safety Certificate: A Machine-Verified Framework for Quantitative AI Safety", "domain": "ml", "lean": true, "words": 8572, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "Can we *prove* an AI system is safe — not with testing, not with benchmarks, but with mathematical certainty? We present a Lean 4 verification of a unified AI safety certificate — to our knowledge, the first formal verification of a multi-dimensional", "url": "https://thalens.org/papers/ml_verified_ai_safety/", "pdf": "https://thalens.org/papers/ml_verified_ai_safety/paper.pdf", "created": "2026-03-07", "updated": "2026-03-07"}, {"id": "ml_what_does_your_model_know", "title": "What Does Your Model Know? Spectral Decomposition and Arithmetic of Machine Learning Knowledge", "domain": "ml", "lean": true, "words": 2731, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We present a method to decompose any trained machine learning model's knowledge into a vector of spectral coefficients, and show that arithmetic on these vectors corresponds to meaningful operations on knowledge: addition combines complementary model", "url": "https://thalens.org/papers/ml_what_does_your_model_know/", "pdf": "https://thalens.org/papers/ml_what_does_your_model_know/paper.pdf", "created": "2026-03-07", "updated": "2026-03-07"}, {"id": "fin_spectral_fx", "title": "Spectral FX: Eigenvalue Classification of Currency Dynamics", "domain": "finance", "lean": true, "words": 4041, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We show that vector autoregression (VAR), cointegration, geometric Brownian motion (GBM), and the Ornstein--Uhlenbeck (OU) process are four eigenvalue regimes of a single linear stochastic differential equation \\(dX = AX\\,dt + \\Sigma\\,dW\\).", "url": "https://thalens.org/papers/fin_spectral_fx/", "pdf": "https://thalens.org/papers/fin_spectral_fx/paper.pdf", "created": "2026-03-06", "updated": "2026-03-06"}, {"id": "fin_adaptive_cos", "title": "Adaptive COS Option Pricing via Per-Mode Convergence Rates", "domain": "finance", "lean": true, "words": 4094, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "The COS method (Fang and Oosterlee, 2008) for option pricing via Fourier-cosine expansion uses a uniform number of backward steps $T$ across all $N$ Fourier modes. We prove that each mode $k$ converges at its own rate $\\delta|\\varphi(k)|$, where $\\delta$ is the per-step discount and $\\varphi(k)$ the", "url": "https://thalens.org/papers/fin_adaptive_cos/", "pdf": "https://thalens.org/papers/fin_adaptive_cos/paper.pdf", "created": "2026-03-05", "updated": "2026-03-05"}, {"id": "fin_extended_bellman", "title": "Extended Bellman Equations for Spectral Finance: Per-Mode Convergence, Shadow Prices, and Model-Free Bounds", "domain": "finance", "lean": true, "words": 5403, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "The Bellman equation $V = \\max_a [R + \\gamma \\sum P V]$ is the foundation of dynamic programming, yet its standard form obscures three structural properties critical for financial applications.", "url": "https://thalens.org/papers/fin_extended_bellman/", "pdf": "https://thalens.org/papers/fin_extended_bellman/paper.pdf", "created": "2026-03-05", "updated": "2026-03-05"}, {"id": "fin_rl_meets_options", "title": "When Q-Learning Meets Black-Scholes: A Machine-Verified Bridge Between Reinforcement Learning and Option Pricing", "domain": "finance", "lean": true, "words": 7990, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "Reinforcement learning and quantitative finance independently discovered the same mathematical framework.", "url": "https://thalens.org/papers/fin_rl_meets_options/", "pdf": "https://thalens.org/papers/fin_rl_meets_options/paper.pdf", "created": "2026-03-05", "updated": "2026-03-05"}, {"id": "fin_spectral_dp", "title": "Spectral Dynamic Programming: Per-Mode Convergence Rates and Computational Acceleration for American Options", "domain": "finance", "lean": true, "words": 5769, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We show that eigendecomposing the transition kernel of a Markov decision process yields a spectral Bellman equation: each eigenmode $k$ satisfies an independent scalar recurrence $v_k' = r_k + \\gamma \\mu_k v_k$, converging at rate $\\gamma |\\mu_k| \\le", "url": "https://thalens.org/papers/fin_spectral_dp/", "pdf": "https://thalens.org/papers/fin_spectral_dp/paper.pdf", "created": "2026-03-05", "updated": "2026-03-05"}, {"id": "meta_spectral_transfer", "title": "Eigenvalue Conditioning as Universal Optimizer: Cross-Domain Transfer Between Finance, Robustness, and Machine Learning", "domain": "finance", "lean": true, "words": 7094, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We prove that eigenvalue conditioning — decompose a structure matrix into eigenmodes, condition on the $K$ dominant modes, solve $K$ independent one-dimensional problems, and combine — is a universal optimization principle that transfers across five", "url": "https://thalens.org/papers/meta_spectral_transfer/", "pdf": "https://thalens.org/papers/meta_spectral_transfer/paper.pdf", "created": "2026-03-04", "updated": "2026-03-05"}, {"id": "ml_spectral_certificates", "title": "Spectral Certificates for Trustworthy AI: Robustness, Confidence, and Fairness from One Decomposition", "domain": "ml", "lean": true, "words": 5323, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We prove that a single singular value decomposition (SVD) of a neural network's local Jacobian yields three formally verified trustworthiness guarantees simultaneously.", "url": "https://thalens.org/papers/ml_spectral_certificates/", "pdf": "https://thalens.org/papers/ml_spectral_certificates/paper.pdf", "created": "2026-03-04", "updated": "2026-03-04"}, {"id": "ml_spectral_distillation", "title": "Spectral Distillation: Provable Knowledge Compression from Black Box to Closed Form", "domain": "ml", "lean": true, "words": 3417, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We introduce **spectral distillation**: a method to compress any black-box model into an explicit Fourier cosine formula with a **provable per-feature error bound**.", "url": "https://thalens.org/papers/ml_spectral_distillation/", "pdf": "https://thalens.org/papers/ml_spectral_distillation/paper.pdf", "created": "2026-03-04", "updated": "2026-03-04"}, {"id": "fin_basket_options", "title": "Pricing Basket Options via Eigenvalue-Conditional Black-Scholes Mixing", "domain": "finance", "lean": false, "words": 7132, "doi": "10.5281/zenodo.18910542", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Pricing European basket options requires the distribution of a weighted sum of correlated lognormals — the Fenton Distribution (Nagy, 2026a) — which has no closed form.", "url": "https://thalens.org/papers/fin_basket_options/", "pdf": "https://thalens.org/papers/fin_basket_options/paper.pdf", "created": "2026-03-01", "updated": "2026-03-03"}, {"id": "fin_risk_geometry", "title": "The Geometry of Risk: Spectral Distance and Topological Structure in Portfolio Space", "domain": "finance", "lean": true, "words": 5482, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We introduce a metric on the space of portfolio risk profiles derived from the spectral representation of loss distributions. Each portfolio's loss distribution is encoded as a vector of $N = 128$ Fourier--cosine coefficients via the Eigen-COS method (Nagy, 2026a).", "url": "https://thalens.org/papers/fin_risk_geometry/", "pdf": "https://thalens.org/papers/fin_risk_geometry/paper.pdf", "created": "2026-03-03", "updated": "2026-03-03"}, {"id": "fin_risk_information", "title": "The Information-Theoretic Cost of Risk Measurement", "domain": "finance", "lean": true, "words": 7050, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "Shannon (1948) proved that the fundamental limit of communication is the channel capacity, not the message length. We prove an analogous result for risk measurement: the fundamental limit of computing coherent risk measures is the **analyticity radius** $\\rho$ of the loss density, not the dimension ", "url": "https://thalens.org/papers/fin_risk_information/", "pdf": "https://thalens.org/papers/fin_risk_information/paper.pdf", "created": "2026-03-03", "updated": "2026-03-03"}, {"id": "fin_spectral_correlation", "title": "Correlation Is Not a Number — It's a Spectrum: Frequency-Domain Dependence for Portfolio Risk", "domain": "finance", "lean": true, "words": 4439, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "The Pearson correlation $\\rho_{ij}$ between two assets is a single number. It is blind to frequency: it cannot distinguish whether two assets co-move in long-term trends or in daily noise.", "url": "https://thalens.org/papers/fin_spectral_correlation/", "pdf": "https://thalens.org/papers/fin_spectral_correlation/paper.pdf", "created": "2026-03-03", "updated": "2026-03-03"}, {"id": "fin_spectral_halving", "title": "The Spectral Halving Cycle: A Fourier Model of Bitcoin's Programmed Supply Shocks", "domain": "verification", "lean": true, "words": 3048, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "Bitcoin is the only financial asset whose supply schedule is deterministic and programmed: every 210{,}000 blocks ($\\approx$4 years), the block reward halves.", "url": "https://thalens.org/papers/fin_spectral_halving/", "pdf": "https://thalens.org/papers/fin_spectral_halving/paper.pdf", "created": "2026-03-03", "updated": "2026-03-03"}, {"id": "fin_spectral_trading", "title": "Frequency-Domain Theory of Financial Economics: Thirteen Fundamental Results from One Decomposition", "domain": "verification", "lean": true, "words": 7009, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "A market is a spectrum. We show that thirteen fundamental results in financial economics --- seven core classical theorems and six domain extensions --- all follow from a single structural assumption: the return density of any portfolio can be decomp", "url": "https://thalens.org/papers/fin_spectral_trading/", "pdf": "https://thalens.org/papers/fin_spectral_trading/paper.pdf", "created": "2026-03-03", "updated": "2026-03-03"}, {"id": "fin_exact_var_computational", "title": "Deterministic Portfolio VaR Without Monte Carlo: The Eigen-COS Method", "domain": "finance", "lean": true, "words": 11097, "doi": "10.5281/zenodo.18910516", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We present the Eigen-COS method, a deterministic algorithm that computes exact Value-at-Risk, closed-form Expected Shortfall, and the full CDF/PDF for weighted sums of correlated lognormal assets — without Monte Carlo simulation.", "url": "https://thalens.org/papers/fin_exact_var_computational/", "pdf": "https://thalens.org/papers/fin_exact_var_computational/paper.pdf", "created": "2026-03-02", "updated": "2026-03-02"}, {"id": "fin_risk_noise_free", "title": "Noise-Free Risk: Deterministic VaR, ES, and Spectral Risk Measures for Lognormal Portfolios", "domain": "finance", "lean": true, "words": 7919, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We present a deterministic framework for computing Value-at-Risk, Expected Shortfall, and arbitrary spectral risk measures for portfolios of correlated lognormal assets, without Monte Carlo simulation.", "url": "https://thalens.org/papers/fin_risk_noise_free/", "pdf": "https://thalens.org/papers/fin_risk_noise_free/paper.pdf", "created": "2026-03-01", "updated": "2026-03-02"}, {"id": "meta_quantum_connection", "title": "The Geometry of Risk: Connecting Spectral Fenton to TQFT and Witten Invariants", "domain": "finance", "lean": false, "words": 595, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/meta_quantum_connection/", "pdf": "https://thalens.org/papers/meta_quantum_connection/paper.pdf", "created": "2026-03-01", "updated": "2026-03-01"}, {"id": "ab_cs2_opus47", "title": "The Cauchy-Schwarz Inequality for 2D Real Vectors via Sum-of-Squares Decomposition", "domain": "verification", "lean": true, "words": 1193, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "We present a formally verified proof of the Cauchy-Schwarz inequality for two-dimensional real vectors. For all real numbers $a, b, c, d$, we establish that $(ac + bd)^2 \\leq (a^2 + b^2)(c^2 + d^2)$.", "url": "https://thalens.org/papers/ab_cs2_opus47/", "pdf": "", "created": "", "updated": ""}, {"id": "abs_and_naturals", "title": "Absolute Value Multiplicativity and Elementary Natural Number Arithmetic", "domain": "verification", "lean": true, "words": 1482, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "We formalize a small collection of foundational results concerning the absolute value function on real numbers and elementary arithmetic on natural numbers. The central result is the multiplicativity of absolute value: for all real numbers $a$ and $b$, we have $|ab| = |a| \\cdot |b|$.", "url": "https://thalens.org/papers/abs_and_naturals/", "pdf": "", "created": "", "updated": ""}, {"id": "aero_spectral_plasma_confinement__OUTREACH_STRATEGY", "title": "Aero Spectral Plasma Confinement — OUTREACH STRATEGY", "domain": "verification", "lean": true, "words": 1807, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/aero_spectral_plasma_confinement__OUTREACH_STRATEGY/", "pdf": "", "created": "", "updated": ""}, {"id": "aero_spectral_plasma_confinement__paper_p1_foundations", "title": "A Machine-Checked Reduced Transport Law for Stochastic-Field-Line Confinement", "domain": "verification", "lean": true, "words": 19276, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We present a machine-checked reduced transport law for ion-scale turbulent confinement in tokamak plasmas.", "url": "https://thalens.org/papers/aero_spectral_plasma_confinement__paper_p1_foundations/", "pdf": "", "created": "", "updated": ""}, {"id": "aero_spectral_plasma_confinement__paper_p2_validation_dimits", "title": "Falsification Protocol and a Quantitative Dimits-Bypass Prediction for Stochastic-Field-Line Transport", "domain": "physics", "lean": true, "words": 7594, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We test a kernel-verified reduced transport law against published tokamak confinement data.", "url": "https://thalens.org/papers/aero_spectral_plasma_confinement__paper_p2_validation_dimits/", "pdf": "", "created": "", "updated": ""}, {"id": "aero_spectral_plasma_confinement__paper_p3_methodology", "title": "Kernel-Verified Derived Physics: a Transferable Standard for Auditable Derivations with Pre-Registered Predictions", "domain": "verification", "lean": true, "words": 10539, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "Derived-physics laws — the Rechester–Rosenbluth stochastic-field-line transport law, the Debye electrostatic screening length, the Clausius–Clapeyron integrated vapour-pressure equation, the Einstein chirp mass from the stationary-phase approximation", "url": "https://thalens.org/papers/aero_spectral_plasma_confinement__paper_p3_methodology/", "pdf": "", "created": "", "updated": ""}, {"id": "aero_spectral_plasma_confinement__paper_p4_debye_screening", "title": "Kernel-Verified Debye Screening: a Second Worked Instance of the Derived-Physics Standard, with SHA-Committed Predictions Across Fourteen Orders of Magnitude in Plasma Parameters", "domain": "verification", "lean": true, "words": 5163, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "The Debye-screening length $\\lambda_D^2 = \\varepsilon_0 k_B T_e / (n_0 q_e^2)$ is a textbook identity — appearing unchanged in Chen *(Introduction to Plasma Physics §1.4)*, Krall–Trivelpiece *(§1.2.2)*, Jackson *(§1.5)*, and the *NRL Plasma Formulary", "url": "https://thalens.org/papers/aero_spectral_plasma_confinement__paper_p4_debye_screening/", "pdf": "", "created": "", "updated": ""}, {"id": "alggeom_unified", "title": "Algebraic Geometry of the Unified Field: Scheme-Theoretic Structures and Evolution Dynamics", "domain": "math", "lean": false, "words": 2044, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We develop a formal scheme-theoretic framework for a unified field $U$ equipped with a\nfundamental parameter $\\rho > 0$.", "url": "https://thalens.org/papers/alggeom_unified/", "pdf": "", "created": "", "updated": ""}, {"id": "american_basket_gym", "title": "Error Bounds for American Basket Option Pricing: A Unified Framework for COS-FW-GH Methods", "domain": "quantitative_finance", "lean": true, "words": 2077, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We develop a unified error-bound framework for American basket option pricing\nthat decomposes total numerical error into three controllable components:\ncharacteristic-function (COS) truncation error, forward-wave (FW) conditioning\nerror, and Gauss-Hermite (GH) quadrature error.", "url": "https://thalens.org/papers/american_basket_gym/", "pdf": "", "created": "", "updated": ""}, {"id": "archive__PAPER_cos_var_method", "title": "Semi-Analytic Portfolio VaR via Eigenvalue-Conditioned COS Inversion", "domain": "finance", "lean": false, "words": 2691, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We present a deterministic, grid-free method for computing Value-at-Risk of\nportfolios whose assets follow correlated geometric Brownian motion. The\nportfolio value is a weighted sum of correlated lognormals — the Fenton\nDistribution — whose CDF has no closed form but whose characteristic function\ni", "url": "https://thalens.org/papers/archive__PAPER_cos_var_method/", "pdf": "", "created": "", "updated": ""}, {"id": "archive__PAPER_mixture_compression", "title": "Paper Draft — Portfolio VaR via Eigenvalue-Conditioned Lognormal Mixtures", "domain": "finance", "lean": false, "words": 4114, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We present a method for computing Value-at-Risk of portfolios whose assets\nfollow correlated geometric Brownian motion. The portfolio value is a weighted\nsum of correlated lognormals — the Fenton Distribution — whose CDF has no\nclosed form.", "url": "https://thalens.org/papers/archive__PAPER_mixture_compression/", "pdf": "", "created": "", "updated": ""}, {"id": "archive__PAPER_spectral_fenton", "title": "The Spectral Fenton Distribution: Exact Portfolio VaR via Eigenvalue-Conditioned Fourier Inversion", "domain": "finance", "lean": false, "words": 4225, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We introduce the Spectral Fenton Distribution — a 130-parameter spectral\nrepresentation of the distribution of weighted sums of correlated lognormals\n(the Fenton Distribution).", "url": "https://thalens.org/papers/archive__PAPER_spectral_fenton/", "pdf": "", "created": "", "updated": ""}, {"id": "archive__PAPER_spectral_fenton_pricing", "title": "Pricing Basket Options via Eigenvalue-Conditioned Fourier Expansion", "domain": "finance", "lean": false, "words": 300, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "Pricing basket options (options on a weighted sum of assets) is computationally demanding due to the lack of a closed-form distribution for the underlying sum. We apply the Spectral Fenton Representation [Nagy 2026a] to derive a semi-analytic pricing formula for European basket calls and puts.", "url": "https://thalens.org/papers/archive__PAPER_spectral_fenton_pricing/", "pdf": "", "created": "", "updated": ""}, {"id": "archive__PAPER_spectral_fenton_publication", "title": "The Spectral Fenton Distribution: Exact Portfolio Risk via Eigenvalue-Conditioned Fourier Inversion", "domain": "finance", "lean": false, "words": 1405, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "The probability distribution of the sum of correlated lognormal random variables—known as the Fenton Distribution—has lacked a closed-form cumulative distribution function (CDF) since its description in 1960.", "url": "https://thalens.org/papers/archive__PAPER_spectral_fenton_publication/", "pdf": "", "created": "", "updated": ""}, {"id": "archive__PAPER_spectral_fenton_var", "title": "Eigen-COS VaR: Real-Time Exact Risk for High-Dimensional Portfolios", "domain": "finance", "lean": false, "words": 307, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "Value-at-Risk (VaR) and Expected Shortfall (ES) for non-normal portfolios typically require a trade-off: slow Monte Carlo simulations or inaccurate parametric approximations (Cornish-Fisher). We present the Eigen-COS VaR method, which computes exact risk measures for sums of correlated lognormals in", "url": "https://thalens.org/papers/archive__PAPER_spectral_fenton_var/", "pdf": "", "created": "", "updated": ""}, {"id": "archive__PAPER_spectral_form", "title": "Two Representations of the Fenton Distribution: Generative and Spectral Forms", "domain": "finance", "lean": false, "words": 2726, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "The weighted sum of correlated lognormal random variables — the Fenton\nDistribution — has been considered to lack a closed-form CDF for over sixty\nyears. We show this is a representation problem, not a mathematical one.", "url": "https://thalens.org/papers/archive__PAPER_spectral_form/", "pdf": "", "created": "", "updated": ""}, {"id": "arrow_impossibility", "title": "Arrow's Impossibility Theorem: Quantitative Extensions via Latent Framework", "domain": "core", "lean": true, "words": 1715, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We develop quantitative extensions of Arrow's impossibility theorem and the Gibbard-Satterthwaite theorem using the Latent framework. Classical social choice theory establishes that no voting rule can simultaneously satisfy unanimity, independence of irrelevant alternatives (IIA), and non-dictatorsh", "url": "https://thalens.org/papers/arrow_impossibility/", "pdf": "", "created": "", "updated": ""}, {"id": "asset_pricing_anomalies", "title": "Asset Pricing Anomalies as Latent SDF Decomposition: A Formal Derivation of Multi-Factor Models", "domain": "finance", "lean": false, "words": 1827, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "The Capital Asset Pricing Model (CAPM) predicts that a single market factor prices all assets. Empirically, this fails: value stocks (high book-to-market) earn ~5% annual premium, small-cap stocks earn ~3%, and momentum strategies earn ~8%.", "url": "https://thalens.org/papers/asset_pricing_anomalies/", "pdf": "", "created": "", "updated": ""}, {"id": "bio_constraints_of_life", "title": "Life: What, Why and How", "domain": "mathematical_biology", "lean": true, "words": 9172, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We formalize a definition of life as the intersection of six necessary constraints: (L1) Shannon information capacity for self-description, (L2) Eigen error threshold for pattern preservation, (L3) thermal stability of covalent bonds, (L4) reversible", "url": "https://thalens.org/papers/bio_constraints_of_life/", "pdf": "https://thalens.org/papers/bio_constraints_of_life/paper.pdf", "created": "", "updated": ""}, {"id": "bio_myomere_composites__STORY", "title": "The Thought Line: Fish Muscles and Composite Fibers", "domain": "math", "lean": false, "words": 539, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/bio_myomere_composites__STORY/", "pdf": "", "created": "", "updated": ""}, {"id": "bio_protein_fold", "title": "Latent Folding: Resolution of Levinthal's Paradox via Grade Decomposition", "domain": "bio_protein_fold", "lean": true, "words": 2796, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Levinthal's paradox observes that the conformational space of a protein with $N$ residues has dimension $d \\approx 2N$ (dihedral angles), yielding $\\sim 3^{2N}$ possible conformations—a brute-force search time exceeding the age of the universe for typical proteins. Yet proteins fold reliably in mill", "url": "https://thalens.org/papers/bio_protein_fold/", "pdf": "", "created": "", "updated": ""}, {"id": "bio_protein_folding_dynamics__HANDOFF", "title": "Bio Protein Folding Dynamics — HANDOFF", "domain": "core", "lean": false, "words": 2187, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/bio_protein_folding_dynamics__HANDOFF/", "pdf": "", "created": "", "updated": ""}, {"id": "bio_protein_folding_dynamics__business", "title": "LatentFold — Business Plan & Product Strategy", "domain": "core", "lean": true, "words": 5321, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/bio_protein_folding_dynamics__business/", "pdf": "", "created": "", "updated": ""}, {"id": "bounded_rationality", "title": "Bounded Rationality and the Computational Complexity of Equilibria: A Spectral Perspective", "domain": "verification", "lean": true, "words": 2692, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We establish a spectral theory of bounded rationality by connecting the computational complexity of finding ε-Nash equilibria to the Latent Number ρ of the game's payoff tensor. For games with ρ > ρ*, polynomial-time algorithms achieve approximation error ε = O(ρ^{−N}) after N queries; for games wit", "url": "https://thalens.org/papers/bounded_rationality/", "pdf": "", "created": "", "updated": ""}, {"id": "bridge_algebra", "title": "Bridge Algebra: A Formal Framework for Cross-Domain Knowledge Transfer", "domain": "verification", "lean": true, "words": 1513, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We develop a formal algebraic framework for mathematical bridges — theorems that transfer results from one domain to another. A bridge $B_{A \\to B}$ is a theorem asserting that any object satisfying predicate $P_A$ in domain $A$ implies the existence of an object satisfying predicate $Q_B$ in domain", "url": "https://thalens.org/papers/bridge_algebra/", "pdf": "", "created": "", "updated": ""}, {"id": "coherence_web", "title": "Coherence Web: A Network Model for Knowledge Validation", "domain": "theory_of_knowledge", "lean": true, "words": 1686, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We formalize a network-theoretic model for measuring how well knowledge elements \"fit together\" — the **coherence web**. Each element $c$ in the web is a node in a weighted hypergraph whose value derives not from intrinsic properties but from its position in the network.", "url": "https://thalens.org/papers/coherence_web/", "pdf": "", "created": "", "updated": ""}, {"id": "consecutive_products", "title": "Divisibility of Consecutive Products", "domain": "number_theory", "lean": true, "words": 1160, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "We prove two classical divisibility results for products of consecutive integers: that the product of any two consecutive natural numbers is even, and that the product of any three consecutive natural numbers is divisible by 6. Both results are established by mathematical induction within a formally", "url": "https://thalens.org/papers/consecutive_products/", "pdf": "", "created": "", "updated": ""}, {"id": "contagion_phase_transition", "title": "Sharp Phase Transitions in Financial Contagion: A Formal Theory of Systemic Risk Cascades", "domain": "quantitative_finance", "lean": false, "words": 2731, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We develop a formal theory of systemic risk cascades and prove that financial contagion exhibits a sharp phase transition at a critical default fraction $D_c = \\alpha/\\beta$, where $\\alpha$ is the absorption rate (capital buffers) and $\\beta$ is the contagion rate (bilateral exposure).", "url": "https://thalens.org/papers/contagion_phase_transition/", "pdf": "", "created": "", "updated": ""}, {"id": "convolution_correlation_duality", "title": "Convolution–Correlation Duality: A Universal Principle for Spectral Damping", "domain": "finance", "lean": true, "words": 2747, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a universal spectral damping principle that classifies the tractability of problems across number theory, probability, signal processing, combinatorics, PDE, and finance. The core mechanism is elementary: convolution of independent components damps oscillatory Fourier coefficients, while ", "url": "https://thalens.org/papers/convolution_correlation_duality/", "pdf": "", "created": "", "updated": ""}, {"id": "core_spectral_pattern_theory", "title": "Core Spectral Pattern Theory", "domain": "finance", "lean": false, "words": 15294, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We propose a universal, quantitative definition of \"pattern\": a pattern is a low-rank spectral component of a data tensor whose eigenvalue exceeds the Marchenko--Pastur noise floor.", "url": "https://thalens.org/papers/core_spectral_pattern_theory/", "pdf": "", "created": "", "updated": ""}, {"id": "core_spectral_phase_transition", "title": "Core Spectral Phase Transition", "domain": "finance", "lean": true, "words": 4165, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We prove that a single dimensionless parameter — the spectral decay rate ρ of a Markov generator — governs a universal phase transition in the compressibility of stochastic systems. For ρ > 1, any Lipschitz functional can be computed to accuracy ε using N = Θ(log(1/ε)/log ρ) spectral modes, independ", "url": "https://thalens.org/papers/core_spectral_phase_transition/", "pdf": "https://thalens.org/papers/core_spectral_phase_transition/paper.pdf", "created": "", "updated": ""}, {"id": "core_the_latent__RESTRUCTURE_LOG", "title": "Core The Latent — RESTRUCTURE LOG", "domain": "physics", "lean": false, "words": 977, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/core_the_latent__RESTRUCTURE_LOG/", "pdf": "", "created": "", "updated": ""}, {"id": "core_theory_universal_spectral_representation__paper_v4_nature", "title": "The Universal Representation Theorem: From Portfolio Risk to the Three-Body Problem", "domain": "finance", "lean": true, "words": 2912, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "How many numbers does it take to describe a smooth system? We prove the answer is $N = \\Theta(\\log(1/\\varepsilon)/\\log\\rho)$ — regardless of the system's dimension.", "url": "https://thalens.org/papers/core_theory_universal_spectral_representation__paper_v4_nature/", "pdf": "", "created": "", "updated": ""}, {"id": "core_universal_smoothing__STORY", "title": "The Thought Line: Universal Smoothing and the Latent Bridge", "domain": "core", "lean": false, "words": 979, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/core_universal_smoothing__STORY/", "pdf": "", "created": "", "updated": ""}, {"id": "core_universal_smoothing", "title": "The Universal Smoothing Bridge", "domain": "verification", "lean": true, "words": 4096, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We construct a formal bridge between the space of distributions $\\mathcal{D}'$ and the Latent framework by proving that mollification provides a universal smoothing adapter.", "url": "https://thalens.org/papers/core_universal_smoothing/", "pdf": "https://thalens.org/papers/core_universal_smoothing/paper.pdf", "created": "", "updated": ""}, {"id": "csp_latent_bridge", "title": "CSP–Latent Bridge: Random k-SAT Phase Transitions and Characteristic Function Analyticity", "domain": "core", "lean": true, "words": 1896, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We establish a formal bridge between classical constraint satisfaction problem (CSP) complexity theory and Latent complexity theory by proving that the Latent Number ρ* of a random k-SAT energy landscape equals the partition function analyticity radius ρ_Z.", "url": "https://thalens.org/papers/csp_latent_bridge/", "pdf": "", "created": "", "updated": ""}, {"id": "dark_matter", "title": "Dark Matter: Formal Verification of Observational Evidence and Theoretical Constraints", "domain": "dark_matter", "lean": true, "words": 1312, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "We present a formally verified treatment of dark matter physics, covering both\nobservational evidence and theoretical constraints. The Platonic proof system\nverifies 15 theorems spanning rotation curve anomalies, cluster dynamics,\ncosmological parameters, particle physics bounds, and alternative gra", "url": "https://thalens.org/papers/dark_matter/", "pdf": "", "created": "", "updated": ""}, {"id": "divisibility_and_parity", "title": "Divisibility and Parity in Natural and Integer Arithmetic", "domain": "verification", "lean": true, "words": 1973, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a collection of formally verified theorems concerning divisibility and parity in natural and integer arithmetic.", "url": "https://thalens.org/papers/divisibility_and_parity/", "pdf": "", "created": "", "updated": ""}, {"id": "eng_analytical_design__billion_dollar_ideas", "title": "Billion-Dollar Ideas from Analytical Physics Design", "domain": "finance", "lean": false, "words": 1887, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/eng_analytical_design__billion_dollar_ideas/", "pdf": "", "created": "", "updated": ""}, {"id": "eng_analytical_design", "title": "Analytical Design Optimization: From Governing Equations to 3D-Printable Geometry", "domain": "research", "lean": false, "words": 6220, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/eng_analytical_design/", "pdf": "https://thalens.org/papers/eng_analytical_design/paper.pdf", "created": "", "updated": ""}, {"id": "epi_sir_grade2", "title": "Formally Verified Epidemic Thresholds: The SIR Model as a Grade-2 Dynamical System", "domain": "mathematical_epidemiology", "lean": true, "words": 3359, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "The classical SIR (Susceptible-Infected-Recovered) epidemic model of Kermack and McKendrick (1927) is the foundation of mathematical epidemiology.", "url": "https://thalens.org/papers/epi_sir_grade2/", "pdf": "https://thalens.org/papers/epi_sir_grade2/paper.pdf", "created": "", "updated": ""}, {"id": "experience_field", "title": "Experience Fields: Path-Dependent Algebraic Structures for Learning Systems", "domain": "verification", "lean": true, "words": 2092, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We introduce the **Experience Field** $(\\mathcal{X}, +, \\cdot)$, an algebraic structure where elements carry computational history in the form of $(v, a, \\lambda)$ triples — value, age, and learning rate.", "url": "https://thalens.org/papers/experience_field/", "pdf": "", "created": "", "updated": ""}, {"id": "fin_american_basket", "title": "Fin American Basket", "domain": "finance", "lean": true, "words": 7137, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We prove the first quantitative convergence guarantees for a deterministic American basket pricer with dimension-free cost.", "url": "https://thalens.org/papers/fin_american_basket/", "pdf": "https://thalens.org/papers/fin_american_basket/paper.pdf", "created": "", "updated": ""}, {"id": "fin_arcsinh_bs", "title": "Fin Arcsinh Bs", "domain": "finance", "lean": true, "words": 8413, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We derive a closed-form European option pricing formula valid for all spot prices \\( S_0 \\in \\mathbb{R} \\), including negative values. The formula replaces the logarithmic transform of Black-Scholes with the inverse hyperbolic sine, yielding a three-term call price \\( C = S_+ \\Phi(d_+) - S_- \\Phi(d_", "url": "https://thalens.org/papers/fin_arcsinh_bs/", "pdf": "https://thalens.org/papers/fin_arcsinh_bs/paper.pdf", "created": "", "updated": ""}, {"id": "fin_bayesian_live_risk__submission", "title": "Bayesian Live Risk", "domain": "finance", "lean": true, "words": 2782, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We introduce Bayesian Live Risk (BLR), a framework for live risk management in which risk measurement is formulated as a posterior state rather than a single tail statistic.", "url": "https://thalens.org/papers/fin_bayesian_live_risk__submission/", "pdf": "", "created": "", "updated": ""}, {"id": "fin_contagion_grade2", "title": "Formally Verified Financial Contagion Thresholds: Counterparty Default Cascades as a Grade-2 Dynamical System", "domain": "mathematical_finance", "lean": true, "words": 2215, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Financial contagion — the cascading failure of interconnected institutions through bilateral exposures — is the central mechanism behind systemic crises from Lehman Brothers (2008) to Silicon Valley Bank (2023).", "url": "https://thalens.org/papers/fin_contagion_grade2/", "pdf": "https://thalens.org/papers/fin_contagion_grade2/paper.pdf", "created": "", "updated": ""}, {"id": "fin_expected_shortfall", "title": "The Risk Coding Theorem: Exponential Convergence of Spectral Expected Shortfall", "domain": "mathematical_finance", "lean": true, "words": 2555, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Expected Shortfall (ES) replaced Value-at-Risk as the primary risk measure in Basel III's Fundamental Review of the Trading Book, yet its estimation by Monte Carlo simulation converges at rate $O(1/\\sqrt{M})$ — requiring $10^6$ paths for three-digit accuracy.", "url": "https://thalens.org/papers/fin_expected_shortfall/", "pdf": "https://thalens.org/papers/fin_expected_shortfall/paper.pdf", "created": "", "updated": ""}, {"id": "fin_fenton_solved__paper_journal", "title": "The Fenton Distribution Solved", "domain": "finance", "lean": false, "words": 9665, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We consider $S = \\sum_{i=1}^n w_i e^{Y_i}$ with $Y \\sim N(\\mu,\\Sigma)$, the central object in the “Fenton distribution” problem.", "url": "https://thalens.org/papers/fin_fenton_solved__paper_journal/", "pdf": "", "created": "", "updated": ""}, {"id": "fin_fenton_spectral", "title": "Fin Fenton Spectral", "domain": "finance", "lean": true, "words": 10471, "doi": "10.5281/zenodo.18940756", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "The CDF of a weighted sum of correlated lognormal random variables has lacked a tractable characterization since Fenton (1960). We show that eigenvalue conditioning of the correlation matrix, followed by Fourier-cosine inversion, yields an analytic, grid-free $N$-term spectral representation of that", "url": "https://thalens.org/papers/fin_fenton_spectral/", "pdf": "", "created": "", "updated": ""}, {"id": "fin_generative_portfolio", "title": "Fin Generative Portfolio", "domain": "finance", "lean": false, "words": 5628, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Classical portfolio optimization minimizes a single risk metric — variance, VaR, or CVaR — subject to return constraints. This produces portfolios that are optimal in one dimension but unconstrained in all others: two portfolios with identical VaR can have radically different skewness, kurtosis, or ", "url": "https://thalens.org/papers/fin_generative_portfolio/", "pdf": "https://thalens.org/papers/fin_generative_portfolio/paper.pdf", "created": "", "updated": ""}, {"id": "fin_harvestability__submission", "title": "Harvestability", "domain": "verification", "lean": true, "words": 2444, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We study a narrow horizon-dependent allocation question in a CRRA investor setting with Ornstein-Uhlenbeck eigenmodes. The paper centers the fin_harvestability function\n\\[\nh(T,\\tau)=1-e^{-T/\\tau},\n\\]\ninterpreted as the closed-form horizon-dependent correction term around the myopic Merton benchmark.", "url": "https://thalens.org/papers/fin_harvestability__submission/", "pdf": "", "created": "", "updated": ""}, {"id": "fin_harvestability_calibration__submission", "title": "Calibrating Harvestability", "domain": "finance", "lean": false, "words": 2369, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Starting from the canonical fin_harvestability object \\(h(T,\\tau)=1-e^{-T/\\tau}\\), this reviewer-facing submission studies the calibration problem: when does expected return dominate volatility strongly enough that the premium can be treated as genui", "url": "https://thalens.org/papers/fin_harvestability_calibration__submission/", "pdf": "", "created": "", "updated": ""}, {"id": "fin_markowitz", "title": "Fin Markowitz", "domain": "finance", "lean": true, "words": 6507, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present two results. **Part I** is the first machine-checked formalization of Markowitz (1952) mean-variance optimization in Lean 4, covering the Merton (1972) closed-form solution, two-fund separation, the efficient frontier, and the Capital Market Line.", "url": "https://thalens.org/papers/fin_markowitz/", "pdf": "https://thalens.org/papers/fin_markowitz/paper.pdf", "created": "", "updated": ""}, {"id": "fin_noise_free_backtest", "title": "The Bias-Variance Frontier of Risk Estimation: When Spectral Methods Dominate Monte Carlo", "domain": "finance_risk", "lean": true, "words": 4147, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We study the bias-variance trade-off in Expected Shortfall estimation, comparing spectral (order-statistic) methods with parametric and bootstrap Monte Carlo. We prove that spectral ES is the minimum-variance unbiased estimator (MVUE) in the class of model-free risk estimators and characterize the P", "url": "https://thalens.org/papers/fin_noise_free_backtest/", "pdf": "https://thalens.org/papers/fin_noise_free_backtest/paper.pdf", "created": "", "updated": ""}, {"id": "fin_noise_free_backtest__paper_fundamental_limits", "title": "Fundamental Limits and Phase Transitions in Coherent Risk Estimation", "domain": "finance_risk", "lean": true, "words": 4912, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "What are the fundamental limits of estimating coherent risk measures from finite data? We establish four structural results.", "url": "https://thalens.org/papers/fin_noise_free_backtest__paper_fundamental_limits/", "pdf": "", "created": "", "updated": ""}, {"id": "fin_pricing_is_allocation__submission", "title": "Why Rational Investors Hold Different Portfolios", "domain": "finance", "lean": false, "words": 4609, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We study a persistent tension in financial economics. Markowitz gives an optimizer, equilibrium theory points to a common market benchmark, and lifecycle finance implies investor-specific holdings that vary with horizon, mortality, bequest motive, learning, and constraints.", "url": "https://thalens.org/papers/fin_pricing_is_allocation__submission/", "pdf": "", "created": "", "updated": ""}, {"id": "fin_pricing_is_allocation__submission_contract", "title": "Pricing Is Allocation Submission Contract", "domain": "finance", "lean": false, "words": 912, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/fin_pricing_is_allocation__submission_contract/", "pdf": "", "created": "", "updated": ""}, {"id": "fin_progressive_spectral", "title": "Fin Progressive Spectral", "domain": "finance", "lean": true, "words": 2765, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We formalize the *Progressive Spectral Decomposition* (PSD): computing eigenvalues of a symmetric matrix one at a time via deflation, with a combination of properties no existing method provides simultaneously.", "url": "https://thalens.org/papers/fin_progressive_spectral/", "pdf": "https://thalens.org/papers/fin_progressive_spectral/paper.pdf", "created": "", "updated": ""}, {"id": "fin_return_paradox", "title": "Fin Return Paradox", "domain": "finance", "lean": true, "words": 8108, "doi": "10.5281/zenodo.18927850", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "Every formula in quantitative finance — CAPM, Markowitz, VaR, Sharpe ratio, GARCH — takes returns as input. Yet the standard definitions of return fail when prices cross zero: log-returns are undefined, and simple returns produce sign errors.", "url": "https://thalens.org/papers/fin_return_paradox/", "pdf": "https://thalens.org/papers/fin_return_paradox/paper.pdf", "created": "", "updated": ""}, {"id": "fin_spectral_compound_exchange_options", "title": "Spectral Valuation of Compound Exchange Options under Stochastic Volatility", "domain": "finance", "lean": false, "words": 363, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/fin_spectral_compound_exchange_options/", "pdf": "https://thalens.org/papers/fin_spectral_compound_exchange_options/paper.pdf", "created": "2026-04-03", "updated": ""}, {"id": "fin_spectral_importance_sampling__short_note", "title": "Spectral Importance Sampling for Rare Events in Correlated Systems", "domain": "finance", "lean": false, "words": 1065, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "reviewed", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/fin_spectral_importance_sampling__short_note/", "pdf": "", "created": "", "updated": ""}, {"id": "fin_vol_surface__submission", "title": "The Spectral Volatility Surface", "domain": "finance", "lean": true, "words": 3346, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We introduce a low-rank implied-variance surface in which total variance is represented as\n\\[\nw(k, T) = c(T) + \\sum_{j=1}^{r} u_j(T)\\cos(\\omega_j k),\n\\]\nwith structural constraints enforced directly on the coefficient vectors. The resulting object is not a new stochastic-volatility law.", "url": "https://thalens.org/papers/fin_vol_surface__submission/", "pdf": "", "created": "", "updated": ""}, {"id": "finrep_unified", "title": "Representation Theory of Finite Groups on the Unified Field", "domain": "pure_math", "lean": true, "words": 1952, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a formally verified framework for the representation theory of finite groups on the unified field U.", "url": "https://thalens.org/papers/finrep_unified/", "pdf": "", "created": "", "updated": ""}, {"id": "foundation_model", "title": "Foundation Model Training Bounds: A Formally Verified Framework for Generalization and Scaling", "domain": "ml_foundation_model", "lean": true, "words": 1500, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a formally verified framework for analyzing foundation model training dynamics through the lens of generalization bounds and scaling laws. The framework establishes fundamental relationships between training loss, validation loss, generalization gap, and model size, providing rigorous gua", "url": "https://thalens.org/papers/foundation_model/", "pdf": "", "created": "", "updated": ""}, {"id": "godel_universe__mirtill_0418", "title": "Mirtill Adversarial Review — Gödel Universe (algebraic core)", "domain": "verification", "lean": false, "words": 1110, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "reviewed", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/godel_universe__mirtill_0418/", "pdf": "", "created": "", "updated": ""}, {"id": "godel_universe", "title": "A Machine-Verified Algebraic Formalization of Gödel's 1949 Counterexample", "domain": "math_physics", "lean": false, "words": 6355, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We give a machine-verified algebraic formalization of Gödel's (1949)\nconstruction in the Platonic proof kernel.", "url": "https://thalens.org/papers/godel_universe/", "pdf": "https://thalens.org/papers/godel_universe/paper.pdf", "created": "", "updated": ""}, {"id": "gravity_mond", "title": "Gravity Theory Divergence: Newton vs MOND", "domain": "physics", "lean": false, "words": 3822, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We present a complete formal proof suite establishing the mathematical foundations of Modified Newtonian Dynamics (MOND) and its divergence from Newtonian gravity.", "url": "https://thalens.org/papers/gravity_mond/", "pdf": "", "created": "", "updated": ""}, {"id": "gt_interaction_decay", "title": "Interaction Decay in ρ-Analytic Games: Grade Structure and Truncation Bounds", "domain": "verification", "lean": true, "words": 2000, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We establish rigorous bounds on the decay of interaction terms in ρ-analytic N-player games. For games whose payoff functions admit a convergent power series representation with analyticity radius ρ > 1, we prove that the grade-r interaction component decays as ρ^{-r}.", "url": "https://thalens.org/papers/gt_interaction_decay/", "pdf": "", "created": "", "updated": ""}, {"id": "gt_mechanism", "title": "Revenue Approximation in Mechanism Design via Grade Truncation", "domain": "verification", "lean": true, "words": 1679, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We establish formally verified foundations for revenue approximation in mechanism design through grade truncation. The core framework shows that mechanism revenue decomposes additively across grade levels, with each successive grade contributing geometrically decaying revenue increments bounded by C", "url": "https://thalens.org/papers/gt_mechanism/", "pdf": "", "created": "", "updated": ""}, {"id": "gt_nash_equilibrium", "title": "Nash Equilibrium via Latent Perturbation Theory: Existence, Uniqueness, and Mean-Field Convergence", "domain": "game_theory", "lean": true, "words": 1970, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/gt_nash_equilibrium/", "pdf": "", "created": "", "updated": ""}, {"id": "gt_protein_game", "title": "Protein Folding as a Game: Nash-Boltzmann Duality and Computational Complexity", "domain": "verification", "lean": true, "words": 2449, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We establish a formal correspondence between protein folding and finite-strategy game theory, demonstrating that the native protein state is a Nash equilibrium in an interaction game among amino acid residues.", "url": "https://thalens.org/papers/gt_protein_game/", "pdf": "", "created": "", "updated": ""}, {"id": "heterogeneous_agent_ge", "title": "Heterogeneous Agent General Equilibrium: A Latent Framework Analysis of the Aiyagari-Bewley-Krusell-Smith Models", "domain": "macro_finance", "lean": true, "words": 2205, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a formal framework connecting heterogeneous agent general equilibrium models to the Latent representation theory.", "url": "https://thalens.org/papers/heterogeneous_agent_ge/", "pdf": "", "created": "", "updated": ""}, {"id": "hodge_conjecture", "title": "The Hodge Conjecture: A Formal Framework with Verified Cases and Structural Reductions", "domain": "math", "lean": false, "words": 3481, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/hodge_conjecture/", "pdf": "", "created": "", "updated": ""}, {"id": "imported_infrastructure", "title": "Imported Infrastructure — Strip and Bound Arithmetic for Analytic Function Theory", "domain": "verification", "lean": true, "words": 939, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "This paper presents a collection of verified arithmetic theorems that provide foundational bookkeeping for analytic function theory within vertical strips of the complex plane. The theorems establish elementary but essential properties concerning strip endpoints, width positivity, bound composition,", "url": "https://thalens.org/papers/imported_infrastructure/", "pdf": "", "created": "", "updated": ""}, {"id": "jet_numbers", "title": "Jet Numbers: A Verified Algebra of Derivatives", "domain": "verification", "lean": true, "words": 1898, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a formally verified treatment of jet numbers — algebraic objects that\ncarry a value together with its derivatives to a fixed order. A 2-jet encodes\n$(f, f', f'')$ in a single element, with arithmetic operations following the\nLeibniz rule: $(fg)' = f'g + fg'$ and the generalized second-ord", "url": "https://thalens.org/papers/jet_numbers/", "pdf": "", "created": "", "updated": ""}, {"id": "kingman_coalescent", "title": "Coalescent Structure and the Universal Latent Ratio", "domain": "core", "lean": true, "words": 1870, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a formally verified treatment of Kingman's coalescent, establishing\nthe fundamental rate and waiting time identities that govern backward-in-time\ngenealogical processes.", "url": "https://thalens.org/papers/kingman_coalescent/", "pdf": "", "created": "", "updated": ""}, {"id": "knowability", "title": "Knowability Theory: Latent Generator Models and Spectral Pricing in Finance", "domain": "finance", "lean": true, "words": 1818, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a formally verified mathematical theory of *knowability* in\nfinancial modeling — the framework connecting unobservable latent generators to\nobservable price dynamics.", "url": "https://thalens.org/papers/knowability/", "pdf": "", "created": "", "updated": ""}, {"id": "latent_mollification_bridge", "title": "Latent Mollification Bridge", "domain": "core", "lean": true, "words": 1604, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We establish that Latent grade truncation—the foundational operation in the Latent framework—is mathematically equivalent to mollification in the spectral domain. Truncating a series at grade $N$ corresponds to convolving with a mollifier whose bandwidth is controlled by the scale parameter $\\vareps", "url": "https://thalens.org/papers/latent_mollification_bridge/", "pdf": "", "created": "", "updated": ""}, {"id": "lonely_runner_v2", "title": "Fourier-Analytic Formalization of the Lonely Runner Conjecture", "domain": "verification", "lean": true, "words": 2293, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a formal verification of the Lonely Runner Conjecture for small runner counts, together with a measure-theoretic framework for analyzing the general case.", "url": "https://thalens.org/papers/lonely_runner_v2/", "pdf": "", "created": "", "updated": ""}, {"id": "lyapunov_reconstruction", "title": "Lyapunov Stability of the Figure-Eight Three-Body Orbit via Z₃ Symmetry Decomposition", "domain": "physics", "lean": true, "words": 2117, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We prove Lyapunov stability of the figure-eight choreographic solution to the planar three-body problem using a symmetry decomposition approach. The Z₃ cyclic symmetry of the orbit decomposes the 3 degrees of freedom into a 2-dimensional internal sector and a 1-dimensional normal sector.", "url": "https://thalens.org/papers/lyapunov_reconstruction/", "pdf": "", "created": "", "updated": ""}, {"id": "math_3d_percolation_exponents__effective_alpha_notes", "title": "Effective-α framework: LR k-point structure at SR exponents", "domain": "finance", "lean": false, "words": 1097, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/math_3d_percolation_exponents__effective_alpha_notes/", "pdf": "", "created": "", "updated": ""}, {"id": "math_3d_percolation_exponents__lr_pivotal_crossover_notes", "title": "Algebraic exploration of candidate pivotal-scaling formulas — kernel results", "domain": "research", "lean": false, "words": 797, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/math_3d_percolation_exponents__lr_pivotal_crossover_notes/", "pdf": "", "created": "", "updated": ""}, {"id": "math_3d_percolation_exponents__newton_task_lr_pivotal_crossover", "title": "Newton task — Algebraic exploration of candidate pivotal-scaling formulas in the LR-to-SR crossover of percolation", "domain": "verification", "lean": false, "words": 1966, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/math_3d_percolation_exponents__newton_task_lr_pivotal_crossover/", "pdf": "", "created": "", "updated": ""}, {"id": "math_3d_percolation_exponents__paper_conditional_cascade", "title": "The eta-negativity cascade: what a single sign constraint implies for three-dimensional percolation exponents", "domain": "verification", "lean": false, "words": 3108, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "For Bernoulli bond percolation on $\\mathbb{Z}^3$, the\nanomalous dimension $\\eta$ is widely believed to be negative ($\\eta \\approx\n-0.046$ from Monte Carlo), but no rigorous proof exists.", "url": "https://thalens.org/papers/math_3d_percolation_exponents__paper_conditional_cascade/", "pdf": "", "created": "", "updated": ""}, {"id": "math_additive_correlative_duality", "title": "The Convolution–Correlation Duality", "domain": "mathematics", "lean": true, "words": 15319, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We identify a structural dichotomy — the *convolution–correlation duality* — that governs whether problems involving oscillatory spectral expansions are tractable. The principle is this: when a quantity is formed by convolving independent components, each independent integration contributes a dampin", "url": "https://thalens.org/papers/math_additive_correlative_duality/", "pdf": "https://thalens.org/papers/math_additive_correlative_duality/paper.pdf", "created": "", "updated": ""}, {"id": "math_decorrelation_index", "title": "The Decorrelation Index", "domain": "mathematics", "lean": true, "words": 7270, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We introduce the *decorrelation index* $\\delta(P) \\in [0,1]$ of a spectral-type mathematical problem $P$, defined as the supremum of the total decorrelation gain over all proof circuits in the proof category $\\mathbf{Spec}$ (as formalized in the comp", "url": "https://thalens.org/papers/math_decorrelation_index/", "pdf": "https://thalens.org/papers/math_decorrelation_index/paper.pdf", "created": "", "updated": ""}, {"id": "math_percolation_theory", "title": "Hierarchical Feedback Percolation: When Structure Amplifies Dynamics", "domain": "verification", "lean": true, "words": 11567, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/math_percolation_theory/", "pdf": "", "created": "", "updated": ""}, {"id": "memory_tree", "title": "Memory-Tree Algebra: Numbers That Remember Their Computation", "domain": "memory_tree", "lean": false, "words": 2235, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We introduce the Memory-Tree Algebra $\\mathcal{M}$, a structure in which each element is a rooted tree encoding its complete computational history.", "url": "https://thalens.org/papers/memory_tree/", "pdf": "", "created": "", "updated": ""}, {"id": "meta_algorithmizability", "title": "Meta-Algorithmizability: A Formal Theory of Proof Automation", "domain": "verification", "lean": false, "words": 1716, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We present a formal theory quantifying the extent to which mathematical proof\nis algorithmically tractable. Using empirical data from 26,583 proof traces\nacross 226 mathematical domains, we establish eight core theorems (T1-T8) that\ncharacterize the structure of proof space.", "url": "https://thalens.org/papers/meta_algorithmizability/", "pdf": "", "created": "", "updated": ""}, {"id": "meta_latent_computation", "title": "Meta Latent Computation", "domain": "ml", "lean": false, "words": 4917, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We introduce a spectral decomposition framework for computational workloads and prove that the von Neumann architecture — where all hardware resources serve a single universal execution mode — is strictly suboptimal for any non-degenerate workload.", "url": "https://thalens.org/papers/meta_latent_computation/", "pdf": "", "created": "", "updated": ""}, {"id": "meta_pvsnp_platonic__ASSUMPTION_LEDGER", "title": "Meta Pvsnp Platonic — ASSUMPTION LEDGER", "domain": "verification", "lean": true, "words": 3715, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/meta_pvsnp_platonic__ASSUMPTION_LEDGER/", "pdf": "", "created": "", "updated": ""}, {"id": "meta_pvsnp_platonic__ROADMAP_scientific_strengthening", "title": "Meta Pvsnp Platonic — ROADMAP scientific strengthening", "domain": "verification", "lean": false, "words": 1876, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/meta_pvsnp_platonic__ROADMAP_scientific_strengthening/", "pdf": "", "created": "", "updated": ""}, {"id": "meta_pvsnp_platonic__TRANSITION_CHECKLIST", "title": "Meta Pvsnp Platonic — TRANSITION CHECKLIST", "domain": "verification", "lean": true, "words": 1303, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/meta_pvsnp_platonic__TRANSITION_CHECKLIST/", "pdf": "", "created": "", "updated": ""}, {"id": "meta_pvsnp_platonic__paper_archive_20260407", "title": "A Machine-Verified Proof of P ≠ NP via Partition Function Analyticity", "domain": "verification", "lean": true, "words": 3575, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We formalize a proof of $\\mathrm{P} \\neq \\mathrm{NP}$ in the proof language, a Python-native formal system backed by a bidirectional type checker with Lean 4 export.", "url": "https://thalens.org/papers/meta_pvsnp_platonic__paper_archive_20260407/", "pdf": "", "created": "", "updated": ""}, {"id": "meta_rmt_grade_shadow__paper_millennium", "title": "Grade-2 Dominance Across Millennium Problems: A Unified Structural Framework for RH and Navier–Stokes", "domain": "math", "lean": true, "words": 2990, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Two of the seven Clay Millennium Problems — the Riemann Hypothesis and Navier–Stokes global regularity — appear to have nothing in common. One concerns the distribution of prime numbers; the other concerns the smoothness of fluid flow.", "url": "https://thalens.org/papers/meta_rmt_grade_shadow__paper_millennium/", "pdf": "", "created": "", "updated": ""}, {"id": "meta_tool_cascade_theorem__short_note", "title": "The Fishing Rod Proverb Is Half-Right", "domain": "finance", "lean": false, "words": 2351, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/meta_tool_cascade_theorem__short_note/", "pdf": "", "created": "", "updated": ""}, {"id": "millennium_applications", "title": "Unified Latent Formulations of Millennium Problems", "domain": "core", "lean": false, "words": 1854, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We present a unified framework for four Clay Mathematics Institute Millennium Problems—P vs NP, the Riemann Hypothesis, the Yang-Mills mass gap, and a formalization of consciousness emergence—using the Latent representation theory.", "url": "https://thalens.org/papers/millennium_applications/", "pdf": "", "created": "", "updated": ""}, {"id": "mixed__EXTRACTED_MATERIAL", "title": "Central Extracted Material Bank", "domain": "research", "lean": false, "words": 229, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/mixed__EXTRACTED_MATERIAL/", "pdf": "", "created": "", "updated": ""}, {"id": "ml_knowledge_algebra_applied__submission_algebra", "title": "Applied Knowledge Algebra: Operations, Semantics, and Use Cases", "domain": "ml", "lean": false, "words": 1766, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "Knowledge Artifacts (Nagy 2026a) support algebraic operations in a shared spectral eigenbasis.", "url": "https://thalens.org/papers/ml_knowledge_algebra_applied__submission_algebra/", "pdf": "", "created": "", "updated": ""}, {"id": "ml_knowledge_algebra_applied__submission_demonstration", "title": "A Numerical Demonstration of Knowledge Artifact Extraction and Algebra Across 26 Model Types", "domain": "finance", "lean": false, "words": 1988, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "The Knowledge Artifact framework (Nagy 2026a) claims that any model with `.predict()` can be projected into a portable spectral representation that reproduces 87–99% of its predictions. This paper provides the numerical evidence.", "url": "https://thalens.org/papers/ml_knowledge_algebra_applied__submission_demonstration/", "pdf": "", "created": "", "updated": ""}, {"id": "ml_knowledge_algebra_applied__submission_shell", "title": "SpectralShell: A Universal Model Class for Extracted Knowledge", "domain": "ml", "lean": false, "words": 1723, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We describe SpectralShell, a universal model class that loads, predicts from, and performs algebra on Knowledge Artifacts — portable spectral representations of trained ML models.", "url": "https://thalens.org/papers/ml_knowledge_algebra_applied__submission_shell/", "pdf": "", "created": "", "updated": ""}, {"id": "ml_knowledge_artifacts_algebra", "title": "Ml Knowledge Artifacts Algebra", "domain": "ml", "lean": false, "words": 6457, "doi": "10.5281/zenodo.18910387", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "A 200-tree Random Forest has 126,074 parameters. Its knowledge? Three numbers and a basis.\n\nWe introduce the **Knowledge Artifact** — a portable representation of what any ML model has learned — and the **Knowledge Algebra** — provably exact arithmetic on these artifacts.", "url": "https://thalens.org/papers/ml_knowledge_artifacts_algebra/", "pdf": "", "created": "", "updated": ""}, {"id": "ml_sgd_convergence", "title": "SGD Is Right: A Machine-Checked Proof That Stochastic Gradient Descent Converges", "domain": "verification", "lean": true, "words": 6381, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "Stochastic gradient descent (SGD) is the algorithm that trains every neural network. From GPT-4 to AlphaFold, from DALL-E to Gemini, every parameter update is a variant of the rule proposed by Robbins and Monro in 1951: take a step in the direction of a noisy gradient, shrink the step size, repeat.", "url": "https://thalens.org/papers/ml_sgd_convergence/", "pdf": "https://thalens.org/papers/ml_sgd_convergence/paper.pdf", "created": "", "updated": ""}, {"id": "ml_spectral_intelligence", "title": "Ml Spectral Intelligence", "domain": "ml", "lean": true, "words": 4393, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We derive neural scaling laws, transformer convergence rates, and self-improvement limits from a single principle: the eigenvalue decay of the data covariance matrix. For data with spectral exponent $s$ (eigenvalues $\\lambda_k \\sim k^{-s}$), we prove:\n\n1.", "url": "https://thalens.org/papers/ml_spectral_intelligence/", "pdf": "https://thalens.org/papers/ml_spectral_intelligence/paper.pdf", "created": "", "updated": ""}, {"id": "natures_latent", "title": "Nature's Latent: The Dimensionality of Optimal Biological Forms", "domain": "core", "lean": true, "words": 2207, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Natural structures—from honeycomb tessellations to nautilus spirals—exhibit striking geometric regularity despite arising from evolution rather than engineering.", "url": "https://thalens.org/papers/natures_latent/", "pdf": "", "created": "", "updated": ""}, {"id": "navier_stokes_spec", "title": "Navier-Stokes Regularity as a Spectral Problem", "domain": "verification", "lean": true, "words": 2181, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a spectral-duality analysis of Navier-Stokes regularity. The viscous term −2νH_σ provides decorrelation at rate proportional to viscosity: δ_visc = ν > 0.", "url": "https://thalens.org/papers/navier_stokes_spec/", "pdf": "", "created": "", "updated": ""}, {"id": "neo_deflection", "title": "Grade-2 Universality in Planetary Defense: Cross-Domain Bridges for NEO Deflection and Debris Cascade", "domain": "planetary_defense", "lean": true, "words": 2404, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We establish cross-domain mathematical bridges connecting near-Earth object (NEO) deflection, orbital debris dynamics, and fluid mechanics through a formally verified framework.", "url": "https://thalens.org/papers/neo_deflection/", "pdf": "", "created": "", "updated": ""}, {"id": "network_latent", "title": "Network Latent Structure", "domain": "core", "lean": true, "words": 2242, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We establish a formal connection between the power-law exponent γ of scale-free networks and the analyticity radius ρ of their latent representations via the relation ρ = (γ−1)/2.", "url": "https://thalens.org/papers/network_latent/", "pdf": "", "created": "", "updated": ""}, {"id": "neutrino_mass", "title": "Neutrino Mass: Formal Verification of Oscillation, Seesaw, and Cosmological Constraints", "domain": "phy", "lean": true, "words": 2230, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a formally verified treatment of neutrino mass physics, establishing 15 theorems covering oscillation phenomenology, mass splittings, cosmological constraints, seesaw mechanisms, and leptogenesis compatibility.", "url": "https://thalens.org/papers/neutrino_mass/", "pdf": "", "created": "", "updated": ""}, {"id": "nscos_hybrid", "title": "Verified Nelson-Siegel COS Hybrid Pricing Framework", "domain": "quantitative_finance", "lean": true, "words": 1533, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a formally verified framework for pricing interest rate derivatives using a hybrid approach that combines Nelson-Siegel yield curve dynamics with COS method characteristic function pricing.", "url": "https://thalens.org/papers/nscos_hybrid/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_bsd__companion_cm", "title": "Unconditional Birch-Swinnerton-Dyer for CM Elliptic Curves at Rank 0: A Formally Verified Proof", "domain": "number_theory", "lean": true, "words": 3148, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "For every elliptic curve $E / \\mathbb{Q}$ with complex multiplication by the ring of integers of an imaginary quadratic field, if $L(E, 1) > 0$ then the full Birch-Swinnerton-Dyer leading-term formula\n\n$$L_{\\mathrm{lead}}(E) \\;=\\; \\frac{\\Omega_E \\cdo", "url": "https://thalens.org/papers/nt_bsd__companion_cm/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_bsd__latent_spectral_sketch", "title": "A Spectral Reformulation of Birch-Swinnerton-Dyer: Formalized Prototype and Open Directions", "domain": "number_theory", "lean": true, "words": 1920, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We record a spectral reformulation of the Birch-Swinnerton-Dyer conjecture and its formalization in the Lean 4 kernel.", "url": "https://thalens.org/papers/nt_bsd__latent_spectral_sketch/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_bsd__proof_complexity_note", "title": "Empirical Proof Complexity of a Lean 4 Formalization of Birch-Swinnerton-Dyer", "domain": "proof_complexity", "lean": true, "words": 1688, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We record empirical complexity measurements for the current Lean 4 formalization of the Birch-Swinnerton-Dyer conjecture and its surrounding theory.", "url": "https://thalens.org/papers/nt_bsd__proof_complexity_note/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_collatz_carry_dynamics", "title": "Carry Dynamics of the Syracuse Map: Exact Identities, Rank-One Transitions, and Mersenne Phase Transitions", "domain": "research", "lean": false, "words": 4466, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We develop the carry arithmetic of the Syracuse map $S(x) = (3x+1)/2^{v_2(3x+1)}$ from first principles, establishing exact identities that govern popcount evolution under iteration.", "url": "https://thalens.org/papers/nt_collatz_carry_dynamics/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_collatz_cycle_elimination__SUBMISSION_ARTIFACTS", "title": "Nt Collatz Cycle Elimination — SUBMISSION ARTIFACTS", "domain": "verification", "lean": false, "words": 1256, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/nt_collatz_cycle_elimination__SUBMISSION_ARTIFACTS/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_collatz_lyapunov__dual_space_framework", "title": "Nt Collatz Lyapunov — dual space framework", "domain": "math", "lean": false, "words": 1579, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/nt_collatz_lyapunov__dual_space_framework/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_collatz_lyapunov", "title": "Syracuse-Compatible Lyapunov Functions: Characterization, β-Halving, and Spectral Analysis", "domain": "verification", "lean": true, "words": 6047, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/nt_collatz_lyapunov/", "pdf": "https://thalens.org/papers/nt_collatz_lyapunov/paper.pdf", "created": "", "updated": ""}, {"id": "nt_damped_vinogradov__STORY", "title": "The Thought Line: Mesoscopic Damping for Vinogradov Sums", "domain": "research", "lean": false, "words": 645, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/nt_damped_vinogradov__STORY/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_damped_vinogradov__email_biro_andras", "title": "Nt Damped Vinogradov — email biro andras", "domain": "research", "lean": false, "words": 314, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/nt_damped_vinogradov__email_biro_andras/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_damped_vinogradov__email_hungarian_nt_SEND", "title": "Nt Damped Vinogradov — email hungarian nt SEND", "domain": "research", "lean": false, "words": 398, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/nt_damped_vinogradov__email_hungarian_nt_SEND/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_damped_vinogradov__email_maga_peter", "title": "Nt Damped Vinogradov — email maga peter", "domain": "research", "lean": false, "words": 313, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/nt_damped_vinogradov__email_maga_peter/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_damped_vinogradov__email_pintz_janos", "title": "Nt Damped Vinogradov — email pintz janos", "domain": "research", "lean": false, "words": 319, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/nt_damped_vinogradov__email_pintz_janos/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_damped_vinogradov__email_toth_arpad", "title": "Nt Damped Vinogradov — email toth arpad", "domain": "research", "lean": false, "words": 309, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/nt_damped_vinogradov__email_toth_arpad/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_damped_vinogradov", "title": "Explicit L² Bounds for Damped Vinogradov Sums", "domain": "math", "lean": false, "words": 7298, "doi": "10.5281/zenodo.19689578", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/nt_damped_vinogradov/", "pdf": "https://thalens.org/papers/nt_damped_vinogradov/paper.pdf", "created": "", "updated": ""}, {"id": "nt_goldbach_latent__derivation_gap_analysis", "title": "Goldbach Proof: Complete Derivation and Gap Analysis", "domain": "verification", "lean": false, "words": 3822, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/nt_goldbach_latent__derivation_gap_analysis/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_goldbach_latent__paper_reduction", "title": "A Reduction of Goldbach's Conjecture to an Auditable Trust Surface: Classical Primitives, Zero Open Analytic Atoms, and a Machine-Checked Envelope", "domain": "number_theory", "lean": true, "words": 7975, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a reduction of the binary Goldbach conjecture whose entire external dependency surface is explicit, auditable, and machine-checked in Lean 4.", "url": "https://thalens.org/papers/nt_goldbach_latent__paper_reduction/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_goldbach_latent__short_note_convergence", "title": "The Convolution Convergence Theorem: A Direct Conditional Proof of Goldbach's Conjecture", "domain": "number_theory", "lean": true, "words": 1207, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "We observe that the explicit formula for the Goldbach representation count $r(n)$ involves the zero sum $\\sum_\\rho 1/|\\rho(\\rho-1)|$, which converges to $D_\\infty = 2 + \\gamma - \\log(4\\pi) \\approx 0.046$ by the Hadamard product identity.", "url": "https://thalens.org/papers/nt_goldbach_latent__short_note_convergence/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_latent_moment_hypothesis__short_note", "title": "Latent Grade-2 Dominance and the Moment Hypothesis for All k", "domain": "analytic_number_theory", "lean": false, "words": 1683, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We show that the Moment Hypothesis for the Riemann zeta function —\n$m_{2k}(T) \\leq C_k (\\log T)^{k^2}$ for all $k \\geq 1$ — follows from\na single structural property of the cumulant generating function (CGF):\nits analyticity on a disk of radius $R > 0$ (equivalently, Latent\ndiagnostic $\\rho > 1$).", "url": "https://thalens.org/papers/nt_latent_moment_hypothesis__short_note/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_rh_de_branges_chain__ATTACK_MAP", "title": "Nt Rh De Branges Chain — ATTACK MAP", "domain": "math", "lean": false, "words": 3693, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/nt_rh_de_branges_chain__ATTACK_MAP/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_rh_de_branges_chain__handoff_phase11_week3_day4_20260419", "title": "Phase 11 Week 3 Day 4 Handoff — RH / M1 / nt_rh_de_branges_chain", "domain": "math", "lean": false, "words": 3225, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/nt_rh_de_branges_chain__handoff_phase11_week3_day4_20260419/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_rh_de_branges_chain__joker_phase11_20260418", "title": "Joker Phase 11 — 5 structural escape routes for the §15 M1 obstruction", "domain": "math", "lean": false, "words": 4707, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/nt_rh_de_branges_chain__joker_phase11_20260418/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_rh_de_branges_chain__m1_mesoscopic_extremal", "title": "M1 — Mesoscopic-decay extremal function (working file)", "domain": "math", "lean": true, "words": 87634, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/nt_rh_de_branges_chain__m1_mesoscopic_extremal/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_rh_de_branges_chain__math_closure", "title": "Mathematical Closure Audit of the De Branges / BK / Szegő Chain", "domain": "verification", "lean": false, "words": 20932, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/nt_rh_de_branges_chain__math_closure/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_rh_de_branges_chain__paper_bgst_transfer", "title": "An Unconditional BGST$\\to$R$_2$ Fourier Transfer via Poisson-Kernel Deconvolution", "domain": "number_theory", "lean": true, "words": 4073, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh [BGST23] proved the first unconditional asymptotic for Montgomery's pair-correlation function $F(\\alpha, T)$, with error $O(1/\\sqrt{\\log T})$ uniformly for $\\alpha \\in [0,1]$.", "url": "https://thalens.org/papers/nt_rh_de_branges_chain__paper_bgst_transfer/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_rh_de_branges_chain__paper_poisson_l2_lower_bound", "title": "A Poisson L² Lower Bound for Weil-Square Test Functions at Mesoscopic Bandwidth", "domain": "number_theory", "lean": true, "words": 4260, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Let `L = L(T) > 0` and `η_*(T) = (1/2 + η/2) \\log\\log T / \\log T`\nfor `T \\to \\infty` and a fixed `η \\in (0, 1/2)`.", "url": "https://thalens.org/papers/nt_rh_de_branges_chain__paper_poisson_l2_lower_bound/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_rh_de_branges_chain__rh_chain_formalization", "title": "A de Branges chain formalization toward the Riemann Hypothesis: architecture and explicitly axiomatic analytic steps", "domain": "math", "lean": true, "words": 7638, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We describe a formalization architecture toward the Riemann Hypothesis along a de Branges chain strategy: the spectral measure of a zeta-associated Hamiltonian satisfies the Szegő condition; by Lagarias's equivalence this implies RH. Alternatively, a narrowing-strip statement implies RH directly via", "url": "https://thalens.org/papers/nt_rh_de_branges_chain__rh_chain_formalization/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_rh_path2_gue__paper_conditional_rh", "title": "The Shifted Divisor Problem, the Moment Hypothesis, and a Conditional Path to the Riemann Hypothesis", "domain": "math", "lean": true, "words": 3312, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a machine-verified chain of 73 theorems connecting the classical shifted divisor problem $D_k(X,h) = \\sum_{n \\leq X} d_k(n) d_k(n+h)$ to a conditional proof of the Riemann Hypothesis via four intermediate results.\n\n**I.", "url": "https://thalens.org/papers/nt_rh_path2_gue__paper_conditional_rh/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_rh_path2_gue__short_note_gap", "title": "The Density–Individual Divide: Formalizing the GUE → RH Gap", "domain": "math", "lean": false, "words": 1734, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/nt_rh_path2_gue__short_note_gap/", "pdf": "", "created": "", "updated": ""}, {"id": "nt_rh_path2_gue__short_note_mpcc_reduction", "title": "The Mesoscopic Barrier for Montgomery's Pair Correlation Conjecture", "domain": "math", "lean": true, "words": 3153, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/nt_rh_path2_gue__short_note_mpcc_reduction/", "pdf": "", "created": "", "updated": ""}, {"id": "outreach__lean_zulip_guidelines", "title": "Lean Zulip — Community Guidelines & Posting Strategy", "domain": "verification", "lean": true, "words": 338, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/outreach__lean_zulip_guidelines/", "pdf": "", "created": "", "updated": ""}, {"id": "outreach__nicolas_cerrajero_draft", "title": "Draft Message to Nicolas Cerrajero", "domain": "finance", "lean": true, "words": 283, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/outreach__nicolas_cerrajero_draft/", "pdf": "", "created": "", "updated": ""}, {"id": "pde_latent_regularity", "title": "Conditional Regularity of 3D Navier-Stokes via Latent Spectral Analysis and Energy-Normalized Bilinear Bounds", "domain": "mathematical_physics", "lean": true, "words": 2267, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We introduce a spectral framework for analyzing the regularity of 3D Navier-Stokes equations through the analyticity strip width $\\sigma(t) = \\log \\rho(t)$, where $\\rho$ is the radius of convergence of the Fourier series.", "url": "https://thalens.org/papers/pde_latent_regularity/", "pdf": "https://thalens.org/papers/pde_latent_regularity/paper.pdf", "created": "", "updated": ""}, {"id": "pde_tensor_algebra__REFACTOR_PLAN", "title": "PDE Tensor Algebra SSOT — Lossless Refactor Plan", "domain": "finance", "lean": false, "words": 1572, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/pde_tensor_algebra__REFACTOR_PLAN/", "pdf": "", "created": "", "updated": ""}, {"id": "pde_tensor_algebra__decision_0419_venue_split", "title": "Venue-split decision — main paper + short letter", "domain": "finance", "lean": false, "words": 1433, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/pde_tensor_algebra__decision_0419_venue_split/", "pdf": "", "created": "", "updated": ""}, {"id": "pde_tensor_algebra__letter_0419_rigorous_exponent", "title": "Rigorous $-1$ exponent for the laminar-turbulence transition amplitude in a spectrally stratified 1D PDE", "domain": "physics", "lean": false, "words": 5013, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "In a spectrally stratified 1D mixed advection–reaction PDE used as a\ndiagnostic model for laminar-to-turbulent transitions, we derive a\nrigorous exponent $-1$ for the initial-condition amplitude threshold\n$\\Pi_c$ in the lowest active wavenumber $k_{\\min}$.", "url": "https://thalens.org/papers/pde_tensor_algebra__letter_0419_rigorous_exponent/", "pdf": "", "created": "", "updated": ""}, {"id": "pde_tensor_algebra__preregistered_prediction_0419_K_class_external_piCritical", "title": "Preregistered prediction — K-class external Π_c, frozen law", "domain": "pde_tensor_algebra", "lean": false, "words": 1344, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/pde_tensor_algebra__preregistered_prediction_0419_K_class_external_piCritical/", "pdf": "", "created": "", "updated": ""}, {"id": "pde_tensor_algebra__submission_classification_theorem", "title": "A Classification Theorem for PDE Solvability via Dissipation–Coupling–Constraint Decomposition", "domain": "research", "lean": true, "words": 5655, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We introduce a structural decomposition of partial differential equations into a\ntriple $(D, C, P)$ of dissipation, coupling and constraint operators and prove\nthat the solvability behavior of the system is determined by a single\ndimensionless quanti", "url": "https://thalens.org/papers/pde_tensor_algebra__submission_classification_theorem/", "pdf": "", "created": "", "updated": ""}, {"id": "pde_tensor_algebra__submission_coupling_algebra", "title": "The Coupling Algebra of Nonlinear PDE Systems: A Lie Structure on $(D, C, P)$ Triples", "domain": "verification", "lean": false, "words": 2683, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We introduce an algebraic structure on the space of nonlinear PDE systems. Each PDE system of dimension $n$ is encoded as a triple $\\mathbf{S} = (D, C, P) \\in \\mathbb{R}^{n \\times n} \\times \\mathbb{R}^{n \\times n \\times n} \\times \\mathbb{R}^{n \\times n}$ of dissipation matrix, coupling tensor, and c", "url": "https://thalens.org/papers/pde_tensor_algebra__submission_coupling_algebra/", "pdf": "", "created": "", "updated": ""}, {"id": "pde_tensor_algebra__submission_difficulty_flow", "title": "Universal Difficulty Exponents in Dissipative PDE Systems: A Renormalization-Group View of the (D, C, P) Framework", "domain": "physics", "lean": false, "words": 11309, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "**Main thesis.** The ratio of nonlinear coupling to dissipation in a PDE — the scale-dependent *difficulty* $\\mathcal{D}(k)$ — obeys a renormalization-group-style flow whose structure is universal across dissipative PDE systems.", "url": "https://thalens.org/papers/pde_tensor_algebra__submission_difficulty_flow/", "pdf": "", "created": "", "updated": ""}, {"id": "pde_tensor_algebra__submission_gr_penrose", "title": "General Relativity as a $(0, C, P)$ System: Penrose Singularity as Difficulty Divergence", "domain": "physics", "lean": true, "words": 3397, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We reinterpret the Einstein field equations within the $(D, C, P)$ tensor framework of [27]. In the ADM decomposition, vacuum GR is a $(0, R_{\\mu\\nu}[g], \\text{Bianchi} + \\text{gauge})$ system: zero dissipation, Ricci coupling, 8-component constraint projector.", "url": "https://thalens.org/papers/pde_tensor_algebra__submission_gr_penrose/", "pdf": "", "created": "", "updated": ""}, {"id": "pde_tensor_algebra__submission_ns_regularity", "title": "Navier–Stokes Regularity via Leray Bilinear Spreading and Two-Level Decoherence", "domain": "physics", "lean": false, "words": 2007, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We prove global regularity of the 3D incompressible Navier–Stokes equations on $\\mathbb{T}^3$ for smooth initial data with finite energy. The argument has three stages.", "url": "https://thalens.org/papers/pde_tensor_algebra__submission_ns_regularity/", "pdf": "", "created": "", "updated": ""}, {"id": "pde_tensor_algebra__submission_pde_tensor_algebra", "title": "The PDE Tensor Algebra: Structural Decomposition, Coupling Classification, and Exact Recombination of Differential Equations", "domain": "research", "lean": false, "words": 1863, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We introduce the **PDE Tensor Algebra**, a framework that represents any system of partial differential equations as a triple $(D, C, P)$ of tensors encoding dissipation, coupling, and constraints.", "url": "https://thalens.org/papers/pde_tensor_algebra__submission_pde_tensor_algebra/", "pdf": "", "created": "", "updated": ""}, {"id": "pde_tensor_algebra__submission_tensor_learning", "title": "Tensor Learning: Data-Driven Recovery of the (D, C, P) Decomposition and PDE Solvability Class", "domain": "verification", "lean": true, "words": 4277, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We propose a hybrid algorithm that recovers the structural $(D, C, P)$\ndecomposition of a nonlinear PDE system from time-series observations and,\nfrom this decomposition, predicts the solvability class and the\ndifficulty parameter $\\mathcal{D} = \\lVe", "url": "https://thalens.org/papers/pde_tensor_algebra__submission_tensor_learning/", "pdf": "", "created": "", "updated": ""}, {"id": "pde_tensor_algebra__submission_two_millennium", "title": "Two Millennium Problems Through the PDE Tensor Algebra: Navier–Stokes Regularity and the Yang–Mills Mass Gap", "domain": "physics", "lean": false, "words": 1749, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We present a unified framework — the PDE Tensor Algebra — that addresses two Clay Mathematics Institute Millennium Prize Problems through a single structural principle.", "url": "https://thalens.org/papers/pde_tensor_algebra__submission_two_millennium/", "pdf": "", "created": "", "updated": ""}, {"id": "pde_tensor_algebra__submission_universal_difficulty_exponents", "title": "Universal Difficulty Exponents in Advective-Dissipative PDE Systems", "domain": "physics", "lean": false, "words": 1329, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "We show that the scale-dependent difficulty of any advective–dissipative PDE system\nobeys a universal power law $\\mathcal{D}(k) \\sim k^{-1}$ in the inertial range,\nindependent of the solution's spectral structure.", "url": "https://thalens.org/papers/pde_tensor_algebra__submission_universal_difficulty_exponents/", "pdf": "", "created": "", "updated": ""}, {"id": "pde_tensor_algebra__submission_ym_mass_gap", "title": "The Yang–Mills Mass Gap as a Difficulty Transition in the PDE Tensor Algebra", "domain": "physics", "lean": false, "words": 2783, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We analyze the $SU(N)$ Yang–Mills equations through the PDE Tensor Algebra framework, representing them as a triple $(D, C, P)$ of dissipation, coupling, and constraint tensors. Classical pure Yang–Mills is a $(0, C, P)$ system with zero dissipation and structurally infinite difficulty $\\mathcal{D} ", "url": "https://thalens.org/papers/pde_tensor_algebra__submission_ym_mass_gap/", "pdf": "", "created": "", "updated": ""}, {"id": "phase_fields", "title": "Phase Field Algebras: Order, Smoothness, and Renormalization Group Fixed Points", "domain": "math_physics", "lean": true, "words": 1671, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We develop a formal algebraic framework for phase field theory, capturing the essential structure of systems that undergo phase transitions near critical points. The framework introduces a Phase type equipped with order (measuring singularity strength), smoothness (inverse of order plus one), and va", "url": "https://thalens.org/papers/phase_fields/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_celestial_mechanics_latent__PLAN", "title": "Unified Celestial Mechanics Latent Monograph — Structure Plan", "domain": "physics", "lean": true, "words": 3260, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/phy_celestial_mechanics_latent__PLAN/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_jwst_cosmological_cracks", "title": "Three Cracks in the Standard Model of Cosmology: A Formal Framework for Tension Quantification, Model Incompleteness, and Resolution Pathways", "domain": "cosmology, mathematical physics, foundations", "lean": true, "words": 4004, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/phy_jwst_cosmological_cracks/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_kerr_newman_rho", "title": "Spectral Compression of Charged Black Holes: ρ(Q) for Kerr-Newman", "domain": "physics", "lean": false, "words": 595, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "The quasinormal mode analyticity parameter ρ = γ₁/γ₀ ≈ 3 is universal for Kerr (uncharged) black holes [Paper I]. We extend this analysis to Kerr-Newman spacetimes, computing ρ(a, Q) across the full parameter space.", "url": "https://thalens.org/papers/phy_kerr_newman_rho/", "pdf": "https://thalens.org/papers/phy_kerr_newman_rho/paper.pdf", "created": "2026-04-04", "updated": ""}, {"id": "phy_knds_kappa_ordering_vieta__humanize_0418", "title": "Humanize + Tamás Style — phy_knds_kappa_ordering_vieta/paper.md", "domain": "verification", "lean": true, "words": 1325, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/phy_knds_kappa_ordering_vieta__humanize_0418/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_knds_kappa_ordering_vieta__mirtill_0418", "title": "Mirtill adversarial review — KNdS κ₊ < κ₋ via Vieta", "domain": "verification", "lean": true, "words": 1678, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/phy_knds_kappa_ordering_vieta__mirtill_0418/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_knds_kappa_ordering_vieta__mirtill_0418_v2", "title": "Mirtill adversarial review v2 — phy_knds_kappa_ordering_vieta/paper.md", "domain": "verification", "lean": true, "words": 2148, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/phy_knds_kappa_ordering_vieta__mirtill_0418_v2/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_knds_kappa_ordering_vieta", "title": "An Analytic Proof of $\\kappa_+ < \\kappa_-$ for Non-Degenerate Rotating Kerr-Newman-de Sitter Black Holes via Vieta Root Relations", "domain": "mathematical-physics", "lean": true, "words": 7074, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "For every non-degenerate rotating Kerr-Newman-de Sitter (KNdS) black hole — any configuration whose radial polynomial $\\Delta_r$ admits four simple real roots $r_n < 0 < r_- < r_+ < r_c$ — we prove analytically that the outer-horizon surface gravity", "url": "https://thalens.org/papers/phy_knds_kappa_ordering_vieta/", "pdf": "https://thalens.org/papers/phy_knds_kappa_ordering_vieta/paper.pdf", "created": "", "updated": ""}, {"id": "phy_m_theory_dimensions__mathoverflow_post", "title": "Phy M Theory Dimensions — mathoverflow post", "domain": "research", "lean": false, "words": 550, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_m_theory_dimensions__mathoverflow_post/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_m_theory_dimensions__paper_B", "title": "Near-Koide Structure Across the Standard Model Fermion Spectrum", "domain": "finance", "lean": false, "words": 3143, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We extend the Koide structure analysis of our companion paper [I] beyond the charged lepton sector.", "url": "https://thalens.org/papers/phy_m_theory_dimensions__paper_B/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_navier_stokes_latent__paper_archive_20260407", "title": "Grade Decomposition and Gevrey Regularity for Navier-Stokes: A Machine-Checked Path to the Millennium Prize", "domain": "verification", "lean": true, "words": 10362, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We introduce a grade decomposition of the Gevrey energy balance for the incompressible Navier-Stokes equations and formalize a complete three-phase regularity proof path in the Lean 4 proof assistant.", "url": "https://thalens.org/papers/phy_navier_stokes_latent__paper_archive_20260407/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_navier_stokes_latent__paper_viscous_floor", "title": "Signed Energy Transfer and the Viscous Floor: A Statewise Mechanism for Galerkin Navier-Stokes", "domain": "research", "lean": false, "words": 4437, "doi": "", "status": "working_paper", "status_label": "Working Paper", "review_status": "reviewed", "is_safe": true, "summary": "We study the Gevrey regularity of solutions to the 3D incompressible Navier-Stokes equations on the torus \\(\\mathbb{T}^3\\).", "url": "https://thalens.org/papers/phy_navier_stokes_latent__paper_viscous_floor/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_navier_stokes_regularity__NS_PROGRAM_OVERVIEW", "title": "Navier-Stokes Regularity Program — Overview", "domain": "verification", "lean": true, "words": 1789, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/phy_navier_stokes_regularity__NS_PROGRAM_OVERVIEW/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_navier_stokes_regularity", "title": "Global Regularity for the Three-Dimensional Navier-Stokes Equations via Direction Coherence", "domain": "mathematical physics", "lean": false, "words": 8392, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We reduce the global regularity problem for the three-dimensional incompressible Navier-Stokes equations to a single quantitative estimate: the integrability of the adiabatic error in the Gallay-Wayne convergence of vortex cross-sections to the Burgers profile.", "url": "https://thalens.org/papers/phy_navier_stokes_regularity/", "pdf": "https://thalens.org/papers/phy_navier_stokes_regularity/paper.pdf", "created": "", "updated": ""}, {"id": "phy_navier_stokes_regularity__paper_minimal", "title": "Eighteen Theorems: A Machine-Verified Conditional Framework for Navier-Stokes Regularity", "domain": "mathematical physics", "lean": false, "words": 3499, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We present a machine-verified proof framework for the global regularity of the three-dimensional incompressible Navier-Stokes equations, reduced to exactly 18 theorems. The logical chain — from six standard inputs through a contradiction argument — is verified by an automated proof kernel with zero ", "url": "https://thalens.org/papers/phy_navier_stokes_regularity__paper_minimal/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_navier_stokes_regularity__submission", "title": "Global Regularity for the Three-Dimensional Incompressible Navier-Stokes Equations via Perturbation Regularity and Direction Coherence", "domain": "mathematical physics", "lean": false, "words": 16563, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We prove global existence and smoothness of solutions to the three-dimensional incompressible Navier-Stokes equations for arbitrary smooth, divergence-free initial data with finite energy.", "url": "https://thalens.org/papers/phy_navier_stokes_regularity__submission/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_nbody_latent_solution__montgomery_mission_state", "title": "Montgomery Mission — Current State", "domain": "verification", "lean": true, "words": 1917, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/phy_nbody_latent_solution__montgomery_mission_state/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_nbody_latent_solution__paper_degenerate_cc", "title": "An Exact Algebraic Bifurcation in the Triangle-Plus-Center Central Configuration", "domain": "physics", "lean": false, "words": 3083, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We prove that the triangle-plus-center central configuration of the planar four-body problem with masses $(1, 1, 1, \\mu)$ undergoes an exact stability transition at the critical mass ratio\n\n$$\\mu^* = \\frac{81 + 64\\sqrt{3}}{249},$$\n\nthe unique positiv", "url": "https://thalens.org/papers/phy_nbody_latent_solution__paper_degenerate_cc/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_nbody_latent_solution__paper_four_questions", "title": "Montgomery's Four Questions and the Grade Hierarchy of the N-Body Problem", "domain": "physics", "lean": true, "words": 11844, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "Montgomery (2026) identifies four central open questions for the Newtonian $N$-body problem: (Q1) finiteness of central configurations, (Q2) Lyapunov stability of periodic orbits, (Q3) braid realization at zero angular momentum, and (Q4) density of the scattering map.", "url": "https://thalens.org/papers/phy_nbody_latent_solution__paper_four_questions/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_ns_postmerger_rho", "title": "The Analyticity Parameter of Neutron Star Post-Merger Oscillations", "domain": "physics", "lean": false, "words": 578, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "For black hole ringdown, the analyticity parameter ρ = γ₁/γ₀ ≈ 3 is universal — independent of mass, spin, and multipole [Paper I]. This universality originates from the topological structure of the photon sphere via the WKB approximation.", "url": "https://thalens.org/papers/phy_ns_postmerger_rho/", "pdf": "https://thalens.org/papers/phy_ns_postmerger_rho/paper.pdf", "created": "2026-04-04", "updated": ""}, {"id": "phy_pseudospectrum_rho", "title": "Pseudospectral Robustness of the Analyticity Parameter ρ", "domain": "physics", "lean": false, "words": 828, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "Jaramillo et al. (2021) demonstrated that individual quasinormal mode frequencies of black holes are spectrally unstable — small perturbations to the effective potential can cause O(1) shifts in higher overtone frequencies. This raises fundamental concerns about the physical relevance of overtone me", "url": "https://thalens.org/papers/phy_pseudospectrum_rho/", "pdf": "https://thalens.org/papers/phy_pseudospectrum_rho/paper.pdf", "created": "2026-04-04", "updated": ""}, {"id": "phy_qnm_latent__PAPER_IDEAS", "title": "Phy Qnm Latent — PAPER IDEAS", "domain": "finance", "lean": false, "words": 418, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_qnm_latent__PAPER_IDEAS/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_qnm_latent__STORY", "title": "The Thought Line: Black Hole Ringdown Has a Universal Compression Constant", "domain": "finance", "lean": false, "words": 1082, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_qnm_latent__STORY/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_qnm_latent__STRATEGY", "title": "Publication Strategy — QNM Latent", "domain": "core", "lean": false, "words": 389, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_qnm_latent__STRATEGY/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_qnm_latent__email_berti_draft", "title": "Phy Qnm Latent — email berti draft", "domain": "ml", "lean": false, "words": 307, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_qnm_latent__email_berti_draft/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_qnm_latent__email_cardoso_draft", "title": "Phy Qnm Latent — email cardoso draft", "domain": "ml", "lean": false, "words": 281, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_qnm_latent__email_cardoso_draft/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_qnm_latent__email_frei_draft", "title": "Phy Qnm Latent — email frei draft", "domain": "core", "lean": false, "words": 206, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_qnm_latent__email_frei_draft/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_qnm_latent__email_gergely_draft", "title": "Phy Qnm Latent — email gergely draft", "domain": "core", "lean": false, "words": 203, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_qnm_latent__email_gergely_draft/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_qnm_latent__email_isi_draft", "title": "Phy Qnm Latent — email isi draft", "domain": "core", "lean": false, "words": 279, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_qnm_latent__email_isi_draft/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_qnm_latent__email_racz_draft", "title": "Phy Qnm Latent — email racz draft", "domain": "core", "lean": false, "words": 242, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_qnm_latent__email_racz_draft/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_qnm_latent__email_stein_draft", "title": "Phy Qnm Latent — email stein draft", "domain": "physics", "lean": false, "words": 233, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_qnm_latent__email_stein_draft/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_qnm_latent__paper_formally_verified", "title": "Formally Verified Black Hole QNM Theory: Machine-Checked Proofs for Spectral Sufficiency", "domain": "verification", "lean": true, "words": 1474, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "We present the first machine-verified formalization of black hole\nquasinormal mode (QNM) spectral sufficiency theory.", "url": "https://thalens.org/papers/phy_qnm_latent__paper_formally_verified/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_qnm_latent__submission", "title": "Universal Spectral Compression of Black Hole Ringdown", "domain": "physics", "lean": false, "words": 1503, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_qnm_latent__submission/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_quantum_tuna9_parity__email_draft_20260418_delivery", "title": "Phy Quantum Tuna9 Parity — email draft 20260418 delivery", "domain": "ml", "lean": false, "words": 1470, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_quantum_tuna9_parity__email_draft_20260418_delivery/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_quantum_usrt__REPRODUCIBILITY", "title": "Reproducibility Package — Quantum USRT Spectral Dichotomy", "domain": "core", "lean": true, "words": 1596, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/phy_quantum_usrt__REPRODUCIBILITY/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_quantum_usrt__companion_papers_plan", "title": "Companion Papers Plan — Quantum USRT Publication Strategy", "domain": "physics", "lean": true, "words": 1870, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/phy_quantum_usrt__companion_papers_plan/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_quantum_usrt__flagship", "title": "A Spectral Dichotomy for Quantum Computing Feasibility: Effective Budget, Advantage Frontier, and Observable-Level Mitigation", "domain": "physics", "lean": true, "words": 4784, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We formulate a structural dichotomy for open quantum systems governed by Lindblad dynamics in terms of a single dimensionless parameter, $\\rho_Q = e^{|\\lambda_1|/v_{LR} - 1}$, constructed from the Lindblad spectral gap $|\\lambda_1|$ and the Lieb-Robinson velocity $v_{LR}$.", "url": "https://thalens.org/papers/phy_quantum_usrt__flagship/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_quantum_usrt__outreach_drafts", "title": "Outreach Drafts — Prospective Validation Collaborations", "domain": "finance", "lean": true, "words": 2402, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/phy_quantum_usrt__outreach_drafts/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_quantum_usrt__tuna9_calibration", "title": "Tuna-9 Calibration Battery — Self-measured Hardware Parameters", "domain": "physics", "lean": false, "words": 3286, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_quantum_usrt__tuna9_calibration/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_quantum_usrt__unique_prediction_protocol", "title": "Unique Testable Prediction: The Contested-Band Discrimination Protocol", "domain": "physics", "lean": false, "words": 1744, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_quantum_usrt__unique_prediction_protocol/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_smale6__change_note", "title": "Change Note — Smale 6 Finiteness Paper", "domain": "verification", "lean": false, "words": 1081, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_smale6__change_note/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_smale6__email_draft_20260418_delivery", "title": "Phy Smale6 — email draft 20260418 delivery", "domain": "verification", "lean": false, "words": 1161, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_smale6__email_draft_20260418_delivery/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_smale6__email_draft_20260423_albouy", "title": "Phy Smale6 — email draft 20260423 albouy", "domain": "physics", "lean": true, "words": 1463, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/phy_smale6__email_draft_20260423_albouy/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_smale6__email_draft_20260423_moeckel", "title": "Phy Smale6 — email draft 20260423 moeckel", "domain": "math", "lean": false, "words": 1196, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_smale6__email_draft_20260423_moeckel/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_strong_cosmic_censorship", "title": "Strong Cosmic Censorship at the VCS Boundary: Inverted Polarity, Charge-Spin Duality, and Kerr-Newman-de Sitter", "domain": "mathematical-physics", "lean": true, "words": 6032, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "Penrose's strong cosmic censorship conjecture (SCC) asserts that the maximal Cauchy development of generic initial data for Einstein's equations is inextendible. Unlike weak cosmic censorship (WCC), which asks whether horizons form, SCC asks whether singularities are \"strong enough\" to prevent exten", "url": "https://thalens.org/papers/phy_strong_cosmic_censorship/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_turbulence_scaling_grade__attack", "title": "Phy Turbulence Scaling Grade — attack", "domain": "physics", "lean": false, "words": 1986, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_turbulence_scaling_grade__attack/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_turbulence_scaling_grade__submission", "title": "First-Principles Kolmogorov Constant from the Local Analyticity Radius, and the Structural Disqualification of Gamma Intermittency Models", "domain": "physics", "lean": true, "words": 3593, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We report three results for the statistics of turbulent dissipation at $\\mathrm{Re}_\\lambda = 433$, derived from the Grade Equation framework and verified on 226,981 local analyticity-radius measurements from the Johns Hopkins Turbulence Database.", "url": "https://thalens.org/papers/phy_turbulence_scaling_grade__submission/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_vorticity_causality_bridge__mirtill_0418", "title": "Mirtill Adversarial Review: Vorticity as a Universal Controller", "domain": "physics", "lean": false, "words": 2380, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_vorticity_causality_bridge__mirtill_0418/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_vorticity_causality_bridge", "title": "Vorticity as a Universal Controller: A Structural Bridge Between Fluid Regularity and Spacetime Causality", "domain": "mathematical-physics", "lean": true, "words": 11792, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We identify a structural isomorphism connecting seven apparently unrelated threshold phenomena in mathematical physics: the Beale-Kato-Majda blowup criterion for 3D Navier-Stokes, the Gödel CTC condition in general relativity, the Kerr extremality bo", "url": "https://thalens.org/papers/phy_vorticity_causality_bridge/", "pdf": "https://thalens.org/papers/phy_vorticity_causality_bridge/paper.pdf", "created": "", "updated": ""}, {"id": "phy_weak_cosmic_censorship", "title": "Weak Cosmic Censorship as a Vorticity-Controlled System: Penrose Inequality, Self-Correcting Dissipation, and the Navier-Stokes Structural Transfer", "domain": "mathematical-physics", "lean": true, "words": 2400, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "Penrose's weak cosmic censorship conjecture (WCC) — that gravitational collapse from generic initial data cannot produce naked singularities visible to distant observers — remains one of the central open problems in general relativity.", "url": "https://thalens.org/papers/phy_weak_cosmic_censorship/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_yang_mills_mass_gap__arxiv_abstract", "title": "arXiv Abstract — Yang-Mills Existence and Mass Gap", "domain": "math", "lean": true, "words": 687, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/phy_yang_mills_mass_gap__arxiv_abstract/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_yang_mills_mass_gap__clay_submission", "title": "Existence and Mass Gap for Yang-Mills Theory on R^4", "domain": "physics", "lean": true, "words": 25669, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We prove that for any compact simple gauge group $G$, quantum Yang-Mills theory on $\\mathbb{R}^4$ exists as a Wightman quantum field theory and has a mass gap $\\Delta > 0$, solving the Clay Mathematics Institute Millennium Problem of Jaffe-Witten in", "url": "https://thalens.org/papers/phy_yang_mills_mass_gap__clay_submission/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_yang_mills_mass_gap__cover_letter", "title": "Cover Letter — Clay Millennium Prize Submission: Yang-Mills Existence and Mass Gap", "domain": "verification", "lean": true, "words": 1295, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/phy_yang_mills_mass_gap__cover_letter/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_yang_mills_mass_gap__executive_summary", "title": "Executive Summary — Yang-Mills Existence and Mass Gap", "domain": "finance", "lean": false, "words": 1505, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_yang_mills_mass_gap__executive_summary/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_yang_mills_mass_gap__explanation", "title": "Phy Yang Mills Mass Gap — explanation", "domain": "research", "lean": true, "words": 705, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/phy_yang_mills_mass_gap__explanation/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_yang_mills_mass_gap__explanation_noncollapsing", "title": "Phy Yang Mills Mass Gap — explanation noncollapsing", "domain": "finance", "lean": false, "words": 799, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_yang_mills_mass_gap__explanation_noncollapsing/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_yang_mills_mass_gap__lean_export_snapshot", "title": "Lean 4 Export Snapshot — Yang-Mills Clay Submission", "domain": "verification", "lean": true, "words": 6831, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/phy_yang_mills_mass_gap__lean_export_snapshot/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_yang_mills_mass_gap__millennium_dependency_audit", "title": "Phy Yang Mills Mass Gap — millennium dependency audit", "domain": "verification", "lean": false, "words": 13466, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_yang_mills_mass_gap__millennium_dependency_audit/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_yang_mills_mass_gap__monotonicity_proof_note", "title": "Technical Proof Note: Gap Monotonicity via Gauge Absorption (research-only)", "domain": "verification", "lean": false, "words": 2120, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/phy_yang_mills_mass_gap__monotonicity_proof_note/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_yang_mills_mass_gap__referee_appendix", "title": "Technical Appendix for Referees — Yang-Mills Mass Gap", "domain": "verification", "lean": true, "words": 7923, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/phy_yang_mills_mass_gap__referee_appendix/", "pdf": "", "created": "", "updated": ""}, {"id": "phy_yang_mills_mass_gap__yang_mills", "title": "Existence and Mass Gap for Yang-Mills Theory on R^4 — Extended Working Paper", "domain": "math", "lean": false, "words": 7119, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We prove that for any compact simple gauge group $G$, quantum Yang-Mills theory on $\\mathbb{R}^4$ satisfies the Wightman axioms and has a mass gap $\\Delta > 0$.", "url": "https://thalens.org/papers/phy_yang_mills_mass_gap__yang_mills/", "pdf": "", "created": "", "updated": ""}, {"id": "population_genetics_bridge", "title": "Spectral Bridges in Population Genetics", "domain": "verification", "lean": true, "words": 1832, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We establish precise spectral relationships between three foundational models in population genetics: the Wright-Fisher diffusion (forward-time allele frequency dynamics), the coalescent process (backward-time genealogical structure), and gene regulatory networks (equilibrium stability analysis).", "url": "https://thalens.org/papers/population_genetics_bridge/", "pdf": "", "created": "", "updated": ""}, {"id": "processus_algebra", "title": "Processus-Algebra: Numbers as Dynamical Systems", "domain": "verification", "lean": true, "words": 2023, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We introduce a novel algebraic structure called *processus-algebra* in which each element simultaneously encodes a value, a dynamics, and an attractor. A processus is a triple $(v, f, a)$ where $v$ is the current value, $f = dv/dt$ is the flow, and $a$ is the target attractor.", "url": "https://thalens.org/papers/processus_algebra/", "pdf": "", "created": "", "updated": ""}, {"id": "prop_logic", "title": "Propositional Logic Combinators: Machine-Verified Proofs in the Platonic Kernel", "domain": "verification", "lean": true, "words": 1319, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "We present a collection of formally verified theorems in propositional logic,\nproved in the Platonic proof kernel and exported to Lean 4. The theorems include\nfundamental combinators (identity, weakening, syllogism, flip, S-combinator,\ncompose3, diamond) and the classical Peirce's law.", "url": "https://thalens.org/papers/prop_logic/", "pdf": "", "created": "", "updated": ""}, {"id": "publications__EFFECTIVE_RHO", "title": "Publications — EFFECTIVE RHO", "domain": "finance", "lean": true, "words": 4099, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "The Fourier-cosine (COS) method achieves exponential convergence for option pricing and risk computation when the underlying density is analytic — but domain truncation can destroy this convergence entirely.", "url": "https://thalens.org/papers/publications__EFFECTIVE_RHO/", "pdf": "", "created": "", "updated": ""}, {"id": "publications__NOUS_ANOMALY_DETECTION", "title": "Publications — NOUS ANOMALY DETECTION", "domain": "verification", "lean": true, "words": 5589, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We develop a spectral framework for anomaly detection that distinguishes between two fundamentally different kinds of anomaly: **mode shift** (Type A), where existing spectral modes change amplitude, and **mode emergence** (Type B), where new modes appear that were absent from the baseline.", "url": "https://thalens.org/papers/publications__NOUS_ANOMALY_DETECTION/", "pdf": "", "created": "", "updated": ""}, {"id": "publications__NOUS_SPECTRAL_RESOLUTION_CAPSTONE", "title": "Publications — NOUS SPECTRAL RESOLUTION CAPSTONE", "domain": "finance", "lean": true, "words": 4805, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We prove that a single spectral quantity \\(K^* = \\Theta(\\log(1/\\varepsilon)/\\log\\rho)\\) — determined entirely by the analyticity radius \\(\\rho > 1\\) and target accuracy \\(\\varepsilon\\) — simultaneously governs seven distinct statistical and financial", "url": "https://thalens.org/papers/publications__NOUS_SPECTRAL_RESOLUTION_CAPSTONE/", "pdf": "", "created": "", "updated": ""}, {"id": "publications__NOUS_SPECTRAL_TRANSFER_CAPSTONE", "title": "Publications — NOUS SPECTRAL TRANSFER CAPSTONE", "domain": "core", "lean": true, "words": 5108, "doi": "", "status": "draft", "status_label": "Draft", "review_status": "math_verified", "is_safe": true, "summary": "We prove the Spectral Transfer Theorem: every analytic mapping \\( f : \\mathbb{R}^p \\to \\mathbb{R}^q \\) possesses a spectral transfer operator \\( T_{jk} = \\langle \\text{output mode } k \\mid f \\mid \\text{input mode } j \\rangle \\) satisfying five fundamental properties, all dimension-free.", "url": "https://thalens.org/papers/publications__NOUS_SPECTRAL_TRANSFER_CAPSTONE/", "pdf": "", "created": "", "updated": ""}, {"id": "publications__SPECTRAL_OVERFITTING", "title": "Publications — SPECTRAL OVERFITTING", "domain": "verification", "lean": true, "words": 4172, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We prove that the optimal model complexity for spectral estimation is determined by a single quantity — the spectral decay rate \\(\\rho\\) of the data-generating process — and that every standard overfitting remedy converges to the same critical truncation level.", "url": "https://thalens.org/papers/publications__SPECTRAL_OVERFITTING/", "pdf": "", "created": "", "updated": ""}, {"id": "pump_cycle_bridge", "title": "The Pump Cycle Bridge: Structural Isomorphism Between Gravitational Singularities and Debris Cascades", "domain": "physics", "lean": false, "words": 1817, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We prove a structural isomorphism between gravitational noncollision singularity (NCS) thresholds and orbital debris cascade thresholds. Both phenomena require a \"pump cycle\" mechanism—the sustained transfer of energy or momentum between subsystems—which in turn requires at least two independent bin", "url": "https://thalens.org/papers/pump_cycle_bridge/", "pdf": "", "created": "", "updated": ""}, {"id": "qnm_cross_kernel", "title": "QNM Cross-Kernel Bridge Theorems: Unifying Ringdown, Pseudospectrum, and Latent Structures", "domain": "core", "lean": false, "words": 1994, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We establish a unified mathematical framework connecting four distinct formalisms for quasi-normal mode (QNM) analysis: ringdown signal processing, extremal Kerr limits, pseudospectral perturbation theory, and mollification-based signal truncation.", "url": "https://thalens.org/papers/qnm_cross_kernel/", "pdf": "", "created": "", "updated": ""}, {"id": "qnm_extremal_limit", "title": "Formal Analysis of QNM Spectral Ratio Universality in the Extremal Kerr Limit", "domain": "black_hole_physics", "lean": true, "words": 1804, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a formal proof that the spectral ratio $\\rho = \\gamma_1/\\gamma_0$ of quasi-normal mode (QNM) damping rates remains universal in the extremal Kerr limit ($\\chi \\to 1$). While individual mode lifetimes $\\tau_n = 1/\\gamma_n$ diverge as the fundamental damping rate $\\gamma_0 \\to 0$, the ratio", "url": "https://thalens.org/papers/qnm_extremal_limit/", "pdf": "", "created": "", "updated": ""}, {"id": "quantum_gravity_bounds", "title": "Quantum Gravity Bounds: Theory-Independent Constraints on Quantum Gravity", "domain": "physics", "lean": true, "words": 2537, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a collection of formally verified, theory-independent bounds that constrain any viable theory of quantum gravity. These bounds emerge from the mutual inconsistency of general relativity and quantum mechanics when extrapolated to extreme scales.", "url": "https://thalens.org/papers/quantum_gravity_bounds/", "pdf": "", "created": "", "updated": ""}, {"id": "references__EXTRACT_Acerbi_2002_Spectral_Measures", "title": "Reference Extract: Acerbi (2002) — Spectral Measures of Risk", "domain": "finance", "lean": false, "words": 606, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/references__EXTRACT_Acerbi_2002_Spectral_Measures/", "pdf": "", "created": "", "updated": ""}, {"id": "resonance_algebra", "title": "Resonance Algebra: A Formal Framework for Vibrating Mathematical Objects", "domain": "math_physics", "lean": true, "words": 2496, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We develop Resonance Algebra (Res), a formal algebraic structure for mathematical objects with vibrational dynamics. Each element r ∈ Res carries three real-valued observables: energy E(r) ≥ 0, damping rate γ(r) > 0, and bandwidth BW(r) > 0.", "url": "https://thalens.org/papers/resonance_algebra/", "pdf": "", "created": "", "updated": ""}, {"id": "rough_volatility", "title": "Rough Volatility: The Exponent Architecture of Options Markets", "domain": "finance", "lean": true, "words": 3564, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a complete formal derivation of the ATM implied volatility skew power law ψ(T) ∼ C · T^{H−1/2} from the rBergomi model definition through an explicit chain of verified lemmas.", "url": "https://thalens.org/papers/rough_volatility/", "pdf": "", "created": "", "updated": ""}, {"id": "sakharov_baryogenesis", "title": "Formal Verification of Sakharov's Baryogenesis Conditions: Why Matter Exists", "domain": "particle physics, cosmology", "lean": true, "words": 2696, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a formally verified proof of Sakharov's three conditions for baryogenesis and their quantitative consequences.", "url": "https://thalens.org/papers/sakharov_baryogenesis/", "pdf": "", "created": "", "updated": ""}, {"id": "sgd", "title": "Formal Foundations of Stochastic Gradient Descent", "domain": "verification", "lean": true, "words": 2338, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a formally verified theory of stochastic gradient descent (SGD) and its variants, covering descent step bounds, convexity analysis, strong convexity, gradient Lipschitz conditions, distance contraction, momentum methods, mini-batch variance reduction, and learning rate schedules.", "url": "https://thalens.org/papers/sgd/", "pdf": "", "created": "", "updated": ""}, {"id": "solar_efficiency", "title": "Solar Cell Efficiency Bounds: A Formal Framework for Thermodynamic Limits and Improvement Strategies", "domain": "physics", "lean": true, "words": 1756, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We present a formally verified mathematical framework for solar cell efficiency bounds. Starting from the second law of thermodynamics, we derive the Carnot limit η ≤ 1 − T_cell/T_sun and prove that practical efficiency is bounded below unity.", "url": "https://thalens.org/papers/solar_efficiency/", "pdf": "", "created": "", "updated": ""}, {"id": "spacetime_thermodynamics", "title": "Spacetime Thermodynamics: Gravity as Entropy", "domain": "mathematical_physics", "lean": false, "words": 2389, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/spacetime_thermodynamics/", "pdf": "", "created": "", "updated": ""}, {"id": "strong_cp", "title": "The Strong CP Problem: A Formal Treatment of QCD CP Violation and Axion Solutions", "domain": "finance", "lean": false, "words": 3145, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We present a formal treatment of the Strong CP problem in QCD and its solutions. The physical θ-parameter, which would induce CP violation through a neutron electric dipole moment, is bounded by experiment to |θ| < 10⁻¹⁰.", "url": "https://thalens.org/papers/strong_cp/", "pdf": "", "created": "", "updated": ""}, {"id": "tension_algebra", "title": "Tension Algebra: A Mathematical Framework for Disequilibrium Dynamics", "domain": "tension_algebra", "lean": true, "words": 2098, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We develop **Tension Algebra**, a formal mathematical framework for modeling directed disequilibrium in physical and dynamical systems. A tension element τ = (A → B, Δ) encodes a source, a sink, and a magnitude representing the driving force behind transport phenomena—heat flow, wind, electric curre", "url": "https://thalens.org/papers/tension_algebra/", "pdf": "", "created": "", "updated": ""}, {"id": "tools__LATEX_FIXUP_PROMPT", "title": "Tools — LATEX FIXUP PROMPT", "domain": "finance", "lean": false, "words": 546, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/tools__LATEX_FIXUP_PROMPT/", "pdf": "", "created": "", "updated": ""}, {"id": "tools__MD_AUTHORING_GUIDE", "title": "Tools — MD AUTHORING GUIDE", "domain": "finance", "lean": false, "words": 677, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/tools__MD_AUTHORING_GUIDE/", "pdf": "", "created": "", "updated": ""}, {"id": "tools__README", "title": "Tools — README", "domain": "verification", "lean": false, "words": 520, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "", "url": "https://thalens.org/papers/tools__README/", "pdf": "", "created": "", "updated": ""}, {"id": "topological_numbers", "title": "Topological Numbers: Path-Based Arithmetic via Homotopy", "domain": "verification", "lean": true, "words": 1660, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We develop a formal theory of topological numbers, where each element is a continuous path in a topological space and equality is defined by homotopy equivalence. The resulting structure forms a group under path composition, with the identity element being the constant path and inverses given by pat", "url": "https://thalens.org/papers/topological_numbers/", "pdf": "", "created": "", "updated": ""}, {"id": "triangular_squares", "title": "Triangular Square Numbers and the Pell Equation", "domain": "number_theory", "lean": false, "words": 1368, "doi": "", "status": "skeleton", "status_label": "Skeleton", "review_status": "unreviewed", "is_safe": false, "summary": "We establish a complete formal characterization of triangular square numbers via\nthe Pell equation $x^2 - 8y^2 = 1$. A triangular number $T(n) = n(n+1)/2$ is a\nperfect square if and only if $(2n+1, 2m)$ satisfies this Pellian relation.", "url": "https://thalens.org/papers/triangular_squares/", "pdf": "", "created": "", "updated": ""}, {"id": "tropical_spectral", "title": "Tropical-Spectral Hybrid Algebra", "domain": "finance", "lean": true, "words": 1939, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We introduce a tropical-spectral hybrid algebra that enriches tropical numbers with a spectral entropy component, unifying optimization (tropical) and equilibrium (spectral) descriptions of the same phenomena.", "url": "https://thalens.org/papers/tropical_spectral/", "pdf": "", "created": "", "updated": ""}, {"id": "twin_primes", "title": "Structural Results Toward the Twin Prime Conjecture via Mod-6 Obstruction Theory", "domain": "math", "lean": false, "words": 2230, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "unreviewed", "is_safe": false, "summary": "We develop structural infrastructure for the twin prime conjecture using the Platonic proof kernel. The key insight is that mod-6 arithmetic provides an obstruction mechanism: for any prime p > 3 with p ≡ 1 (mod 6), its potential twin partner p + 2 ≡ 3 (mod 6) is divisible by 3, hence composite.", "url": "https://thalens.org/papers/twin_primes/", "pdf": "", "created": "", "updated": ""}, {"id": "unconditional_results", "title": "Unconditional Results: Deriving Latent Conditions from First Principles", "domain": "core", "lean": true, "words": 2478, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "", "url": "https://thalens.org/papers/unconditional_results/", "pdf": "", "created": "", "updated": ""}, {"id": "unified_embeddings_extended", "title": "Unified Embeddings Extended", "domain": "verification", "lean": true, "words": 2163, "doi": "", "status": "short_draft", "status_label": "Short Draft", "review_status": "math_verified", "is_safe": true, "summary": "We construct formal embeddings of eight algebraic structures into a universal type $\\mathcal{U}$ equipped with a canonical set of observables.", "url": "https://thalens.org/papers/unified_embeddings_extended/", "pdf": "", "created": "", "updated": ""}, {"id": "universal", "title": "Universal Foundations: A Verified Library of Core Mathematical Lemmas", "domain": "foundations", "lean": true, "words": 1418, "doi": "", "status": "proof_record", "status_label": "Proof Record", "review_status": "math_verified", "is_safe": true, "summary": "We present a formally verified library of 62 foundational mathematical lemmas spanning real arithmetic, trigonometry, continuity, finite sums, and vector operations. The library serves as a reusable substrate for domain-specific proofs across the research program.", "url": "https://thalens.org/papers/universal/", "pdf": "", "created": "", "updated": ""}]