Universal Foundations: A Verified Library of Core Mathematical Lemmas

Dimension-Independent Finiteness of Central Configurations f... The Yang-Mills Mass Gap via Gauge Absorption and Perelman W-... The Smooth-Step Spectral Method: Unifying Smooth and Thresho... The PDE Tensor Algebra: Structural Decomposition and Exact R... Machine-Verified Derivation Chain for the Fine Structure Con... A Lean 4 Formalization of the Birch and Swinnerton-Dyer Conj... ε-Removal for Moments of the Riemann Zeta Function via Cumul... An Exact Algebraic Bifurcation in the Triangle-Plus-Center C... No Non-Trivial Collatz Cycle Has Twenty-Three or Fewer Odd S... Turbulence Scaling Laws from the Grade Equation: Kolmogorov ... Critical Exponents for Three-Dimensional Percolation: A Cros... $D_\infty$ and the Goldbach Convergence Hierarchy The Goldbach Conjecture as a Latent Positivity Theorem: Twen... The Quantum Compressibility Threshold Constraint-Forced Absorption: A General Mechanism for Singul... Residual Stream Denoising in Large Language Models: Gradient... An Eigenvalue-Conditioned Copula with Positive Tail Dependen... Proof Circuits Allosteric Regulation — Spectral Communication in Proteins v... Brain Criticality — Phase Transition at the Edge of Chaos vi... Grn Dynamics Neuro Manifold Waddington Landscape Cell Fate via the Latent Framework Spectral Taxation: The Latent Structure of Optimal Income Ta... The Fenton Distribution Solved Grade3 Latent Algebra Grade Method Latent Core Latent Optimization Mollification Nash Boltzmann Bridge The Operational Curry–Howard: Proof Discovery as Navigated S... The Grade-Shadow Correspondence: Random Matrix Universality ... Resonance Goldbach The Riemann Hypothesis via Zeta Moment Hankel Positivity The Exact Latent Solution of the Gravitational Three-Body Pr... Grade Decomposition and Gevrey Regularity for Navier-Stokes:... Braid Realization at Zero Angular Momentum for the Planar N-... Neutron Star Postmerger The Spectral Theory of LLM Understanding: When Can We Trust ... Epidemic Spreading on Networks via the Latent Framework: Spe... Gene Regulatory Network Inference via the Latent Framework Morphogenesis as Spectral Selection Optimal Neural Decoding via the Latent Framework: How Many E... The ATM Skew Power Law: A Machine-Verified Derivation from t... The Langlands Transfer Graph The Latent Path Integral A Unified Resonance Framework for the Riemann Hypothesis The Riemann Hypothesis via Cumulant Independence and de Bran... The Simultaneous Field: A Universal Mathematical Framework f... A Unified Spectral Theory of Machine Learning: Neural Scalin... Spectral Mollification of Singular Distributions: A Latent F... The Latent Number in Biology P ≠ NP from Six Cited Axioms: A Machine-Verified Conditional... The Latent of a Game: Dimension-Free Representations for N-P... Interaction Grade as a Universal Language: The Latent Unific... Nash-Boltzmann Duality: Statistical Mechanics as Game Theory... The Latent of Black Hole Ringdown: Spectral Sufficiency and ... The Kessler Threshold as a Grade-2 Bifurcation: Formally Ver... The Grade Equation: A Universal Structural Law for Smooth Dy... The Latent Solution: A Finite Sufficient Representation Fram... Grade-2 Universality: A Formally Verified Unification of Flu... The Spectral Zeta Function of Markov Generators The Cumulant Bridge: Reducing the Moment Hypothesis to a Sin... The Grade Structure of MHD Conserved Quantities: Effective G... Spectral Market Risk: Deterministic VaR, ES, and Stress Test... Spectral XVA vs Nested Monte Carlo: A Three-Engine Benchmark The Riemann Hypothesis via Fourier-Euler Product: The Shorte... The Riemann Hypothesis via Berry-Keating Spectral Constructi... The Bridge Method: Systematic Cross-Domain Discovery via Sha... Eigenvalue Conditioning: A Universal Computational Primitive... The Grade Method: Structural Decomposition of ODE Vector Fie... One Parameter: How ρ Unifies Computation, Structure, and Saf... Latent Complexity: A Computable Theory of System Difficulty ... Two Lenses, One Invariant: Empirical Confirmation That ρ Is ... Full Density of Zeta Zeros on the Critical Line via GUE Univ... Grade Regularity: A Universal Criterion for Strong Solutions... Reflective Self-Amendment Euler Product Cumulant Bounds, GUE Pair Correlation, and Zer... Why Scale-Free Networks Are Spectrally Compressible: A Laten... Contaminated by Construction: Separating Simulation Noise fr... Reducing the Contamination: Variance Reduction Strategies fo... Emergent Canopy Architecture from Discrete Leaf Dynamics: A ... Plants: The Bidirectional Optimizers The Cascade Depth Theorem Spectral Importance Sampling: Optimal Rare-Event Simulation ... The Manifold Latent Theorem: Finite Spectral Representations... Nonlinear Portfolio Risk in Closed Form Ricci Flow as Spectral Compression: A Latent Interpretation ... The Fine Structure Constant from First Principles: A Two-Axi... The Feynman Integral as a Latent: Constructive Quantum Field... UV-IR Grade Duality: The Fine-Structure Constant and Cosmolo... The Latent Theory of Fusion Plasma Confinement What DESI Tells Us About the Grade Structure of Dark Energy The Grade Structure of Navier–Stokes: Why Blowup Requires Gr... Supercavitation Dynamics from the Grade Equation: The Analyt... Latent Probability: Conditional Dependence as Graded Spectra... Why Seven: Grade-3 Sufficiency and the Dimension of M-Theory The Latent of Latents: Hierarchical Finite Representations o... Emergent Capabilities as Universal Latent Modes The Shadow Theorem The Spectral Generator of the N-Body Latent: Connecting Padé... The Euler Product Smoothness Theorem: Multiplicative Structu... The Universal Padé–Stieltjes Machine: One Algebraic Pipeline... The Riemann Hypothesis as a Latent Existence Theorem Machine-Verified Spectral Intelligence: From Data Geometry t... The Smooth Latent Operator: Parameter-Free Distributional Re... Practical Padé Representations of the Gravitational Three-Bo... Applied Knowledge Algebra: A Collection of Demonstrations an... The Minimal Sufficient Spectral State for Decision and Risk Spectral Causal Identifiability Under Partial and Noisy Obse... Universal Approximation Theorems for Spectral Decision Funct... Spectral Ergodic Control with Provable Regret Guarantees Topological Obstructions in Market Dynamics Bridging Information Geometry and Spectral Risk Geometry Universality Classes of Spectral Learning Dynamics Spectral Phase Transitions in Generalization Spectral Time: Optimal Stopping, First Passage, and Subordin... Emergent Complexity in Reality and Engineered AI Systems The Spectral Information State Spectral Matrix Evolution of the Mandelbrot Iteration: Jacob... Spectral of Spectrals: Second-Order Mode Decomposition for C... Spectral Methods for Bioinformatics and Drug Discovery Intelligence as Organized Difficulty Compression Mathematical Manifestation The Spectral Theory of Observation The Universal Spectral Representation Theorem: Breaking the ... Bayesian Live Risk Harvestability Calibrating Harvestability Common Pricing, Different Portfolios The Duality of Bayesian and Frequentist Statistics The Three-Body Problem Solved Distributionally: Spectral Fok... Catastrophe Is Observer-Relative Knowability What Is ρ in Training? Spectral Density Propagation for Conjunction Assessment: A D... Spectral Cascade Risk: Quantifying the Kessler Syndrome via ... Optimal Collision Avoidance in One Microsecond: Analytical M... Real-Time Space Traffic Management for 50,000 Satellites: A ... Spectral Liquidation Risk: Exact First Passage Times for DeF... K* Modes Are All You Need: Spectral Uncertainty Quantificati... Why Does LoRA Work? The Spectral Theory of Low-Rank Adaptati... Specular Reflection in Spectral Fokker-Planck: A Penalty Met... The α-Continuum: Spectral Gap Controls the Quantum-Classical... There Is No Collapse: Measurement as Spectral Projection The Spectral Theory of Observation: Modes, Collapse, and the... The Spectral Resolution Theorem: Dimension-Free Compression ... Arbitrage-Free Term Structure via Spectral Coefficient Const... Three Numbers for Risk: A Data-Driven Spectral Basis for Por... Spectral Schrödinger Bridges: Optimal Transport Between Port... Spectral Alpha: Trading Signals from Fourier Risk Coefficien... Spectral Insurance: Aggregate Loss Distribution Without Mont... Spectral Kernel Risk: The Fourier--Gaussian Process Duality ... The Spectral Tensor Representation of Stochastic Processes When Simulation Is Unnecessary: An Information-Theoretic Cha... The Spectral Unity: Risk, Pricing, and Hedging from a Single... Verified Adversarial Robustness: Lipschitz Certificates for ... The Self-Modeling Ceiling: Formally Verified Bounds on Machi... Neural Scaling Laws Formalized: Why Chinchilla Works (A Mach... Spectral Knowledge Distillation: From Black Box to Certified... Spectral Model Compression: Provably Optimal Knowledge Extra... What Does Your Model Know? Spectral Decomposition and Arithm... Eigenvalue Conditioning as Universal Optimizer: Cross-Domain... The Geometry of Risk: Spectral Distance and Topological Stru... The Information-Theoretic Cost of Risk Measurement The Spectral Halving Cycle: A Fourier Model of Bitcoin's Pro... Frequency-Domain Theory of Financial Economics: Thirteen Fun... A Machine-Checked Reduced Transport Law for Stochastic-Field... Falsification Protocol and a Quantitative Dimits-Bypass Pred... Kernel-Verified Derived Physics: a Transferable Standard for... Kernel-Verified Debye Screening: a Second Worked Instance of... Algebraic Geometry of the Unified Field: Scheme-Theoretic St... Semi-Analytic Portfolio VaR via Eigenvalue-Conditioned COS I... Paper Draft — Portfolio VaR via Eigenvalue-Conditioned Logno... The Spectral Fenton Distribution: Exact Portfolio VaR via Ei... Arrow's Impossibility Theorem: Quantitative Extensions via L... Life: What, Why and How LatentFold — Business Plan & Product Strategy Convolution–Correlation Duality: A Universal Principle for S... Core Spectral Pattern Theory Core Spectral Phase Transition The Universal Representation Theorem: From Portfolio Risk to... The Thought Line: Universal Smoothing and the Latent Bridge The Universal Smoothing Bridge Divisibility and Parity in Natural and Integer Arithmetic Billion-Dollar Ideas from Analytical Physics Design Analytical Design Optimization: From Governing Equations to ... Formally Verified Epidemic Thresholds: The SIR Model as a Gr... Fin American Basket Bayesian Live Risk Formally Verified Financial Contagion Thresholds: Counterpar... The Risk Coding Theorem: Exponential Convergence of Spectral... The Fenton Distribution Solved Fin Fenton Spectral Harvestability Calibrating Harvestability The Bias-Variance Frontier of Risk Estimation: When Spectral... Fundamental Limits and Phase Transitions in Coherent Risk Es... Why Rational Investors Hold Different Portfolios Pricing Is Allocation Submission Contract Fin Progressive Spectral Fin Return Paradox Spectral Importance Sampling for Rare Events in Correlated S... Representation Theory of Finite Groups on the Unified Field A Machine-Verified Algebraic Formalization of Gödel's 1949 C... Gravity Theory Divergence: Newton vs MOND The Hodge Conjecture: A Formal Framework with Verified Cases... Imported Infrastructure — Strip and Bound Arithmetic for Ana... Coalescent Structure and the Universal Latent Ratio Effective-α framework: LR k-point structure at SR exponents The eta-negativity cascade: what a single sign constraint im... The Convolution–Correlation Duality The Decorrelation Index Hierarchical Feedback Percolation: When Structure Amplifies ... Meta Latent Computation Meta Pvsnp Platonic — ASSUMPTION LEDGER Meta Pvsnp Platonic — ROADMAP scientific strengthening A Machine-Verified Proof of P ≠ NP via Partition Function An... Grade-2 Dominance Across Millennium Problems: A Unified Stru... Unified Latent Formulations of Millennium Problems Applied Knowledge Algebra: Operations, Semantics, and Use Ca... A Numerical Demonstration of Knowledge Artifact Extraction a... SpectralShell: A Universal Model Class for Extracted Knowled... Ml Knowledge Artifacts Algebra Grade-2 Universality in Planetary Defense: Cross-Domain Brid... Network Latent Structure Verified Nelson-Siegel COS Hybrid Pricing Framework Nt Collatz Cycle Elimination — SUBMISSION ARTIFACTS Nt Collatz Lyapunov — dual space framework A Reduction of Goldbach's Conjecture to an Auditable Trust S... M1 — Mesoscopic-decay extremal function (working file) Mathematical Closure Audit of the De Branges / BK / Szegő Ch... A Poisson L² Lower Bound for Weil-Square Test Functions at M... A de Branges chain formalization toward the Riemann Hypothes... The Shifted Divisor Problem, the Moment Hypothesis, and a Co... The Density–Individual Divide: Formalizing the GUE → RH Gap Conditional Regularity of 3D Navier-Stokes via Latent Spectr... PDE Tensor Algebra SSOT — Lossless Refactor Plan Venue-split decision — main paper + short letter Rigorous $-1$ exponent for the laminar-turbulence transition... A Classification Theorem for PDE Solvability via Dissipation... The Coupling Algebra of Nonlinear PDE Systems: A Lie Structu... Universal Difficulty Exponents in Dissipative PDE Systems: A... General Relativity as a $(0, C, P)$ System: Penrose Singular... The PDE Tensor Algebra: Structural Decomposition, Coupling C... Two Millennium Problems Through the PDE Tensor Algebra: Navi... Universal Difficulty Exponents in Advective-Dissipative PDE ... The Yang–Mills Mass Gap as a Difficulty Transition in the PD... Phase Field Algebras: Order, Smoothness, and Renormalization... Unified Celestial Mechanics Latent Monograph — Structure Pla... Spectral Compression of Charged Black Holes: ρ(Q) for Kerr-N... Humanize + Tamás Style — phy_knds_kappa_ordering_vieta/paper... Mirtill adversarial review — KNdS κ₊ < κ₋ via Vieta Mirtill adversarial review v2 — phy_knds_kappa_ordering_viet... An Analytic Proof of $\kappa_+ < \kappa_-$ for Non-Degenerat... Near-Koide Structure Across the Standard Model Fermion Spect... Grade Decomposition and Gevrey Regularity for Navier-Stokes:... Eighteen Theorems: A Machine-Verified Conditional Framework ... Global Regularity for the Three-Dimensional Incompressible N... An Exact Algebraic Bifurcation in the Triangle-Plus-Center C... Montgomery's Four Questions and the Grade Hierarchy of the N... The Analyticity Parameter of Neutron Star Post-Merger Oscill... Pseudospectral Robustness of the Analyticity Parameter ρ The Thought Line: Black Hole Ringdown Has a Universal Compre... Publication Strategy — QNM Latent Phy Qnm Latent — email isi draft Phy Qnm Latent — email stein draft Formally Verified Black Hole QNM Theory: Machine-Checked Pro... Universal Spectral Compression of Black Hole Ringdown Strong Cosmic Censorship at the VCS Boundary: Inverted Polar... First-Principles Kolmogorov Constant from the Local Analytic... Mirtill Adversarial Review: Vorticity as a Universal Control... Vorticity as a Universal Controller: A Structural Bridge Bet... Weak Cosmic Censorship as a Vorticity-Controlled System: Pen... arXiv Abstract — Yang-Mills Existence and Mass Gap Existence and Mass Gap for Yang-Mills Theory on R^4 Executive Summary — Yang-Mills Existence and Mass Gap Phy Yang Mills Mass Gap — millennium dependency audit Technical Proof Note: Gap Monotonicity via Gauge Absorption ... Technical Appendix for Referees — Yang-Mills Mass Gap Existence and Mass Gap for Yang-Mills Theory on R^4 — Extend... Publications — EFFECTIVE RHO Publications — NOUS ANOMALY DETECTION Publications — NOUS SPECTRAL RESOLUTION CAPSTONE Publications — NOUS SPECTRAL TRANSFER CAPSTONE Publications — SPECTRAL OVERFITTING The Pump Cycle Bridge: Structural Isomorphism Between Gravit... QNM Cross-Kernel Bridge Theorems: Unifying Ringdown, Pseudos... Formal Analysis of QNM Spectral Ratio Universality in the Ex... Quantum Gravity Bounds: Theory-Independent Constraints on Qu... Rough Volatility: The Exponent Architecture of Options Marke... Spacetime Thermodynamics: Gravity as Entropy Tension Algebra: A Mathematical Framework for Disequilibrium... Topological Numbers: Path-Based Arithmetic via Homotopy Unified Embeddings Extended