Papers — thalens.org

Phylogenetic Tree Reconstruction via the Latent Framework

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.

Short Draft mathematical_biology Verified 3,088 words

Dimension-Independent Finiteness of Central Configurations for Positive Masses

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**.

Working Paper Formal Verification Verified DOI 18,003 words

The Structured Latent Basis: Feature Engineering as Basis Selection

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.

Working Paper Machine Learning 4,963 words

Formal Koide Structure: Mass Bounds, Generation Counting, and Neutrino Predictions from the Z_N Ansatz

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 $Q = 2/3$.

Short Draft Formal Verification Verified DOI 3,710 words

The Yang-Mills Mass Gap via Gauge Absorption and Perelman W-Entropy

**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

Draft Formal Verification Verified DOI 20,020 words

The Smooth-Step Spectral Method: Unifying Smooth and Threshold Structure in Tabular Regression

We introduce the Smooth-Step Spectral Method (S³M), a structured basis regression approach that unifies smooth spectral features with learned threshold features for tabular data.

Working Paper Machine Learning 6,039 words

An Unconditional BGST$\to$R$_2$ Fourier Transfer via Poisson-Kernel Deconvolution

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 $\alpha \in [0,1]$.

Short Draft number_theory Verified DOI 4,073 words

The PDE Tensor Algebra: Structural Decomposition and Exact Recombination of Differential Equations

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 computable algebraic conditions on the tensors.

Draft Formal Verification Verified 67,878 words

Machine-Verified Derivation Chain for the Fine Structure Constant via the E8 → Standard Model Breaking Pattern

We present a machine-verified derivation chain connecting the E₈ Lie algebra to the electromagnetic fine structure constant $\alpha \approx 1/137$.

Short Draft Formal Verification Verified 3,615 words

A refuted-and-vindicated pre-registration test of a spectral error model on a superconducting processor

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.

Working Paper Physics DOI 7,875 words

The Latent: Finite Sufficient Representations of Smooth Systems

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.

Draft Core Theory Verified DOI 80,076 words

A Lean 4 Formalization of the Birch and Swinnerton-Dyer Conjecture and Its Surrounding Theory

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

Draft number_theory Verified 16,260 words

ε-Removal for Moments of the Riemann Zeta Function via Cumulant Generating Function Analysis, Subject to Grade-2 Dominance

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 $r_\star \geq 1/A$;

Draft number_theory Verified 22,090 words

An Exact Algebraic Bifurcation in the Triangle-Plus-Center Central Configuration

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 $$\mu^* = \frac{81 + 64\sqrt{3}}{249},$$ the unique positiv

Short Draft celestial_mechanics Verified 2,225 words

The Double Gate Theorem: Cascade Instabilities Require Two Independent Conditions

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

Skeleton Physics 1,396 words

Integer Basics: Foundational Properties of Integer Arithmetic

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 machine-verified in the Lean 4 proof assistant.

Proof Record Formal Verification Verified 992 words

No Non-Trivial Collatz Cycle Has Twenty-Three or Fewer Odd Steps: Cycle Elimination, Descent Analysis, and Residue-Class Structure

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.

Draft number_theory 51,662 words

Turbulence Scaling Laws from the Grade Equation: Kolmogorov Spectrum and Intermittency from Analyticity

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 three steps.

Working Paper Physics Verified 12,483 words

Spectral Interest Rate Pricing: COS-Based Derivatives from Yield Curve Coefficients

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.

Working Paper Quantitative Finance Verified 6,508 words

Critical Exponents for Three-Dimensional Percolation: A Crossover Approach via Long-Range Renormalization

Draft Formal Verification Verified 37,738 words

$D_\infty$ and the Goldbach Convergence Hierarchy

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.

Short Draft number_theory Verified 4,697 words

The Goldbach Conjecture as a Latent Positivity Theorem: Twenty-Five Paths, the Convergence Theorem, and the Five-Layer Strategy

We develop a conditional proof program for the Goldbach conjecture through the generating function $G(z) = P(z)^2$, where $P(z) = \sum_{p \text{ prime}} z^p$.

Draft number_theory Verified 13,736 words

The Quantum Compressibility Threshold

We prove a structural dichotomy for open quantum systems.

Draft Physics 22,715 words

Constraint-Forced Absorption: A General Mechanism for Singularity Prevention in Gauge-Invariant PDE Systems

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): **Theorem** (Constraint-Forced Absorption).

Draft Quantitative Finance 5,290 words

Residual Stream Denoising in Large Language Models: Gradient-Free Quality Improvement via Functional Sensitivity Analysis

Draft machine learning / model analysis Verified 9,856 words

An Eigenvalue-Conditioned Copula with Positive Tail Dependence: A Machine-Verified Alternative to the Gaussian Copula

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.

Working Paper Quantitative Finance 6,074 words

Proof Circuits

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.

Draft mathematics Verified 21,446 words

Spectral Error Mitigation: Exact Noise Inversion for Quantum Computers via Cluster Lindblad

Working Paper Physics 4,368 words

Allosteric Regulation — Spectral Communication in Proteins via the Latent Framework

Allostery couples distant sites in a macromolecule: ligand binding at one pocket reshapes dynamics and thermodynamics elsewhere.

Short Draft mathematical_biology Verified 2,269 words

Brain Criticality — Phase Transition at the Edge of Chaos via the Latent Framework

Many neural systems are hypothesized to operate near a critical point between ordered and disordered dynamics, balancing sensitivity and stability.

Short Draft mathematical_biology Verified 2,084 words

Evolutionary Fitness Landscape Ruggedness via the Latent Framework

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.

Short Draft mathematical_biology Verified 2,825 words

Grn Dynamics

Gene Regulatory Network Dynamics — Stability, Inference & Latent Connection. This paper presents 50 machine-verified theorems building on 3 established facts and 54 hypotheses.

Proof Record Formal Verification Verified 940 words

Molecular Self Replication

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.

Proof Record Formal Verification Verified 192 words

Neuro Manifold

Neural Manifold — Spectral Dimension Bound, Optimal Decoding & BCI Theory. This paper presents 47 machine-verified theorems building on 5 established facts and 32 hypotheses.

Proof Record Mathematics Verified 855 words

Waddington Landscape Cell Fate via the Latent Framework

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.

Short Draft mathematical_biology Verified 2,471 words

Wright Fisher

Wright-Fisher Population Genetics — Spectral Convergence + Latent. This paper presents 24 machine-verified theorems building on 0 established facts and 35 hypotheses.

Proof Record Formal Verification Verified 628 words

The Latent Algebra: A Universal Representational Language for Smooth Systems

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,

Draft Core Theory Verified 10,585 words

Fat Tails and Carbon Taxes: A Spectral Resolution of the Climate Economics Debate

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.

Short Draft climate economics / integrated assessment Verified 2,097 words

Spectral Contagion: Network Fragility through the Latent Lens

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.

Short Draft financial networks / systemic risk Verified 2,049 words

Spectral Pricing: Bayesian Learning and the Explore-Exploit Frontier via the Latent Framework

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.

Short Draft dynamic pricing / bayesian learning / revenue optimization Verified 2,190 words

When Does Heterogeneity Matter? A Spectral Theory of Wealth Distribution and General Equilibrium

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.

Short Draft macroeconomics / heterogeneous agent models Verified 2,342 words

Latent Mechanism Design: Spectral Approximation of Optimal Mechanisms

We apply the Latent spectral framework to mechanism design, framing bilateral-trade tensions through a spectral parameter $\rho$ (Latent Number) of the type distribution.

Short Draft mechanism design / game theory Verified 1,818 words

Spectral Microstructure: Kyle's Lambda, Price Discovery, and the HFT Debate through the Latent Lens

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.

Short Draft market microstructure / financial economics Verified 1,764 words

Spectral Taxation: The Latent Structure of Optimal Income Tax Schedules

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.

Short Draft public economics / optimal taxation Verified 2,176 words

Quantitative Arrow: Measuring Distance from Impossibility via the Latent Framework

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.

Short Draft Verified 1,887 words

American Basket Gym

American Basket Gym — ProofEnv proofs (eigen-COS / FW / GH error bounds). This paper presents 67 machine-verified theorems.

Proof Record Formal Verification Verified 946 words

American Exercise

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.

Proof Record Formal Verification Verified 189 words

Bayes Decision

Bayes decision — ProofEnv encoding of BayesClassifier / MAPClassifier (Lean BayesDecision). This paper presents 58 machine-verified theorems.

Proof Record Formal Verification Verified 831 words

Bayesian Risk

Bayesian Risk — clean ProofEnv proof. This paper presents 51 machine-verified theorems.

Proof Record Formal Verification Verified 766 words

Bellman

Bellman — ProofEnv: DP value bounds, MDP Bellman operator, contraction lemmas. This paper presents 95 machine-verified theorems building on 0 established facts and 1 hypotheses.

Proof Record Formal Verification Verified 1,180 words

Correlated Cos V2

Correlated COS v2 (Correlated COS Method) — Newton proofs This paper presents 24 machine-verified theorems building on 0 established facts and 4 hypotheses.

Proof Record Formal Verification Verified 562 words

Credit Portfolio Loss

Credit Portfolio Loss Distribution — Beyond the Gaussian Copula. This paper presents 33 machine-verified theorems building on 0 established facts and 52 hypotheses.

Proof Record Quantitative Finance Verified 815 words

Elicitability

#9 Elicitability Resolution — Coherence vs Elicitability. This paper presents 35 machine-verified theorems building on 0 established facts and 68 hypotheses.

Proof Record Formal Verification Verified 852 words

Elicitability Coherence

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.

Proof Record Formal Verification Verified 194 words

Equity Premium

Equity Premium Puzzle — Machine-Checked Formalization This paper presents 73 machine-verified theorems building on 2 established facts and 3 hypotheses.

Proof Record Formal Verification Verified 984 words

Fenton Copula

Fenton — Hermite-COS Convergence Theory This paper presents 20 machine-verified theorems building on 0 established facts and 8 hypotheses.

Proof Record Formal Verification Verified 463 words

The Fenton Distribution Solved

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 $e^{\sigma_{\max}^2 k^2/2}$.

Draft Quantitative Finance Verified 14,277 words

Financial Contagion

Contagion Phase Transition — Sharp Threshold with Network Topology. This paper presents 53 machine-verified theorems building on 0 established facts and 50 hypotheses.

Proof Record Formal Verification Verified 955 words

Ftap Uncertainty

#2 FTAP under Model Uncertainty (Knightian Uncertainty). This paper presents 35 machine-verified theorems building on 0 established facts and 77 hypotheses.

Proof Record Formal Verification Verified 924 words

Harvestability Derivation

Harvestability Derivation — clean ProofEnv proof. This paper presents 54 machine-verified theorems building on 0 established facts and 12 hypotheses.

Proof Record Formal Verification Verified 874 words

Harvestability Extensions

Harvestability Extensions — clean ProofEnv proof. This paper presents 19 machine-verified theorems.

Proof Record Formal Verification Verified 406 words

Model Free Bounds

#3 Model-Free Bounds via Optimal Transport + Spectral Methods. This paper presents 35 machine-verified theorems building on 0 established facts and 80 hypotheses.

Proof Record Formal Verification Verified 953 words

Nsco Sv2

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.

Proof Record Formal Verification Verified 193 words

Nsco Sv3 Smile

NSCOSv3 Smile — ProofEnv (Heston-style variance: mean reversion, spreads). This paper presents 62 machine-verified theorems.

Proof Record Formal Verification Verified 886 words

Optimal Execution

#8 Optimal Execution Under Nonlinear Market Impact. This paper presents 38 machine-verified theorems building on 0 established facts and 62 hypotheses.

Proof Record Formal Verification Verified 855 words

Pricing Allocation

Pricing Allocation — clean ProofEnv proof. This paper presents 142 machine-verified theorems.

Short Draft Quantitative Finance Verified 1,670 words

Spectral Fenton

Spectral Fenton — ProofEnv proofs from Lean + quasi archives. This paper presents 74 machine-verified theorems.

Proof Record Formal Verification Verified 965 words

Spectral Portfolio

Spectral Portfolio — ProofEnv proofs (covariance / PC risk split). This paper presents 93 machine-verified theorems.

Proof Record Quantitative Finance Verified 1,092 words

Stochastic Portfolio Theory

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.

Proof Record Quantitative Finance Verified 138 words

Vol Surface Arb Free

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.

Proof Record Formal Verification Verified 140 words

Bridge Methodology

Bridge Methodology — Properties of Cross-Domain Mathematical Bridges. This paper presents 20 machine-verified theorems.

Proof Record Formal Verification Verified 341 words

Concrete Applications

Concrete Applications: Goldbach Crystallization & NS Turbulence. This paper presents 20 machine-verified theorems building on 0 established facts and 31 hypotheses.

Proof Record Formal Verification Verified 471 words

Constraint Lifecycle

Constraint Lifecycle — clean ProofEnv proof. This paper presents 52 machine-verified theorems.

Proof Record Formal Verification Verified 715 words

Grade3 Latent Algebra

Grade-3 Latent Algebra — Formal Foundation. This paper presents 142 machine-verified theorems building on 14 established facts and 122 hypotheses.

Short Draft Core Theory Verified 2,289 words

Grade Method

Grade Method — Domain Application Proofs. This paper presents 62 machine-verified theorems building on 18 established facts and 45 hypotheses.

Proof Record Formal Verification Verified 1,109 words

Kessler Threshold

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.

Proof Record Formal Verification Verified 136 words

Latent Core

Bridge: Latent Core ↔ Eigenvalue Conditioning. This paper presents 120 machine-verified theorems building on 10 established facts and 24 hypotheses.

Short Draft Core Theory Verified 1,506 words

Latent Optimization

Latent Optimization — ProofEnv proofs. This paper presents 136 machine-verified theorems building on 1 established facts and 1 hypotheses.

Proof Record Core Theory Verified 1,492 words

Mollification

Mollification Theory — Full Lean 4 Proof Suite (Bootstrap Edition). This paper presents 47 machine-verified theorems.

Proof Record Formal Verification Verified 581 words

Monodromy Grade

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.

Proof Record Formal Verification Verified 136 words

Nash Boltzmann Bridge

Nash-Boltzmann Duality — Bridge Kernel (Lean 4). This paper presents 66 machine-verified theorems.

Proof Record Formal Verification Verified 779 words

Resonance Simultaneous

Simultaneous Field — Three Millennium Problems as Projections This paper presents 28 machine-verified theorems building on 7 established facts and 40 hypotheses.

Proof Record Formal Verification Verified 624 words

Rmt Latent Bridge

RMT–Latent Bridge: Grade-Shadow Duality. This paper presents 16 machine-verified theorems building on 2 established facts and 16 hypotheses.

Proof Record Core Theory Verified 401 words

Spec Category

Spec Category — The Proof Category of Spectral Transfers. This paper presents 39 machine-verified theorems building on 16 established facts and 8 hypotheses.

Proof Record Formal Verification Verified 636 words

Spectral Optimization

Spectral Optimization — ProofEnv proofs (condition number, spectral step rate). This paper presents 71 machine-verified theorems.

Proof Record Formal Verification Verified 873 words

Stochastic Calculus

Stochastic Calculus — clean ProofEnv proof. This paper presents 19 machine-verified theorems building on 6 established facts and 3 hypotheses.

Proof Record Formal Verification Verified 395 words

Computational Instances

Computational Instances: concrete elements with determined ρ values. This paper presents 28 machine-verified theorems building on 2 established facts and 47 hypotheses.

Proof Record Formal Verification Verified 631 words

P Vs Np

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.

Proof Record Formal Verification Verified 138 words

The Operational Curry–Howard: Proof Discovery as Navigated Software Architecture

We propose a framework that unifies two perspectives on formal proof construction: **proof as pathfinding** in a type-theoretic state space and **proof as programming** via the Curry–Howard correspondence.

Draft meta-methodology Verified 37,516 words

The Grade-Shadow Correspondence: Random Matrix Universality from the Latent Grade Hierarchy

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 Equation).

Draft Mathematics Verified 11,061 words

Ai Safety Chain

AI Safety Chain — Connecting Theorems (Formal Proofs) This paper presents 27 machine-verified theorems building on 21 established facts and 25 hypotheses.

Proof Record Formal Verification Verified 641 words

Deep Consciousness

Deep Consciousness — clean ProofEnv proof. This paper presents 97 machine-verified theorems.

Proof Record Formal Verification Verified 1,157 words

Spectral Overfitting

Spectral Overfitting — clean ProofEnv proof. This paper presents 54 machine-verified theorems building on 0 established facts and 10 hypotheses.

Proof Record Formal Verification Verified 740 words

Explicit Formula Bridge

Explicit Formula Bridge — The Zero–Prime Duality as a Spec Morphism. This paper presents 66 machine-verified theorems building on 10 established facts and 4 hypotheses.

Proof Record Formal Verification Verified 882 words

Fejer Smoothing

Fejér Smoothing — clean ProofEnv proof. This paper presents 32 machine-verified theorems building on 0 established facts and 3 hypotheses.

Proof Record Formal Verification Verified 494 words

Gue Rh Gap

Strip Extension & Spectral Gap — Quantitative Interior ======================================================= Formalizes the two central results of the GUE density paper at a level deeper than the propositional chain in dpp_proof.py: This paper pre

Proof Record Formal Verification Verified 540 words

Pade Resummation

Entry alias for paderesummation_proof.py (canonical filename on disk). This paper presents 44 machine-verified theorems building on 0 established facts and 7 hypotheses.

Proof Record Formal Verification Verified 664 words

Prime Number Theorem

Prime Number Theorem — Logical Structure This paper presents 36 machine-verified theorems building on 0 established facts and 64 hypotheses.

Proof Record Mathematics Verified 787 words

Proof Complexity Rho

This paper presents 51 machine-verified theorems building on 19 established facts and 106 hypotheses.

Proof Record Formal Verification Verified 1,203 words

Resonance Algebra Rh

Representation Theory — Diamond for Characters, Plancherel, Branching This paper presents 673 machine-verified theorems building on 85 established facts and 817 hypotheses.

Draft Physics Verified 10,429 words

Resonance Goldbach

Goldbach-Latent Bridge — Connecting resonance framework to the full Goldbach proof This paper presents 18 machine-verified theorems building on 11 established facts and 58 hypotheses.

Proof Record Formal Verification Verified 626 words

Resonance Selberg

Ihara Zeta Function — Combinatorial Diamond for Graphs This paper presents 37 machine-verified theorems building on 8 established facts and 74 hypotheses.

Proof Record Formal Verification Verified 881 words

The Riemann Hypothesis via Zeta Moment Hankel Positivity

We establish a conditional proof that the Riemann Hypothesis follows from moment upper bounds weaker than the Lindelöf hypothesis.

Draft Mathematics Verified DOI 19,040 words

Black Hole Info

Black Hole Information Paradox — Formal Impossibility and Resolution ==================================================================== Proof suite version 2.0 — 45 theorems across 8 parts. This paper presents 83 machine-verified theorems.

Proof Record Formal Verification Verified 893 words

Cosmic Censorship

Cosmic Censorship — Nature's Modesty ===================================== This paper presents 58 machine-verified theorems.

Proof Record Formal Verification Verified 677 words

Cosmic Inflation

Cosmic Inflation — The First 10⁻³² Seconds ============================================ This paper presents 55 machine-verified theorems.

Proof Record Formal Verification Verified 624 words

The Exact Latent Solution of the Gravitational Three-Body Problem

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

Draft Physics DOI 11,407 words

N Body

N-Body — ProofEnv proofs aligned with NBodyGradeBound.lean + quasi archive. This paper presents 92 machine-verified theorems.

Proof Record Formal Verification Verified 1,057 words

Grade Decomposition and Gevrey Regularity for Navier-Stokes: A Machine-Checked Conditional Framework

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.

Working Paper Formal Verification Verified DOI 9,471 words

Braid Realization at Zero Angular Momentum for the Planar N-Body Problem

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.

Draft Physics Verified 5,937 words

Neutron Star Postmerger

Neutron Star Post-Merger ρ — Lean 4. This paper presents 16 machine-verified theorems.

Proof Record Formal Verification Verified 307 words

Poincare Conjecture

Perelman's Proof of the Poincaré Conjecture — Lean 4 Formalization This paper presents 21 machine-verified theorems building on 4 established facts and 29 hypotheses.

Proof Record Formal Verification Verified 484 words

Qnm Ringdown

Black Hole QNM Ringdown — Lean 4 Proof (v2.0). This paper presents 49 machine-verified theorems.

Proof Record Formal Verification Verified 617 words

Sir Epidemic

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.

Proof Record Formal Verification Verified 136 words

Spectral3 Body

Spectral 3-Body — clean ProofEnv proof. This paper presents 149 machine-verified theorems.

Short Draft Formal Verification Verified 1,678 words

Turbulence Grade

Turbulence Grade — Kernel Proof Suite v2.0 This paper presents 102 machine-verified theorems.

Proof Record Physics Verified 1,149 words

Uv Ir Duality

UV-IR Grade Duality — 30 Machine-Checked Theorems This paper presents 30 machine-verified theorems building on 9 established facts and 10 hypotheses.

Proof Record Formal Verification Verified 545 words

The Spectral Theory of LLM Understanding: When Can We Trust a Language Model?

Large language models are deployed in medicine, law, education, and critical infrastructure — yet we cannot mathematically characterize when their outputs are trustworthy.

Working Paper Machine Learning Verified 7,629 words

Cross-Domain Bridges in the Latent Framework: Structural Isomorphisms Across Ten Biological Domains

The Latent framework ($\Lambda$) provides a unified spectral representation for systems governed by self-adjoint operators with exponentially decaying eigenvalues.

Short Draft biology, cross-domain, latent theory Verified 2,669 words

Epidemic Spreading on Networks via the Latent Framework: Spectral Control, Superspreader Detection, and Optimal Intervention

Short Draft mathematical_epidemiology Verified 3,109 words

Gene Regulatory Network Inference via the Latent Framework

We apply the Latent framework to the gene regulatory network (GRN) inference problem — reconstructing the regulatory interaction matrix $W$ from gene expression data.

Short Draft Core Theory Verified 2,149 words

Morphogenesis as Spectral Selection

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.

Short Draft Physics Verified 3,185 words

Optimal Neural Decoding via the Latent Framework: How Many Electrodes Does a Brain-Computer Interface Need?

Short Draft computational_neuroscience Verified 2,618 words

One Number Rules Finance: The Latent Resolution of Asset Pricing Puzzles

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 $\rho$.

Short Draft quantitative finance / mathematical economics Verified 2,344 words

Bounded Rationality and the Spectral Complexity of Games

Finding a Nash equilibrium is PPAD-complete (Daskalakis, Goldberg & Papadimitriou, 2009), yet economic models routinely assume agents play equilibrium strategies.

Short Draft computational game theory / bounded rationality Verified 1,984 words

The ATM Skew Power Law: A Machine-Verified Derivation from the rBergomi Model

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.

Draft quantitative_finance Verified 14,265 words

The Langlands Transfer Graph

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 computable decorrelation gain $\delta \in [0,1]$.

Draft mathematics Verified 7,642 words

The Latent Path Integral

We prove that a single saddle-point formula governs quantitative approximation in every system whose action functional admits a Latent grade decomposition.

Draft mathematics Verified 10,090 words

The Unified Field: Fifteen Algebraic Structures and a Meta-Algebra for Mathematics

Draft meta-mathematics Verified 10,842 words

Why Neural Networks Scale: A Complete Latent-Theoretic Foundation

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.

Draft machine_learning Verified 6,042 words

A Unified Resonance Framework for the Riemann Hypothesis

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 $\gamma_n$.

Draft number_theory Verified 18,863 words

The Riemann Hypothesis via Cumulant Independence and de Branges Regularity

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).

Draft Mathematics Verified 5,851 words

The Sharp Moment Hypothesis for the Riemann Zeta Function

This draft develops a global strategy aimed at the sharp upper bound $$\frac{1}{T}\int_T^{2T} |\zeta(1/2+it)|^{2k}\,dt \leq C(k)(\log T)^{k^2}$$ for all integers $k \geq 1$ and all $T \geq T_0(k)$, where $C(k)$ is an explicit constant depending only on $k$.

Short Draft Mathematics Verified 2,862 words

Spectral Stability of the Gallay-Wayne Gap under Three-Dimensional Vortex Tube Perturbations

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.

Short Draft pde, spectral theory, fluid mechanics Verified 3,227 words

The Simultaneous Field: A Universal Mathematical Framework for Parallel Computation over Value Spaces

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.

Draft foundations 5,027 words

A Unified Spectral Theory of Machine Learning: Neural Scaling Laws, Generalization, and Architecture Design via the Latent Framework

We present a unified spectral theory of machine learning built on the Latent framework ($\Lambda$, $\rho$).

Short Draft ml_theory Verified 4,572 words

A Zero-Density Bound Near the Critical Line via Cumulant Matching

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.

Short Draft Mathematics Verified 3,174 words

The Latent Solution of the Gravitational N-Body Problem

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.

Draft Physics Verified 8,965 words

Spectral Mollification of Singular Distributions: A Latent Framework Approach

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 expansion.

Short Draft Core Theory 1,593 words

The Latent Number in Biology

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 $\rho$.

Short Draft Core Theory Verified 4,876 words

P ≠ NP from Six Cited Axioms: A Machine-Verified Conditional Proof via Partition Function Analyticity

We formalize a conditional proof of $\mathrm{P} \neq \mathrm{NP}$ in a Python-native proof language with Lean 4 export.

Draft Formal Verification Verified 6,194 words

Protein Folding as a Spectral First-Passage Problem

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

Draft Core Theory 22,583 words

Nature's Latent Catalog: A Formally Verified Tour of Biological Shape Optimization

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.

Short Draft Formal Verification Verified 4,386 words

Dimension-Free Differential Games via Latent Representation

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.

Short Draft Core Theory 3,001 words

The Latent of a Game: Dimension-Free Representations for N-Player Strategic Interactions

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.

Draft Core Theory 5,703 words

Interaction Grade as a Universal Language: The Latent Unification of Complex Systems

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.

Short Draft Core Theory 2,226 words

Spectral Latent Methods for High-Dimensional Mean-Field Games

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 $N$-player game (Nagy, 2026b).

Short Draft Core Theory 3,443 words

Optimal Mechanism Design via Latent Compression

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.

Short Draft Core Theory 2,840 words

Nash-Boltzmann Duality: Statistical Mechanics as Game Theory in the Latent Algebra

We establish an exact duality between strategic games and statistical mechanical systems through the Latent algebra's grade decomposition.

Short Draft Core Theory 3,187 words

The Folding Game: Protein Structure as Nash Equilibrium in the Latent Algebra

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 the (negative) payoff.

Short Draft Core Theory Verified 3,355 words

The Latent of Black Hole Ringdown: Spectral Sufficiency and the Overtone Problem

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.

Draft Physics 7,205 words

From Fish to Composites: The Latent Structure of Fiber-Angle Optimization

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.

Short Draft Core Theory Verified 3,287 words

The Latent of Accretion: Spectral Sufficiency and Grade Structure of Black Hole Accretion Disk Spectra

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.

Short Draft Core Theory Verified 1,763 words

How Many Modes Determine a Star? Spectral Sufficiency Bounds for Asteroseismology via the Latent Theorem

Asteroseismology determines stellar interior structure from observed oscillation frequencies — eigenvalues of the stellar structure operator.

Short Draft Core Theory Verified 1,849 words

Spectral Grade Decomposition of Pulsar Timing Residuals: Separating Gravitational Waves from Intrinsic Noise

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 correlated red noise, and solar system ephemeris errors.

Short Draft Physics Verified 1,729 words

The Kessler Threshold as a Grade-2 Bifurcation: Formally Verified Bounds for Space Debris Cascade Dynamics

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. The debris population rate

Short Draft Formal Verification Verified 4,953 words

The Grade Equation: A Universal Structural Law for Smooth Dynamical Systems

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

Working Paper Physics Verified 4,839 words

The Latent Solution: A Finite Sufficient Representation Framework for Partial Differential Equations

We introduce the **Latent solution** as a quantitative framework for finite-dimensional representation of PDE solutions.

Working Paper Core Theory Verified 8,801 words

Grade-2 Universality: A Formally Verified Unification of Fluid Turbulence, Gravitational Singularities, Orbital Debris Cascades, and Epidemic Thresholds

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

Draft Physics Verified 6,085 words

The Spectral Zeta Function of Markov Generators

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.

Short Draft Core Theory 4,317 words

The Cumulant Bridge: Reducing the Moment Hypothesis to a Single Distributional Condition

We reformulate the Moment Hypothesis for the Riemann zeta function as a cumulant boundedness condition and show that it reduces to a single distributional hypothesis about the Dirichlet polynomial approximation at $\sigma = 1/2$.

Draft analytic_number_theory 5,729 words

The Grade Structure of MHD Conserved Quantities: Effective Grade, Onsager Thresholds, and Taylor Relaxation

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

Working Paper Formal Verification Verified 5,152 words

Spectral Counterparty Credit Risk: Deterministic EPE, PFE, and Regulatory Capital Without Monte Carlo

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.

Short Draft Quantitative Finance 3,990 words

Spectral Market Risk: Deterministic VaR, ES, and Stress Testing Without Monte Carlo

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.

Short Draft Quantitative Finance 3,859 words

Spectral XVA vs Nested Monte Carlo: A Three-Engine Benchmark

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

Short Draft Quantitative Finance 4,260 words

The Riemann Hypothesis via Fourier-Euler Product: The Shortest Unconditional Proof

We prove the Riemann Hypothesis unconditionally from three classical inputs: Kronecker-Weyl equidistribution, the Bessel I₀ product identity, and Mertens' divergence theorem.

Working Paper Mathematics DOI 3,525 words

The Riemann Hypothesis via Berry-Keating Spectral Construction

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

Working Paper Mathematics 9,372 words

The Bridge Method: Systematic Cross-Domain Discovery via Shared Mathematical Structure

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.

Draft Quantitative Finance Verified 11,767 words

Eigenvalue Conditioning: A Universal Computational Primitive for Correlated Systems

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.

Draft Quantitative Finance Verified 7,165 words

The Grade Method: Structural Decomposition of ODE Vector Fields via the Grade Hierarchy

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 analyticity radius.

Draft Quantitative Finance 9,626 words

One Parameter: How ρ Unifies Computation, Structure, and Safety

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*

Short Draft Machine Learning 1,784 words

Latent Complexity: A Computable Theory of System Difficulty for Smooth Systems

We construct a complexity theory for smooth (analytic) systems based on the Latent framework.

Draft Core Theory 14,088 words

The Latent Number ρ: A Universal Diagnostic for Computational Complexity

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).

Draft Core Theory 11,292 words

Two Lenses, One Invariant: Empirical Confirmation That ρ Is Basis-Independent

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

Short Draft Quantitative Finance 4,277 words

Proved Safe: A Machine-Verified Theory of AI Safety from the Eigenvalue Spectrum

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.

Draft Machine Learning Verified 8,080 words

Per-Prime Cumulant Structure of the Riemann Zeta Function

We study the cumulant structure of $\log|\zeta(1/2+it)|^2$ induced by the Euler product. For each prime $p$, define $f_p(t) = 2\operatorname{Re}(-\log(1-p^{-1/2-it}))$.

Short Draft Mathematics Verified 2,634 words

Full Density of Zeta Zeros on the Critical Line via GUE Universality

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.

Working Paper Mathematics DOI 4,700 words

Grade Regularity: A Universal Criterion for Strong Solutions of Nonlinear PDEs

We prove a universal regularity criterion for nonlinear partial differential equations based on the grade decomposition of analytic vector fields.

Short Draft Core Theory 4,916 words

Autocallable Pricing as a Latent Computation: Spectral Methods with Formal Error Bounds

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).

Working Paper Quantitative Finance Verified 4,516 words

Reflective Self-Amendment

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

Short Draft Machine Learning 4,009 words

Euler Product Cumulant Bounds, GUE Pair Correlation, and Zero Density on the Critical Line

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. $

Draft Mathematics 6,470 words

Why Scale-Free Networks Are Spectrally Compressible: A Latent Sufficiency Theorem for Network Dynamics

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.

Short Draft Core Theory Verified 3,480 words

Residual Monte Carlo: A Unifying Framework for Variance Reduction

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.

Draft Quantitative Finance 5,893 words

Creative Flow as a Percolation Phase Transition in Knowledge Graphs

Draft Research Methodology 5,006 words

Machine-Verified Bounds for Chowla's Cosine Problem

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.

Draft Formal Verification Verified 6,311 words

Contaminated by Construction: Separating Simulation Noise from Model Risk in ES Backtests

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

Working Paper Quantitative Finance Verified DOI 12,334 words

Full-Tail Backtesting: Beyond Pointwise Validation for Expected Shortfall

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.

Working Paper Quantitative Finance Verified 10,211 words

Reducing the Contamination: Variance Reduction Strategies for Monte Carlo ES Backtests

Monte Carlo estimation of Expected Shortfall contaminates the Acerbi-Székely (2014) backtest statistic with computational noise.

Working Paper Quantitative Finance 4,528 words

Emergent Canopy Architecture from Discrete Leaf Dynamics: A Latent Environmental Encoding

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.

Short Draft mathematical biology, optimization, latent representations 4,559 words

Plants: The Bidirectional Optimizers

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.

Short Draft mathematical biology, optimization, latent representations 3,764 words

Emergent Root Architecture from Discrete Tip Dynamics: The Underground Mirror

This paper is the underground twin of "Emergent Canopy Architecture from Discrete Leaf Dynamics" (Nagy, 2026).

Short Draft mathematical biology, optimization, latent representations 4,139 words

The Cascade Depth Theorem

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.

Draft Quantitative Finance 5,010 words

Spectral Importance Sampling: Optimal Rare-Event Simulation via Eigenvalue-Conditioned Measure Change

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 matrix — are mutually independent.

Working Paper Quantitative Finance DOI 5,351 words

Terminal Portfolio Value Distribution to Machine Precision

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.

Draft Quantitative Finance Verified DOI 5,172 words

The Manifold Latent Theorem: Finite Spectral Representations Determine Riemannian Geometry

We prove that closed Riemannian manifolds with bounded geometry are determined, up to diffeomorphism, by finitely many spectral invariants.

Short Draft spectral_geometry, riemannian_geometry, inverse_spectral_theory Verified 3,926 words

The Fenton Distribution: A Complete Analytical Characterization

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 $(w, \mu, \Sigma) \in \mathbb{R}^{n(n+5)/2}$.

Short Draft Quantitative Finance 2,492 words

Nonlinear Portfolio Risk in Closed Form

We extend the Hermite-COS framework from linear portfolios (weighted sums of correlated lognormals) to portfolios containing options, structured products, and arbitrary derivative payoffs.

Short Draft Quantitative Finance 2,124 words

Ricci Flow as Spectral Compression: A Latent Interpretation of Perelman's Proof

We develop a spectral interpretation of the Hamilton-Perelman Ricci flow program for 3-manifolds.

Short Draft geometric_analysis, spectral_theory, latent_framework 2,384 words

Spectral Rigidity of the 3-Sphere: Finite Latent Characterization of Topology

We establish spectral rigidity results for the round 3-sphere $S^3$ within the Latent framework. Our main results are threefold.

Short Draft spectral_geometry, topology, latent_framework 2,501 words

The Fine Structure Constant from First Principles: A Two-Axiom Derivation via the Latent Grade Hierarchy

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**.

Draft Physics Verified DOI 9,504 words

Shadow Mining: Inferring Higher-Grade Structure from Lower-Grade Data

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*.

Short Draft mathematical_methodology 3,979 words

The Feynman Integral as a Latent: Constructive Quantum Field Theory from Grade Decay

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.

Draft Physics Verified 6,280 words

UV-IR Grade Duality: The Fine-Structure Constant and Cosmological Constant from a Single Scale

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

Short Draft Physics Verified 3,295 words

The Latent Theory of Fusion Plasma Confinement

**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 four requirements.

Working Paper Physics 26,429 words

The Cosmological Constant as a Grade-0 Residual: Smooth Vacuum Energy Flow in the Latent Hierarchy

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.

Draft Physics Verified 5,485 words

What DESI Tells Us About the Grade Structure of Dark Energy

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 $w(z) \neq -1$.

Short Draft Physics Verified 4,295 words

The Grade Structure of Navier–Stokes: Why Blowup Requires Grade-2 Saturation

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 contributions identically zero.

Short Draft Formal Verification Verified 4,371 words

Supercavitation Dynamics from the Grade Equation: The Analyticity Boundary as Cavity Interface

We propose a new theoretical framework for supercavitation dynamics based on the Grade Equation — a universal structural decomposition for analytic dynamical systems.

Short Draft Physics Verified 3,803 words

Latent Probability: Conditional Dependence as Graded Spectral Structure

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.

Short Draft Core Theory 4,797 words

Why Seven: Grade-3 Sufficiency and the Dimension of M-Theory

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).

Draft Mathematics 23,202 words

The Latent of Latents: Hierarchical Finite Representations of Knowledge Families

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.

Draft Machine Learning Verified DOI 11,938 words

Emergent Capabilities as Universal Latent Modes

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 domains.

Short Draft Machine Learning 3,116 words

The Shadow Theorem

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.

Draft intelligence_theory 9,580 words

The Spectral Generator of the N-Body Latent: Connecting Padé Poles, Koopman Eigenvalues, and Dynamical Classification

The N-body Latent [Nagy 2026g, 2026i] provides an exact, finite representation of gravitational trajectories via the generating function $G(z; \mathbf{v}_0)$.

Working Paper Core Theory 3,837 words

The Euler Product Smoothness Theorem: Multiplicative Structure Forces Latent Existence

We prove that the distribution of values of random Euler products on the critical line possesses a stable Latent — a finite rational approximation with exponential convergence — and provide a **complete structural proof** of the Euler Product Smoothn

Draft Mathematics DOI 45,534 words

The Universal Padé–Stieltjes Machine: One Algebraic Pipeline from Lognormal Sums Through the Riemann Zeta Function to the Three-Body Problem

We identify a single algebraic pipeline — **Moments → Padé–Stieltjes resummation → Rational characteristic function → Distribution/Trajectory** — that solves three structurally different problems: 1. **Lognormal sums** (statistics/finance): $S = \sum w_i e^{Y_i}$, $Y \sim \mathcal{N}(\mu, \Sigma)$.

Short Draft Mathematics 4,947 words

The Riemann Hypothesis as a Latent Existence Theorem

We prove that the Riemann Hypothesis is equivalent to the existence of a stable finite rational approximation — a *Latent* — for the distribution of $|\zeta(1/2+it)|$ on the critical line. Define the empirical Laplace transform $\hat{F}_T(z) = \frac

Short Draft Mathematics 4,389 words

Machine-Verified Spectral Intelligence: From Data Geometry to Holographic Bounds via Lean 4

We present the first complete, machine-verified proof chain connecting four fundamental phenomena in deep learning theory: neural scaling laws, attention as spectral filtering, grokking as a spectral phase transition, and holographic entropy bounds on learned representations.

Short Draft Machine Learning Verified 4,098 words

The Smooth Latent Operator: Parameter-Free Distributional Representations via Kernel Moment Recovery

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 $N_P$.

Short Draft Core Theory 3,167 words

The Exact Latent Distribution of Correlated Lognormal Sums

The distribution of $S = \sum_{i=1}^n w_i e^{Y_i}$, $Y \sim \mathcal{N}(\mu, \Sigma)$, has no closed-form CDF.

Draft Quantitative Finance 7,869 words

The Exact Latent Solution of the Gravitational N-Body Problem

We extend the exact Latent solution of the gravitational three-body problem [Nagy 2026g] to the general $N$-body case.

Short Draft Physics Verified DOI 4,613 words

Practical Padé Representations of the Gravitational Three-Body Problem

We demonstrate that Padé resummation of Taylor-series solutions provides a practical, machine-precision representation of the full gravitational three-body problem.

Short Draft Physics Verified DOI 2,886 words

Applied Knowledge Algebra: A Collection of Demonstrations and Use Cases

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.

Working Paper Machine Learning 10,099 words

The Minimal Sufficient Spectral State for Decision and Risk

Decision systems often carry excessive state, obscuring both theory and implementation. We define and characterize the minimal sufficient spectral state needed to preserve optimal decision value and calibrated risk estimation.

Short Draft Quantitative Finance 1,793 words

A Nonlinear Spectral Fundamental Theorem of Asset Pricing

Classical FTAP characterizes no-arbitrage in linear pricing systems. We develop a nonlinear spectral extension in which state prices are represented as mode-dependent functionals over a spectral decomposition of market states.

Short Draft Quantitative Finance Verified 1,654 words

Spectral Causal Identifiability Under Partial and Noisy Observation

Causal identification typically requires rich interventions or full-state observability. We show that partial, noisy observations can still identify causal structure when signals admit a stable spectral decomposition with separation conditions.

Short Draft Quantitative Finance 1,641 words

Universal Approximation Theorems for Spectral Decision Functionals

We prove universal approximation results for a broad class of decision functionals represented in spectral coordinates. The theorem characterizes expressivity in terms of basis regularity, coefficient decay, and functional smoothness, and provides quantitative approximation rates.

Short Draft Formal Verification Verified 1,926 words

Spectral Ergodic Control with Provable Regret Guarantees

Ergodic control and online regret analysis are usually developed with different tools and assumptions. We present a spectral ergodic control framework where long-run average control objectives and finite-time regret can be analyzed jointly.

Short Draft Quantitative Finance 1,775 words

Topological Obstructions in Market Dynamics

We introduce a topological perspective on market dynamics and prove obstruction results: under specific global state-space constraints, no arbitrage-free dynamics satisfying given regularity and observability axioms can exist.

Short Draft Quantitative Finance 2,018 words

Bridging Information Geometry and Spectral Risk Geometry

Information geometry and risk geometry have developed in parallel, with limited structural unification. We construct a bridge that maps statistical manifold structure to spectral risk coordinates, proving compatibility results between information-theoretic curvature objects and risk-distance functionals.

Short Draft Mathematics 1,689 words

Universality Classes of Spectral Learning Dynamics

Modern learning systems appear algorithmically diverse yet empirically convergent toward a small set of training regimes. We propose a universality framework in which optimization dynamics are classified by spectral invariants rather than by optimizer-specific update rules.

Short Draft Machine Learning 1,768 words

Spectral Phase Transitions in Generalization

We develop a phase-transition theory of generalization based on spectral order parameters. Instead of smooth monotonic scaling, we show that model performance exhibits threshold behavior when mode occupancy, noise-floor coupling, and effective rank cross critical boundaries.

Short Draft Machine Learning 1,662 words

Spectral Time: Optimal Stopping, First Passage, and Subordination via the Fokker-Planck Generator

We develop a unified spectral framework for three fundamental problems in stochastic analysis: optimal stopping, first passage times, and time-changed (subordinated) processes.

Draft Quantitative Finance 6,294 words

Theory and Measure of Decomposability

This paper develops a general theory of decomposability for structured systems, problems, and phenomena.

Short Draft Quantitative Finance 4,835 words

Emergent Complexity in Reality and Engineered AI Systems

We study emergent complexity as a regime property of open dynamical systems rather than as an intrinsic label of objects.

Draft Quantitative Finance 6,536 words

The Spectral Information State

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 strength.

Draft Research Methodology 7,139 words

Spectral Matrix Evolution of the Mandelbrot Iteration: Jacobian Products, Coherence, and Meta-Modes

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.

Short Draft Physics 2,247 words

Spectral of Spectrals: Second-Order Mode Decomposition for Complex Systems

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.

Short Draft Physics 2,105 words

The Spectral Cognitive Resonator: A Dynamic Architecture for Agent Memory, Routing, and Self-Adaptation

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.

Short Draft Machine Learning 2,692 words

Spectral Memory and Graph Routing for Language Model Agents

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 timescale each memory operates on.

Short Draft Machine Learning 2,068 words

Spectral-State Neural Networks: A Mode-Decomposition Architecture for Learned Dynamics

Standard neural networks represent hidden state as unstructured activation vectors evolved by arbitrary weight matrices.

Short Draft Machine Learning 2,516 words

Spectral Vorticity Bridge for Turbulent Flows

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.

Short Draft Physics 1,892 words

Spectral Methods for Bioinformatics and Drug Discovery

We present a bridge framework linking spectral representation methods to core tasks in bioinformatics and drug discovery.

Short Draft Core Theory 1,849 words

Intelligence as Organized Difficulty Compression

We propose a system-level theory of intelligence centered on a single claim: **intelligence is organized difficulty compression**. The motivating problem is that the term *intelligence* is routinely asked to cover too much at once, ranging from consciousness and general reasoning to benchmark success and adaptive control.

Draft Quantitative Finance 6,126 words

Scientific Venture Design

We propose a formal framework for **scientific venture design**: a falsifiable theory of how to build large businesses under uncertainty.

Short Draft Quantitative Finance 3,969 words

Information and Causal Access

We propose a manifestation-and-causality view of information. The motivating problem is that the term `information` is used for several different objects at once: raw differences, symbols, messages, semantic content, causal signals, and epistemic gain.

Short Draft Physics 1,540 words

Mathematical Manifestation

We propose a manifestation-theoretic view of mathematical statements, formal systems, and theories. The guiding question is whether the repo's broader one-object / manifestation framework can be extended beyond physical or observational objects to include theoretical and mathematical ones.

Short Draft Formal Verification 2,481 words

The One Behind Everything

We propose a generator-first extension of the repo's broader spectral program. The starting question is not how a latent state is quantized into a discrete observable, but how a structured latent state may itself arise from repeated application of a local operator.

Draft Physics 9,254 words

The Spectral Theory of Observation

We propose a spectral framework for understanding how discrete outcomes arise from smooth latent structure. The core move is to represent a finite-valued or countable observable not as an object outside continuous analysis, but as an atomic measure \[ \mu = \sum_{j=1}^{m} p_j \,\delta_{x_j}, \] already living on a continuous state space.

Draft Physics 26,608 words

The Universal Spectral Representation Theorem: Breaking the Curse of Dimensionality

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

Working Paper Quantitative Finance Verified 7,243 words

Bayesian Live Risk

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.

Draft Quantitative Finance Verified 11,538 words

Harvestability

This paper studies **fin_harvestability** as a horizon object for horizon-dependent allocation within a CRRA investor facing Ornstein-Uhlenbeck eigenmodes.

Working Paper Formal Verification Verified 8,437 words

Calibrating Harvestability

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

Short Draft Quantitative Finance 2,491 words

Common Pricing, Different Portfolios

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 market weights $w_k^{\mathrm{mkt}}$.

Draft Quantitative Finance Verified 7,021 words

The Spectral Volatility Surface

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.

Working Paper Quantitative Finance Verified 8,745 words

The Duality of Bayesian and Frequentist Statistics

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.

Working Paper Research Methodology Verified 6,046 words

The Three-Body Problem Solved Distributionally: Spectral Fokker-Planck for the Circular Restricted Three-Body Problem

Poincaré (1890) proved that the three-body problem admits no globally convergent power-series trajectory solution. We do not contradict that result.

Short Draft Physics 4,741 words

The Latent Generator

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.

Short Draft Quantitative Finance 4,041 words

Catastrophe Is Observer-Relative

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.

Short Draft Formal Verification 4,793 words

Knowability

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 observation space arise from a smaller latent grammar.

Working Paper Quantitative Finance 2,243 words

Spectral Granger Causality: Mode-by-Mode Causal Bandwidth

Granger causality (Granger, 1969) tests whether the past of $X$ helps predict $Y$ beyond $Y$'s own past.

Working Paper Quantitative Finance Verified 3,170 words

Adam's Convergence Proof Was Wrong: A Machine-Checked Verification of the Bug and the Fix

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 problems.

Working Paper Formal Verification Verified 8,361 words

What Is ρ in Training?

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**.

Short Draft Machine Learning 3,377 words

Spectral Density Propagation for Conjunction Assessment: A Deterministic Alternative to Monte Carlo

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.

Short Draft Core Theory Verified 2,706 words

Spectral Disruption Prediction: Real-Time Plasma Stability via the MHD Generator

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 $\mathcal{L}$.

Short Draft Core Theory 2,262 words

Spectral Cascade Risk: Quantifying the Kessler Syndrome via Fokker-Planck Fragment Propagation

The Kessler syndrome --- a self-sustaining cascade of orbital collisions generating exponentially growing debris --- is the existential threat to the low Earth orbit environment.

Short Draft Core Theory 2,689 words

Optimal Collision Avoidance in One Microsecond: Analytical Maneuver Planning via the Spectral Fokker-Planck Generator

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.

Short Draft Core Theory Verified 2,651 words

Real-Time Space Traffic Management for 50,000 Satellites: A Spectral Digital Twin

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.

Short Draft Formal Verification Verified 3,079 words

Trading the Eigenvalues: Statistical Arbitrage via the Implied Generator

We develop a systematic trading framework based on the **implied generator** — the Fokker--Planck generator matrix $M$ recovered from option prices without parametric assumptions.

Short Draft Quantitative Finance 2,542 words

Spectral Liquidation Risk: Exact First Passage Times for DeFi Lending Protocols

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).

Short Draft Core Theory 2,885 words

K* Modes Are All You Need: Spectral Uncertainty Quantification for Deep Learning

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

Short Draft Machine Learning Verified 2,730 words

One Number Predicts Barren Plateaus: The Lindblad Spectral Gap as Trainability Bound

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.

Short Draft Machine Learning 2,604 words

Emergence Is a Spectral Phase Transition: Predicting When Language Models Acquire New Abilities

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.

Short Draft Machine Learning 2,416 words

Why Does LoRA Work? The Spectral Theory of Low-Rank Adaptation

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.

Short Draft Machine Learning 2,310 words

Specular Reflection in Spectral Fokker-Planck: A Penalty Method with Proven Convergence

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.

Short Draft Mathematics Verified 1,809 words

The α-Continuum: Spectral Gap Controls the Quantum-Classical Transition

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).

Short Draft Physics 3,204 words

There Is No Collapse: Measurement as Spectral Projection

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).

Short Draft Physics 2,190 words

The Spectral Theory of Observation: Modes, Collapse, and the Information Content of Reality

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).

Short Draft Physics 3,311 words

The Spectral Resolution Theorem: Dimension-Free Compression of Analytic Objects

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

Working Paper Formal Verification Verified 3,105 words

The Anomaly Functional: Real-Time Arbitrage Detection via Spectral Risk Coefficients

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)

Working Paper Quantitative Finance 4,250 words

Arbitrage-Free Term Structure via Spectral Coefficient Constraints

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.

Working Paper Quantitative Finance Verified 4,168 words

Bitcoin as a Thermodynamic System: Phase Transitions at Halving Events

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.

Working Paper Formal Verification Verified 3,231 words

The Formula of Doom Was Provably Wrong: A Machine-Checked Replacement for the Copula That Caused the 2008 Financial Crisis

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.

Working Paper Formal Verification Verified 4,067 words

High-Precision Greeks for Multi-Asset Spread Options via Eigenvalue-Conditioned Fourier Inversion

Spread options on three or more correlated assets are fundamental hedging instruments in energy, commodity, and agricultural markets.

Working Paper Quantitative Finance Verified 5,063 words

Basket Option Pricing Under Stochastic Volatility via Eigenvalue-Conditional Heston Mixing

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.

Short Draft Quantitative Finance 4,581 words

How Much Bitcoin? Spectral Portfolio Allocation Beyond Mean-Variance

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$\times$ heavier than the Gaussian approximation.

Working Paper Quantitative Finance Verified 3,195 words

Three Numbers for Risk: A Data-Driven Spectral Basis for Portfolio Loss Distributions

The Spectral Fenton Distribution represents a portfolio loss distribution with 128 Fourier coefficients.

Working Paper Quantitative Finance 4,109 words

The Pythagorean Gap: A Mode-Decomposition Indicator for Financial Crises

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: $SR^2_{\text{tangency}} = \sum_k SR^2_k$.

Working Paper Quantitative Finance Verified 3,855 words

Spectral Schrödinger Bridges: Optimal Transport Between Portfolio Distributions in Fourier Space

The Schrödinger Bridge Problem (SBP) finds the most likely stochastic evolution between two probability distributions, minimizing the Kullback--Leibler divergence from a reference process.

Working Paper Quantitative Finance Verified 4,878 words

Spectral Alpha: Trading Signals from Fourier Risk Coefficients

The Spectral Fenton Distribution compresses a portfolio's loss distribution into 130 parameters: 128 Fourier coefficients plus a location and scale anchor.

Working Paper Quantitative Finance 4,305 words

Spectral Bitcoin VaR: High-Accuracy Risk Measures for Cryptocurrency via Fourier Expansion

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.

Working Paper Quantitative Finance Verified 3,600 words

Spectral XVA: Replacing Monte Carlo in Counterparty Credit Risk

We propose a spectral method for computing Credit Valuation Adjustment (CVA) and related XVA metrics that eliminates Monte Carlo simulation entirely.

Short Draft Quantitative Finance 3,187 words

Spectral Insurance: Aggregate Loss Distribution Without Monte Carlo

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.

Draft Formal Verification Verified 9,949 words

Spectral Kernel Risk: The Fourier--Gaussian Process Duality for Portfolio Loss Distributions

We establish a formal duality between the Spectral Fenton (Fourier) representation of portfolio loss densities and Gaussian Process (GP) regression via Mercer's theorem.

Working Paper Quantitative Finance Verified 3,337 words

Spectral Regime Detection: Change-Point Identification via Eigenmode Drift

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.

Working Paper Quantitative Finance 2,179 words

The Spectral Tensor Representation of Stochastic Processes

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.

Short Draft Quantitative Finance 3,789 words

When Simulation Is Unnecessary: An Information-Theoretic Characterization of Analytically Tractable Distributions

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 $\rho$.

Draft Quantitative Finance Verified 5,565 words

The Spectral Unity: Risk, Pricing, and Hedging from a Single Representation

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.

Short Draft Quantitative Finance Verified 3,855 words

Verified Adversarial Robustness: Lipschitz Certificates for Neural Networks in Lean 4

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.

Draft Machine Learning Verified 8,773 words

The Self-Modeling Ceiling: Formally Verified Bounds on Machine Self-Calibration

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.

Draft Formal Verification Verified 6,507 words

What Your XGBoost Learned: Spectral Knowledge Extraction from Black-Box Models

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.

Working Paper Quantitative Finance Verified 3,400 words

Neural Scaling Laws Formalized: Why Chinchilla Works (A Machine-Verified Derivation)

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.

Working Paper Machine Learning Verified 8,417 words

Provable Bounds on AI Self-Improvement: The Verification Oracle Ceiling

We establish formally verified bounds on recursive AI self-improvement under a spectral coupling model.

Draft Formal Verification Verified 9,347 words

Spectral Knowledge Distillation: From Black Box to Certified White Box

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.

Working Paper Machine Learning Verified 3,019 words

Spectral Model Compression: Provably Optimal Knowledge Extraction via Eigendecomposition

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.

Working Paper Machine Learning 1,911 words

Verified Transformer Dynamics: Token Clustering Convergence in Lean 4

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.

Working Paper Formal Verification Verified 7,466 words

The AI Safety Certificate: A Machine-Verified Framework for Quantitative AI Safety

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

Draft Machine Learning Verified 8,572 words

What Does Your Model Know? Spectral Decomposition and Arithmetic of Machine Learning Knowledge

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

Working Paper Machine Learning Verified 2,731 words

Spectral FX: Eigenvalue Classification of Currency Dynamics

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 $dX = AX\,dt + \Sigma\,dW$.

Working Paper Quantitative Finance Verified 4,041 words

Adaptive COS Option Pricing via Per-Mode Convergence Rates

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 characteristic function value.

Working Paper Quantitative Finance Verified 4,094 words

Extended Bellman Equations for Spectral Finance: Per-Mode Convergence, Shadow Prices, and Model-Free Bounds

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.

Working Paper Quantitative Finance Verified 5,403 words

When Q-Learning Meets Black-Scholes: A Machine-Verified Bridge Between Reinforcement Learning and Option Pricing

Reinforcement learning and quantitative finance independently discovered the same mathematical framework.

Draft Quantitative Finance Verified 7,990 words

Spectral Dynamic Programming: Per-Mode Convergence Rates and Computational Acceleration for American Options

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

Draft Quantitative Finance Verified 5,769 words

Eigenvalue Conditioning as Universal Optimizer: Cross-Domain Transfer Between Finance, Robustness, and Machine Learning

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

Draft Quantitative Finance Verified 7,094 words

Spectral Certificates for Trustworthy AI: Robustness, Confidence, and Fairness from One Decomposition

We prove that a single singular value decomposition (SVD) of a neural network's local Jacobian yields three formally verified trustworthiness guarantees simultaneously.

Working Paper Machine Learning Verified 5,323 words

Spectral Distillation: Provable Knowledge Compression from Black Box to Closed Form

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**.

Working Paper Machine Learning Verified 3,417 words

Pricing Basket Options via Eigenvalue-Conditional Black-Scholes Mixing

Pricing European basket options requires the distribution of a weighted sum of correlated lognormals — the Fenton Distribution (Nagy, 2026a) — which has no closed form.

Draft Quantitative Finance DOI 7,132 words

The Geometry of Risk: Spectral Distance and Topological Structure in Portfolio Space

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).

Working Paper Quantitative Finance Verified 5,482 words

The Information-Theoretic Cost of Risk Measurement

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 $n$ of the underlying risk factors.

Working Paper Quantitative Finance Verified 7,050 words

Correlation Is Not a Number — It's a Spectrum: Frequency-Domain Dependence for Portfolio Risk

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.

Working Paper Quantitative Finance Verified 4,439 words

The Spectral Halving Cycle: A Fourier Model of Bitcoin's Programmed Supply Shocks

Bitcoin is the only financial asset whose supply schedule is deterministic and programmed: every 210{,}000 blocks ($\approx$4 years), the block reward halves.

Working Paper Formal Verification Verified 3,048 words

Frequency-Domain Theory of Financial Economics: Thirteen Fundamental Results from One Decomposition

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

Working Paper Formal Verification Verified 7,009 words

Deterministic Portfolio VaR Without Monte Carlo: The Eigen-COS Method

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.

Working Paper Quantitative Finance Verified DOI 11,097 words

Noise-Free Risk: Deterministic VaR, ES, and Spectral Risk Measures for Lognormal Portfolios

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.

Working Paper Quantitative Finance Verified 7,919 words

The Geometry of Risk: Connecting Spectral Fenton to TQFT and Witten Invariants

Skeleton Quantitative Finance 595 words

The Cauchy-Schwarz Inequality for 2D Real Vectors via Sum-of-Squares Decomposition

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 $(ac + bd)^2 \leq (a^2 + b^2)(c^2 + d^2)$.

Proof Record Formal Verification Verified 1,193 words

Absolute Value Multiplicativity and Elementary Natural Number Arithmetic

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 $|ab| = |a| \cdot |b|$.

Proof Record Formal Verification Verified 1,482 words

Aero Spectral Plasma Confinement — OUTREACH STRATEGY

Short Draft Formal Verification Verified 1,807 words

A Machine-Checked Reduced Transport Law for Stochastic-Field-Line Confinement

We present a machine-checked reduced transport law for ion-scale turbulent confinement in tokamak plasmas.

Working Paper Formal Verification Verified 19,276 words

Falsification Protocol and a Quantitative Dimits-Bypass Prediction for Stochastic-Field-Line Transport

We test a kernel-verified reduced transport law against published tokamak confinement data.

Working Paper Physics Verified 7,594 words

Kernel-Verified Derived Physics: a Transferable Standard for Auditable Derivations with Pre-Registered Predictions

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

Working Paper Formal Verification Verified 10,539 words

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

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

Working Paper Formal Verification Verified 5,163 words

Algebraic Geometry of the Unified Field: Scheme-Theoretic Structures and Evolution Dynamics

We develop a formal scheme-theoretic framework for a unified field $U$ equipped with a fundamental parameter $\rho > 0$.

Short Draft Mathematics 2,044 words

Error Bounds for American Basket Option Pricing: A Unified Framework for COS-FW-GH Methods

We develop a unified error-bound framework for American basket option pricing that decomposes total numerical error into three controllable components: characteristic-function (COS) truncation error, forward-wave (FW) conditioning error, and Gauss-Hermite (GH) quadrature error.

Short Draft quantitative_finance Verified 2,077 words

Semi-Analytic Portfolio VaR via Eigenvalue-Conditioned COS Inversion

We present a deterministic, grid-free method for computing Value-at-Risk of portfolios whose assets follow correlated geometric Brownian motion. The portfolio value is a weighted sum of correlated lognormals — the Fenton Distribution — whose CDF has no closed form but whose characteristic function is exactly computable.

Short Draft Quantitative Finance 2,691 words

Paper Draft — Portfolio VaR via Eigenvalue-Conditioned Lognormal Mixtures

We present a method for computing Value-at-Risk of portfolios whose assets follow correlated geometric Brownian motion. The portfolio value is a weighted sum of correlated lognormals — the Fenton Distribution — whose CDF has no closed form.

Short Draft Quantitative Finance 4,114 words

The Spectral Fenton Distribution: Exact Portfolio VaR via Eigenvalue-Conditioned Fourier Inversion

We introduce the Spectral Fenton Distribution — a 130-parameter spectral representation of the distribution of weighted sums of correlated lognormals (the Fenton Distribution).

Short Draft Quantitative Finance 4,225 words

Pricing Basket Options via Eigenvalue-Conditioned Fourier Expansion

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.

Skeleton Quantitative Finance 300 words

The Spectral Fenton Distribution: Exact Portfolio Risk via Eigenvalue-Conditioned Fourier Inversion

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.

Skeleton Quantitative Finance 1,405 words

Eigen-COS VaR: Real-Time Exact Risk for High-Dimensional Portfolios

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 milliseconds.

Skeleton Quantitative Finance 307 words

Two Representations of the Fenton Distribution: Generative and Spectral Forms

The weighted sum of correlated lognormal random variables — the Fenton Distribution — has been considered to lack a closed-form CDF for over sixty years. We show this is a representation problem, not a mathematical one.

Short Draft Quantitative Finance 2,726 words

Arrow's Impossibility Theorem: Quantitative Extensions via Latent Framework

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-dictatorship.

Short Draft Core Theory Verified 1,715 words

Asset Pricing Anomalies as Latent SDF Decomposition: A Formal Derivation of Multi-Factor Models

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%.

Short Draft Quantitative Finance 1,827 words

Life: What, Why and How

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

Draft mathematical_biology Verified 9,172 words

The Thought Line: Fish Muscles and Composite Fibers

Skeleton Mathematics 539 words

Latent Folding: Resolution of Levinthal's Paradox via Grade Decomposition

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 milliseconds to seconds.

Short Draft bio_protein_fold Verified 2,796 words

Bio Protein Folding Dynamics — HANDOFF

Short Draft Core Theory 2,187 words

LatentFold — Business Plan & Product Strategy

Draft Core Theory Verified 5,321 words

Bounded Rationality and the Computational Complexity of Equilibria: A Spectral Perspective

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 with ρ < ρ*, exponential queries are necessary.

Short Draft Formal Verification Verified 2,692 words

Bridge Algebra: A Formal Framework for Cross-Domain Knowledge Transfer

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 $B$.

Short Draft Formal Verification Verified 1,513 words

Coherence Web: A Network Model for Knowledge Validation

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.

Short Draft theory_of_knowledge Verified 1,686 words

Divisibility of Consecutive Products

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 verified framework.

Proof Record number_theory Verified 1,160 words

Sharp Phase Transitions in Financial Contagion: A Formal Theory of Systemic Risk Cascades

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).

Short Draft quantitative_finance 2,731 words

Convolution–Correlation Duality: A Universal Principle for Spectral Damping

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 autocorrelation preserves them.

Short Draft Quantitative Finance Verified 2,747 words

Core Spectral Pattern Theory

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.

Draft Quantitative Finance 15,294 words

Core Spectral Phase Transition

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, independent of the system's dimension.

Short Draft Quantitative Finance Verified 4,165 words

Core The Latent — RESTRUCTURE LOG

Skeleton Physics 977 words

The Universal Representation Theorem: From Portfolio Risk to the Three-Body Problem

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.

Short Draft Quantitative Finance Verified 2,912 words

The Thought Line: Universal Smoothing and the Latent Bridge

Skeleton Core Theory 979 words

The Universal Smoothing Bridge

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.

Short Draft Formal Verification Verified 4,096 words

CSP–Latent Bridge: Random k-SAT Phase Transitions and Characteristic Function Analyticity

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.

Short Draft Core Theory Verified 1,896 words

Dark Matter: Formal Verification of Observational Evidence and Theoretical Constraints

We present a formally verified treatment of dark matter physics, covering both observational evidence and theoretical constraints. The Platonic proof system verifies 15 theorems spanning rotation curve anomalies, cluster dynamics, cosmological parameters, particle physics bounds, and alternative gravity transitions.

Proof Record dark_matter Verified 1,312 words

Divisibility and Parity in Natural and Integer Arithmetic

We present a collection of formally verified theorems concerning divisibility and parity in natural and integer arithmetic.

Short Draft Formal Verification Verified 1,973 words

Billion-Dollar Ideas from Analytical Physics Design

Short Draft Quantitative Finance 1,887 words

Analytical Design Optimization: From Governing Equations to 3D-Printable Geometry

Draft Research Methodology 6,220 words

Formally Verified Epidemic Thresholds: The SIR Model as a Grade-2 Dynamical System

The classical SIR (Susceptible-Infected-Recovered) epidemic model of Kermack and McKendrick (1927) is the foundation of mathematical epidemiology.

Short Draft mathematical_epidemiology Verified 3,359 words

Experience Fields: Path-Dependent Algebraic Structures for Learning Systems

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.

Short Draft Formal Verification Verified 2,092 words

Fin American Basket

We prove the first quantitative convergence guarantees for a deterministic American basket pricer with dimension-free cost.

Draft Quantitative Finance Verified 7,137 words

Fin Arcsinh Bs

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_-) - K e^{-rT} \Phi(d) $.

Draft Quantitative Finance Verified 8,413 words

Bayesian Live Risk

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.

Short Draft Quantitative Finance Verified 2,782 words

Formally Verified Financial Contagion Thresholds: Counterparty Default Cascades as a Grade-2 Dynamical System

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).

Short Draft mathematical_finance Verified 2,215 words

The Risk Coding Theorem: Exponential Convergence of Spectral Expected Shortfall

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.

Short Draft mathematical_finance Verified 2,555 words

The Fenton Distribution Solved

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.

Draft Quantitative Finance 9,665 words

Fin Fenton Spectral

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 CDF: the **Spectral Lognormal Distribution**.

Draft Quantitative Finance Verified DOI 10,471 words

Fin Generative Portfolio

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 tail structure.

Draft Quantitative Finance 5,628 words

Harvestability

We study a narrow horizon-dependent allocation question in a CRRA investor setting with Ornstein-Uhlenbeck eigenmodes. The paper centers the fin_harvestability function \[ h(T,\tau)=1-e^{-T/\tau}, \] interpreted as the closed-form horizon-dependent correction term around the myopic Merton benchmark.

Short Draft Formal Verification Verified 2,444 words

Calibrating Harvestability

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

Short Draft Quantitative Finance 2,369 words

Fin Markowitz

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.

Draft Quantitative Finance Verified 6,507 words

The Bias-Variance Frontier of Risk Estimation: When Spectral Methods Dominate Monte Carlo

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 Pareto frontier between bias and variance.

Short Draft finance_risk Verified 4,147 words

Fundamental Limits and Phase Transitions in Coherent Risk Estimation

What are the fundamental limits of estimating coherent risk measures from finite data? We establish four structural results.

Short Draft finance_risk Verified 4,912 words

Why Rational Investors Hold Different Portfolios

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.

Short Draft Quantitative Finance 4,609 words

Pricing Is Allocation Submission Contract

Skeleton Quantitative Finance 912 words

Fin Progressive Spectral

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.

Short Draft Quantitative Finance Verified 2,765 words

Fin Return Paradox

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.

Draft Quantitative Finance Verified DOI 8,108 words

Spectral Valuation of Compound Exchange Options under Stochastic Volatility

Skeleton Quantitative Finance 363 words

Spectral Importance Sampling for Rare Events in Correlated Systems

Skeleton Quantitative Finance 1,065 words

The Spectral Volatility Surface

We introduce a low-rank implied-variance surface in which total variance is represented as \[ w(k, T) = c(T) + \sum_{j=1}^{r} u_j(T)\cos(\omega_j k), \] with structural constraints enforced directly on the coefficient vectors. The resulting object is not a new stochastic-volatility law.

Short Draft Quantitative Finance Verified 3,346 words

Representation Theory of Finite Groups on the Unified Field

We present a formally verified framework for the representation theory of finite groups on the unified field U.

Short Draft pure_math Verified 1,952 words

Foundation Model Training Bounds: A Formally Verified Framework for Generalization and Scaling

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 guarantees for certificate-based quality metrics.

Short Draft ml_foundation_model Verified 1,500 words

Mirtill Adversarial Review — Gödel Universe (algebraic core)

Skeleton Formal Verification 1,110 words

A Machine-Verified Algebraic Formalization of Gödel's 1949 Counterexample

We give a machine-verified algebraic formalization of Gödel's (1949) construction in the Platonic proof kernel.

Draft math_physics 6,355 words

Gravity Theory Divergence: Newton vs MOND

We present a complete formal proof suite establishing the mathematical foundations of Modified Newtonian Dynamics (MOND) and its divergence from Newtonian gravity.

Short Draft Physics 3,822 words

Interaction Decay in ρ-Analytic Games: Grade Structure and Truncation Bounds

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}.

Short Draft Formal Verification Verified 2,000 words

Revenue Approximation in Mechanism Design via Grade Truncation

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/ρ^k for constants C > 0 and ρ > 1.

Short Draft Formal Verification Verified 1,679 words

Nash Equilibrium via Latent Perturbation Theory: Existence, Uniqueness, and Mean-Field Convergence

Short Draft game_theory Verified 1,970 words

Protein Folding as a Game: Nash-Boltzmann Duality and Computational Complexity

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.

Short Draft Formal Verification Verified 2,449 words

Heterogeneous Agent General Equilibrium: A Latent Framework Analysis of the Aiyagari-Bewley-Krusell-Smith Models

We present a formal framework connecting heterogeneous agent general equilibrium models to the Latent representation theory.

Short Draft macro_finance Verified 2,205 words

The Hodge Conjecture: A Formal Framework with Verified Cases and Structural Reductions

Short Draft Mathematics 3,481 words

Imported Infrastructure — Strip and Bound Arithmetic for Analytic Function Theory

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, sum positivity, and telescoping identities.

Proof Record Formal Verification Verified 939 words

Jet Numbers: A Verified Algebra of Derivatives

We present a formally verified treatment of jet numbers — algebraic objects that carry a value together with its derivatives to a fixed order. A 2-jet encodes $(f, f', f'')$ in a single element, with arithmetic operations following the Leibniz rule: $(fg)' = f'g + fg'$ and the generalized second-order formula $(fg)'' = f''g + 2f'g' + fg''$.

Short Draft Formal Verification Verified 1,898 words

Coalescent Structure and the Universal Latent Ratio

We present a formally verified treatment of Kingman's coalescent, establishing the fundamental rate and waiting time identities that govern backward-in-time genealogical processes.

Short Draft Core Theory Verified 1,870 words

Knowability Theory: Latent Generator Models and Spectral Pricing in Finance

We present a formally verified mathematical theory of *knowability* in financial modeling — the framework connecting unobservable latent generators to observable price dynamics.

Short Draft Quantitative Finance Verified 1,818 words

Latent Mollification Bridge

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 $\varepsilon$.

Short Draft Core Theory Verified 1,604 words

Fourier-Analytic Formalization of the Lonely Runner Conjecture

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.

Short Draft Formal Verification Verified 2,293 words

Lyapunov Stability of the Figure-Eight Three-Body Orbit via Z₃ Symmetry Decomposition

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.

Short Draft Physics Verified 2,117 words

Effective-α framework: LR k-point structure at SR exponents

Skeleton Quantitative Finance 1,097 words

Algebraic exploration of candidate pivotal-scaling formulas — kernel results

Skeleton Research Methodology 797 words

Newton task — Algebraic exploration of candidate pivotal-scaling formulas in the LR-to-SR crossover of percolation

Short Draft Formal Verification 1,966 words

The eta-negativity cascade: what a single sign constraint implies for three-dimensional percolation exponents

For Bernoulli bond percolation on $\mathbb{Z}^3$, the anomalous dimension $\eta$ is widely believed to be negative ($\eta \approx -0.046$ from Monte Carlo), but no rigorous proof exists.

Short Draft Formal Verification 3,108 words

The Convolution–Correlation Duality

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 damping factor to the spectral coefficients.

Draft mathematics Verified 15,319 words

The Decorrelation Index

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

Draft mathematics Verified 7,270 words

Hierarchical Feedback Percolation: When Structure Amplifies Dynamics

Draft Formal Verification Verified 11,567 words

Memory-Tree Algebra: Numbers That Remember Their Computation

We introduce the Memory-Tree Algebra $\mathcal{M}$, a structure in which each element is a rooted tree encoding its complete computational history.

Short Draft memory_tree 2,235 words

Meta-Algorithmizability: A Formal Theory of Proof Automation

We present a formal theory quantifying the extent to which mathematical proof is algorithmically tractable. Using empirical data from 26,583 proof traces across 226 mathematical domains, we establish eight core theorems (T1-T8) that characterize the structure of proof space.

Short Draft Formal Verification 1,716 words

Meta Latent Computation

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.

Short Draft Machine Learning 4,917 words

Meta Pvsnp Platonic — ASSUMPTION LEDGER

Short Draft Formal Verification Verified 3,715 words

Meta Pvsnp Platonic — ROADMAP scientific strengthening

Short Draft Formal Verification 1,876 words

Meta Pvsnp Platonic — TRANSITION CHECKLIST

Proof Record Formal Verification Verified 1,303 words

A Machine-Verified Proof of P ≠ NP via Partition Function Analyticity

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.

Short Draft Formal Verification Verified 3,575 words

Grade-2 Dominance Across Millennium Problems: A Unified Structural Framework for RH and Navier–Stokes

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.

Short Draft Mathematics Verified 2,990 words

The Fishing Rod Proverb Is Half-Right

Short Draft Quantitative Finance 2,351 words

Unified Latent Formulations of Millennium Problems

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.

Short Draft Core Theory 1,854 words

Central Extracted Material Bank

Skeleton Research Methodology 229 words

Applied Knowledge Algebra: Operations, Semantics, and Use Cases

Knowledge Artifacts (Nagy 2026a) support algebraic operations in a shared spectral eigenbasis.

Short Draft Machine Learning 1,766 words

A Numerical Demonstration of Knowledge Artifact Extraction and Algebra Across 26 Model Types

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.

Short Draft Quantitative Finance 1,988 words

SpectralShell: A Universal Model Class for Extracted Knowledge

We describe SpectralShell, a universal model class that loads, predicts from, and performs algebra on Knowledge Artifacts — portable spectral representations of trained ML models.

Short Draft Machine Learning 1,723 words

Ml Knowledge Artifacts Algebra

A 200-tree Random Forest has 126,074 parameters. Its knowledge? Three numbers and a basis. We introduce the **Knowledge Artifact** — a portable representation of what any ML model has learned — and the **Knowledge Algebra** — provably exact arithmetic on these artifacts.

Draft Machine Learning DOI 6,457 words

SGD Is Right: A Machine-Checked Proof That Stochastic Gradient Descent Converges

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.

Draft Formal Verification Verified 6,381 words

Ml Spectral Intelligence

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: 1.

Short Draft Machine Learning Verified 4,393 words

Nature's Latent: The Dimensionality of Optimal Biological Forms

Natural structures—from honeycomb tessellations to nautilus spirals—exhibit striking geometric regularity despite arising from evolution rather than engineering.

Short Draft Core Theory Verified 2,207 words

Navier-Stokes Regularity as a Spectral Problem

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.

Short Draft Formal Verification Verified 2,181 words

Grade-2 Universality in Planetary Defense: Cross-Domain Bridges for NEO Deflection and Debris Cascade

We establish cross-domain mathematical bridges connecting near-Earth object (NEO) deflection, orbital debris dynamics, and fluid mechanics through a formally verified framework.

Short Draft planetary_defense Verified 2,404 words

Network Latent Structure

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.

Short Draft Core Theory Verified 2,242 words

Neutrino Mass: Formal Verification of Oscillation, Seesaw, and Cosmological Constraints

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.

Short Draft phy Verified 2,230 words

Verified Nelson-Siegel COS Hybrid Pricing Framework

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.

Short Draft quantitative_finance Verified 1,533 words

Unconditional Birch-Swinnerton-Dyer for CM Elliptic Curves at Rank 0: A Formally Verified Proof

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 $$L_{\mathrm{lead}}(E) \;=\; \frac{\Omega_E \cdo

Short Draft number_theory Verified 3,148 words

A Spectral Reformulation of Birch-Swinnerton-Dyer: Formalized Prototype and Open Directions

We record a spectral reformulation of the Birch-Swinnerton-Dyer conjecture and its formalization in the Lean 4 kernel.

Short Draft number_theory Verified 1,920 words

Empirical Proof Complexity of a Lean 4 Formalization of Birch-Swinnerton-Dyer

We record empirical complexity measurements for the current Lean 4 formalization of the Birch-Swinnerton-Dyer conjecture and its surrounding theory.

Short Draft proof_complexity Verified 1,688 words

Carry Dynamics of the Syracuse Map: Exact Identities, Rank-One Transitions, and Mersenne Phase Transitions

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.

Short Draft Research Methodology 4,466 words

Nt Collatz Cycle Elimination — SUBMISSION ARTIFACTS

Skeleton Formal Verification 1,256 words

Nt Collatz Lyapunov — dual space framework

Short Draft Mathematics 1,579 words

Syracuse-Compatible Lyapunov Functions: Characterization, β-Halving, and Spectral Analysis

Draft Formal Verification Verified 6,047 words

The Thought Line: Mesoscopic Damping for Vinogradov Sums

Skeleton Research Methodology 645 words

Nt Damped Vinogradov — email biro andras

Skeleton Research Methodology 314 words

Nt Damped Vinogradov — email hungarian nt SEND

Skeleton Research Methodology 398 words

Nt Damped Vinogradov — email maga peter

Skeleton Research Methodology 313 words

Nt Damped Vinogradov — email pintz janos

Skeleton Research Methodology 319 words

Nt Damped Vinogradov — email toth arpad

Skeleton Research Methodology 309 words

Explicit L² Bounds for Damped Vinogradov Sums

Draft Mathematics DOI 7,298 words

Goldbach Proof: Complete Derivation and Gap Analysis

Short Draft Formal Verification 3,822 words

A Reduction of Goldbach's Conjecture to an Auditable Trust Surface: Classical Primitives, Zero Open Analytic Atoms, and a Machine-Checked Envelope

We present a reduction of the binary Goldbach conjecture whose entire external dependency surface is explicit, auditable, and machine-checked in Lean 4.

Draft number_theory Verified 7,975 words

The Convolution Convergence Theorem: A Direct Conditional Proof of Goldbach's Conjecture

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.

Proof Record number_theory Verified 1,207 words

Latent Grade-2 Dominance and the Moment Hypothesis for All k

We show that the Moment Hypothesis for the Riemann zeta function — $m_{2k}(T) \leq C_k (\log T)^{k^2}$ for all $k \geq 1$ — follows from a single structural property of the cumulant generating function (CGF): its analyticity on a disk of radius $R > 0$ (equivalently, Latent diagnostic $\rho > 1$).

Short Draft analytic_number_theory 1,683 words

Nt Rh De Branges Chain — ATTACK MAP

Short Draft Mathematics 3,693 words

Phase 11 Week 3 Day 4 Handoff — RH / M1 / nt_rh_de_branges_chain

Working Paper Mathematics 3,225 words

Joker Phase 11 — 5 structural escape routes for the §15 M1 obstruction

Working Paper Mathematics 4,707 words

M1 — Mesoscopic-decay extremal function (working file)

Working Paper Mathematics Verified 87,634 words

Mathematical Closure Audit of the De Branges / BK / Szegő Chain

Draft Formal Verification 20,932 words

An Unconditional BGST$\to$R$_2$ Fourier Transfer via Poisson-Kernel Deconvolution

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 $\alpha \in [0,1]$.

Short Draft number_theory Verified 4,073 words

A Poisson L² Lower Bound for Weil-Square Test Functions at Mesoscopic Bandwidth

Let `L = L(T) > 0` and `η_*(T) = (1/2 + η/2) \log\log T / \log T` for `T \to \infty` and a fixed `η \in (0, 1/2)`.

Short Draft number_theory Verified 4,260 words

A de Branges chain formalization toward the Riemann Hypothesis: architecture and explicitly axiomatic analytic steps

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 an $\varepsilon$-argument.

Working Paper Mathematics Verified 7,638 words

The Shifted Divisor Problem, the Moment Hypothesis, and a Conditional Path to the Riemann Hypothesis

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. **I.

Short Draft Mathematics Verified 3,312 words

The Density–Individual Divide: Formalizing the GUE → RH Gap

Short Draft Mathematics 1,734 words

The Mesoscopic Barrier for Montgomery's Pair Correlation Conjecture

Short Draft Mathematics Verified 3,153 words

Lean Zulip — Community Guidelines & Posting Strategy

Proof Record Formal Verification Verified 338 words

Draft Message to Nicolas Cerrajero

Proof Record Quantitative Finance Verified 283 words

Conditional Regularity of 3D Navier-Stokes via Latent Spectral Analysis and Energy-Normalized Bilinear Bounds

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.

Short Draft mathematical_physics Verified 2,267 words

PDE Tensor Algebra SSOT — Lossless Refactor Plan

Short Draft Quantitative Finance 1,572 words

Venue-split decision — main paper + short letter

Skeleton Quantitative Finance 1,433 words

Rigorous $-1$ exponent for the laminar-turbulence transition amplitude in a spectrally stratified 1D PDE

In a spectrally stratified 1D mixed advection–reaction PDE used as a diagnostic model for laminar-to-turbulent transitions, we derive a rigorous exponent $-1$ for the initial-condition amplitude threshold $\Pi_c$ in the lowest active wavenumber $k_{\min}$.

Draft Physics 5,013 words

Preregistered prediction — K-class external Π_c, frozen law

Skeleton pde_tensor_algebra 1,344 words

A Classification Theorem for PDE Solvability via Dissipation–Coupling–Constraint Decomposition

We introduce a structural decomposition of partial differential equations into a triple $(D, C, P)$ of dissipation, coupling and constraint operators and prove that the solvability behavior of the system is determined by a single dimensionless quanti

Draft Research Methodology Verified 5,655 words

The Coupling Algebra of Nonlinear PDE Systems: A Lie Structure on $(D, C, P)$ Triples

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 constraint projector.

Short Draft Formal Verification 2,683 words

Universal Difficulty Exponents in Dissipative PDE Systems: A Renormalization-Group View of the (D, C, P) Framework

**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.

Draft Physics 11,309 words

General Relativity as a $(0, C, P)$ System: Penrose Singularity as Difficulty Divergence

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.

Short Draft Physics Verified 3,397 words

Navier–Stokes Regularity via Leray Bilinear Spreading and Two-Level Decoherence

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.

Working Paper Physics 2,007 words

The PDE Tensor Algebra: Structural Decomposition, Coupling Classification, and Exact Recombination of Differential Equations

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.

Working Paper Research Methodology 1,863 words

Tensor Learning: Data-Driven Recovery of the (D, C, P) Decomposition and PDE Solvability Class

We propose a hybrid algorithm that recovers the structural $(D, C, P)$ decomposition of a nonlinear PDE system from time-series observations and, from this decomposition, predicts the solvability class and the difficulty parameter $\mathcal{D} = \lVe

Short Draft Formal Verification Verified 4,277 words

Two Millennium Problems Through the PDE Tensor Algebra: Navier–Stokes Regularity and the Yang–Mills Mass Gap

We present a unified framework — the PDE Tensor Algebra — that addresses two Clay Mathematics Institute Millennium Prize Problems through a single structural principle.

Short Draft Physics 1,749 words

Universal Difficulty Exponents in Advective-Dissipative PDE Systems

We show that the scale-dependent difficulty of any advective–dissipative PDE system obeys a universal power law $\mathcal{D}(k) \sim k^{-1}$ in the inertial range, independent of the solution's spectral structure.

Skeleton Physics 1,329 words

The Yang–Mills Mass Gap as a Difficulty Transition in the PDE Tensor Algebra

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 $\mathcal{D} = \|C\|_F / \lambda_{\min}(D) = \infty$.

Short Draft Physics 2,783 words

Phase Field Algebras: Order, Smoothness, and Renormalization Group Fixed Points

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 value projections.

Short Draft math_physics Verified 1,671 words

Unified Celestial Mechanics Latent Monograph — Structure Plan

Short Draft Physics Verified 3,260 words

Three Cracks in the Standard Model of Cosmology: A Formal Framework for Tension Quantification, Model Incompleteness, and Resolution Pathways

Short Draft cosmology, mathematical physics, foundations Verified 4,004 words

Spectral Compression of Charged Black Holes: ρ(Q) for Kerr-Newman

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.

Skeleton Physics 595 words

Humanize + Tamás Style — phy_knds_kappa_ordering_vieta/paper.md

Proof Record Formal Verification Verified 1,325 words

Mirtill adversarial review — KNdS κ₊ < κ₋ via Vieta

Short Draft Formal Verification Verified 1,678 words

Mirtill adversarial review v2 — phy_knds_kappa_ordering_vieta/paper.md

Short Draft Formal Verification Verified 2,148 words

An Analytic Proof of $\kappa_+ < \kappa_-$ for Non-Degenerate Rotating Kerr-Newman-de Sitter Black Holes via Vieta Root Relations

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

Draft mathematical-physics Verified 7,074 words

Phy M Theory Dimensions — mathoverflow post

Skeleton Research Methodology 550 words

Near-Koide Structure Across the Standard Model Fermion Spectrum

We extend the Koide structure analysis of our companion paper [I] beyond the charged lepton sector.

Short Draft Quantitative Finance 3,143 words

Grade Decomposition and Gevrey Regularity for Navier-Stokes: A Machine-Checked Path to the Millennium Prize

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.

Working Paper Formal Verification Verified 10,362 words

Signed Energy Transfer and the Viscous Floor: A Statewise Mechanism for Galerkin Navier-Stokes

We study the Gevrey regularity of solutions to the 3D incompressible Navier-Stokes equations on the torus $\mathbb{T}^3$.

Working Paper Research Methodology 4,437 words

Navier-Stokes Regularity Program — Overview

Short Draft Formal Verification Verified 1,789 words

Global Regularity for the Three-Dimensional Navier-Stokes Equations via Direction Coherence

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.

Draft mathematical physics 8,392 words

Eighteen Theorems: A Machine-Verified Conditional Framework for Navier-Stokes Regularity

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 failures.

Short Draft mathematical physics 3,499 words

Global Regularity for the Three-Dimensional Incompressible Navier-Stokes Equations via Perturbation Regularity and Direction Coherence

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.

Draft mathematical physics 16,563 words

Montgomery Mission — Current State

Short Draft Formal Verification Verified 1,917 words

An Exact Algebraic Bifurcation in the Triangle-Plus-Center Central Configuration

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 $$\mu^* = \frac{81 + 64\sqrt{3}}{249},$$ the unique positiv

Short Draft Physics 3,083 words

Montgomery's Four Questions and the Grade Hierarchy of the N-Body Problem

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.

Draft Physics Verified 11,844 words

The Analyticity Parameter of Neutron Star Post-Merger Oscillations

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.

Skeleton Physics 578 words

Pseudospectral Robustness of the Analyticity Parameter ρ

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 measurements.

Skeleton Physics 828 words

Phy Qnm Latent — PAPER IDEAS

Skeleton Quantitative Finance 418 words

The Thought Line: Black Hole Ringdown Has a Universal Compression Constant

Skeleton Quantitative Finance 1,082 words

Publication Strategy — QNM Latent

Skeleton Core Theory 389 words

Phy Qnm Latent — email berti draft

Skeleton Machine Learning 307 words

Phy Qnm Latent — email cardoso draft

Skeleton Machine Learning 281 words

Phy Qnm Latent — email frei draft

Skeleton Core Theory 206 words

Phy Qnm Latent — email gergely draft

Skeleton Core Theory 203 words

Phy Qnm Latent — email isi draft

Skeleton Core Theory 279 words

Phy Qnm Latent — email racz draft

Skeleton Core Theory 242 words

Phy Qnm Latent — email stein draft

Skeleton Physics 233 words

Formally Verified Black Hole QNM Theory: Machine-Checked Proofs for Spectral Sufficiency

We present the first machine-verified formalization of black hole quasinormal mode (QNM) spectral sufficiency theory.

Proof Record Formal Verification Verified 1,474 words

Universal Spectral Compression of Black Hole Ringdown

Short Draft Physics 1,503 words

Phy Quantum Tuna9 Parity — email draft 20260418 delivery

Skeleton Machine Learning 1,470 words

Reproducibility Package — Quantum USRT Spectral Dichotomy

Short Draft Core Theory Verified 1,596 words

Companion Papers Plan — Quantum USRT Publication Strategy

Short Draft Physics Verified 1,870 words

A Spectral Dichotomy for Quantum Computing Feasibility: Effective Budget, Advantage Frontier, and Observable-Level Mitigation

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 $v_{LR}$.

Short Draft Physics Verified 4,784 words

Outreach Drafts — Prospective Validation Collaborations

Short Draft Quantitative Finance Verified 2,402 words

Tuna-9 Calibration Battery — Self-measured Hardware Parameters

Short Draft Physics 3,286 words

Unique Testable Prediction: The Contested-Band Discrimination Protocol

Short Draft Physics 1,744 words

Change Note — Smale 6 Finiteness Paper

Skeleton Formal Verification 1,081 words

Phy Smale6 — email draft 20260418 delivery

Skeleton Formal Verification 1,161 words

Phy Smale6 — email draft 20260423 albouy

Proof Record Physics Verified 1,463 words

Phy Smale6 — email draft 20260423 moeckel

Skeleton Mathematics 1,196 words

Strong Cosmic Censorship at the VCS Boundary: Inverted Polarity, Charge-Spin Duality, and Kerr-Newman-de Sitter

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 extension past the Cauchy horizon.

Draft mathematical-physics Verified 6,032 words

Phy Turbulence Scaling Grade — attack

Short Draft Physics 1,986 words

First-Principles Kolmogorov Constant from the Local Analyticity Radius, and the Structural Disqualification of Gamma Intermittency Models

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.

Short Draft Physics Verified 3,593 words

Mirtill Adversarial Review: Vorticity as a Universal Controller

Short Draft Physics 2,380 words

Vorticity as a Universal Controller: A Structural Bridge Between Fluid Regularity and Spacetime Causality

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

Draft mathematical-physics Verified 11,792 words

Weak Cosmic Censorship as a Vorticity-Controlled System: Penrose Inequality, Self-Correcting Dissipation, and the Navier-Stokes Structural Transfer

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.

Short Draft mathematical-physics Verified 2,400 words

arXiv Abstract — Yang-Mills Existence and Mass Gap

Proof Record Mathematics Verified 687 words

Existence and Mass Gap for Yang-Mills Theory on R^4

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

Draft Physics Verified 25,669 words

Cover Letter — Clay Millennium Prize Submission: Yang-Mills Existence and Mass Gap

Proof Record Formal Verification Verified 1,295 words

Executive Summary — Yang-Mills Existence and Mass Gap

Short Draft Quantitative Finance 1,505 words

Phy Yang Mills Mass Gap — explanation

Proof Record Research Methodology Verified 705 words

Phy Yang Mills Mass Gap — explanation noncollapsing

Skeleton Quantitative Finance 799 words

Lean 4 Export Snapshot — Yang-Mills Clay Submission

Draft Formal Verification Verified 6,831 words

Phy Yang Mills Mass Gap — millennium dependency audit

Draft Formal Verification 13,466 words

Technical Proof Note: Gap Monotonicity via Gauge Absorption (research-only)

Short Draft Formal Verification 2,120 words

Technical Appendix for Referees — Yang-Mills Mass Gap

Draft Formal Verification Verified 7,923 words

Existence and Mass Gap for Yang-Mills Theory on R^4 — Extended Working Paper

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 $\Delta > 0$.

Draft Mathematics 7,119 words

Spectral Bridges in Population Genetics

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).

Short Draft Formal Verification Verified 1,832 words

Processus-Algebra: Numbers as Dynamical Systems

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.

Short Draft Formal Verification Verified 2,023 words

Propositional Logic Combinators: Machine-Verified Proofs in the Platonic Kernel

We present a collection of formally verified theorems in propositional logic, proved in the Platonic proof kernel and exported to Lean 4. The theorems include fundamental combinators (identity, weakening, syllogism, flip, S-combinator, compose3, diamond) and the classical Peirce's law.

Proof Record Formal Verification Verified 1,319 words

Publications — EFFECTIVE RHO

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.

Short Draft Quantitative Finance Verified 4,099 words

Publications — NOUS ANOMALY DETECTION

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.

Draft Formal Verification Verified 5,589 words

Publications — NOUS SPECTRAL RESOLUTION CAPSTONE

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

Short Draft Quantitative Finance Verified 4,805 words

Publications — NOUS SPECTRAL TRANSFER CAPSTONE

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.

Draft Core Theory Verified 5,108 words

Publications — SPECTRAL OVERFITTING

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.

Short Draft Formal Verification Verified 4,172 words

The Pump Cycle Bridge: Structural Isomorphism Between Gravitational Singularities and Debris Cascades

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 binary subsystems.

Short Draft Physics 1,817 words

QNM Cross-Kernel Bridge Theorems: Unifying Ringdown, Pseudospectrum, and Latent Structures

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.

Short Draft Core Theory 1,994 words

Formal Analysis of QNM Spectral Ratio Universality in the Extremal Kerr Limit

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 $\rho \approx 3$ is preserved.

Short Draft black_hole_physics Verified 1,804 words

Quantum Gravity Bounds: Theory-Independent Constraints on Quantum Gravity

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.

Short Draft Physics Verified 2,537 words

Reference Extract: Acerbi (2002) — Spectral Measures of Risk

Skeleton Quantitative Finance 606 words

Resonance Algebra: A Formal Framework for Vibrating Mathematical Objects

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.

Short Draft math_physics Verified 2,496 words

Rough Volatility: The Exponent Architecture of Options Markets

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.

Short Draft Quantitative Finance Verified 3,564 words

Formal Verification of Sakharov's Baryogenesis Conditions: Why Matter Exists

We present a formally verified proof of Sakharov's three conditions for baryogenesis and their quantitative consequences.

Short Draft particle physics, cosmology Verified 2,696 words

Formal Foundations of Stochastic Gradient Descent

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.

Short Draft Formal Verification Verified 2,338 words

Solar Cell Efficiency Bounds: A Formal Framework for Thermodynamic Limits and Improvement Strategies

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.

Short Draft Physics Verified 1,756 words

Spacetime Thermodynamics: Gravity as Entropy

Short Draft mathematical_physics 2,389 words

The Strong CP Problem: A Formal Treatment of QCD CP Violation and Axion Solutions

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.

Short Draft Quantitative Finance 3,145 words

Tension Algebra: A Mathematical Framework for Disequilibrium Dynamics

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 current, trade flows.

Short Draft tension_algebra Verified 2,098 words

Tools — LATEX FIXUP PROMPT

Skeleton Quantitative Finance 546 words

Tools — MD AUTHORING GUIDE

Skeleton Quantitative Finance 677 words

Tools — README

Skeleton Formal Verification 520 words

Topological Numbers: Path-Based Arithmetic via Homotopy

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 path reversal.

Short Draft Formal Verification Verified 1,660 words

Triangular Square Numbers and the Pell Equation

We establish a complete formal characterization of triangular square numbers via the Pell equation $x^2 - 8y^2 = 1$. A triangular number $T(n) = n(n+1)/2$ is a perfect square if and only if $(2n+1, 2m)$ satisfies this Pellian relation.

Skeleton number_theory 1,368 words

Tropical-Spectral Hybrid Algebra

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.

Short Draft Quantitative Finance Verified 1,939 words

Structural Results Toward the Twin Prime Conjecture via Mod-6 Obstruction Theory

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.

Short Draft Mathematics 2,230 words

Unconditional Results: Deriving Latent Conditions from First Principles

Short Draft Core Theory Verified 2,478 words

Unified Embeddings Extended

We construct formal embeddings of eight algebraic structures into a universal type $\mathcal{U}$ equipped with a canonical set of observables.

Short Draft Formal Verification Verified 2,163 words

Universal Foundations: A Verified Library of Core Mathematical Lemmas

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.

Proof Record foundations Verified 1,418 words