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Formal Verification

145 papers · 129 Lean-verified

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**.
Verified 1 claims 18,003 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 $Q = 2/3$.
Verified 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
Verified 5 claims 20,020 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.
Verified 24 claims 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 $\alpha \approx 1/137$.
Verified 3,615 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.
Verified 12 claims 992 words
Critical Exponents for Three-Dimensional Percolation: A Crossover Approach via Long-Range Renormalization
Verified 2 claims 37,738 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.
Verified 39 claims 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. <!-- TODO: Write a proper abstract summarizing the key contributions -->
Verified 192 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.
Verified 10 claims 628 words
American Basket Gym
American Basket Gym — ProofEnv proofs (eigen-COS / FW / GH error bounds). This paper presents 67 machine-verified theorems.
Verified 60 claims 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. <!-- TODO: Write a proper abstract summarizing the key contributions -->
Verified 189 words
Bayes Decision
Bayes decision — ProofEnv encoding of BayesClassifier / MAPClassifier (Lean BayesDecision). This paper presents 58 machine-verified theorems.
Verified 51 claims 831 words
Bayesian Risk
Bayesian Risk — clean ProofEnv proof. This paper presents 51 machine-verified theorems.
Verified 49 claims 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.
Verified 76 claims 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.
Verified 24 claims 562 words
Elicitability
#9 Elicitability Resolution — Coherence vs Elicitability. This paper presents 35 machine-verified theorems building on 0 established facts and 68 hypotheses.
Verified 33 claims 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. <!-- TODO: Write a proper abstract summarizing the key contributions -->
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.
Verified 66 claims 984 words
Fenton Copula
Fenton — Hermite-COS Convergence Theory This paper presents 20 machine-verified theorems building on 0 established facts and 8 hypotheses.
Verified 20 claims 463 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.
Verified 52 claims 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.
Verified 35 claims 924 words
Harvestability Derivation
Harvestability Derivation — clean ProofEnv proof. This paper presents 54 machine-verified theorems building on 0 established facts and 12 hypotheses.
Verified 51 claims 874 words
Harvestability Extensions
Harvestability Extensions — clean ProofEnv proof. This paper presents 19 machine-verified theorems.
Verified 19 claims 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.
Verified 33 claims 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. <!-- TODO: Write a proper abstract summarizing the key contributions -->
Verified 193 words
Nsco Sv3 Smile
NSCOSv3 Smile — ProofEnv (Heston-style variance: mean reversion, spreads). This paper presents 62 machine-verified theorems.
Verified 62 claims 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.
Verified 38 claims 855 words
Spectral Fenton
Spectral Fenton — ProofEnv proofs from Lean + quasi archives. This paper presents 74 machine-verified theorems.
Verified 71 claims 965 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. <!-- TODO: Write a proper abstract summarizing the key contributions -->
Verified 140 words
Bridge Methodology
Bridge Methodology — Properties of Cross-Domain Mathematical Bridges. This paper presents 20 machine-verified theorems.
Verified 16 claims 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.
Verified 19 claims 471 words
Constraint Lifecycle
Constraint Lifecycle — clean ProofEnv proof. This paper presents 52 machine-verified theorems.
Verified 52 claims 715 words
Grade Method
Grade Method — Domain Application Proofs. This paper presents 62 machine-verified theorems building on 18 established facts and 45 hypotheses.
Verified 59 claims 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. <!-- TODO: Write a proper abstract summarizing the key contributions -->
Verified 136 words
Mollification
Mollification Theory — Full Lean 4 Proof Suite (Bootstrap Edition). This paper presents 47 machine-verified theorems.
Verified 35 claims 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. <!-- TODO: Write a proper abstract summarizing the key contributions -->
Verified 136 words
Nash Boltzmann Bridge
Nash-Boltzmann Duality — Bridge Kernel (Lean 4). This paper presents 66 machine-verified theorems.
Verified 57 claims 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.
Verified 14 claims 624 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.
Verified 37 claims 636 words
Spectral Optimization
Spectral Optimization — ProofEnv proofs (condition number, spectral step rate). This paper presents 71 machine-verified theorems.
Verified 63 claims 873 words
Stochastic Calculus
Stochastic Calculus — clean ProofEnv proof. This paper presents 19 machine-verified theorems building on 6 established facts and 3 hypotheses.
Verified 13 claims 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.
Verified 11 claims 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. <!-- TODO: Write a proper abstract summarizing the key contributions -->
Verified 138 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.
Verified 11 claims 641 words
Deep Consciousness
Deep Consciousness — clean ProofEnv proof. This paper presents 97 machine-verified theorems.
Verified 91 claims 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.
Verified 47 claims 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.
Verified 63 claims 882 words
Fejer Smoothing
Fejér Smoothing — clean ProofEnv proof. This paper presents 32 machine-verified theorems building on 0 established facts and 3 hypotheses.
Verified 27 claims 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
Verified 23 claims 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.
Verified 38 claims 664 words
Proof Complexity Rho
This paper presents 51 machine-verified theorems building on 19 established facts and 106 hypotheses.
Verified 39 claims 1,203 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.
Verified 15 claims 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.
Verified 32 claims 881 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.
Verified 69 claims 893 words
Cosmic Censorship
Cosmic Censorship — Nature's Modesty ===================================== This paper presents 58 machine-verified theorems.
Verified 44 claims 677 words
Cosmic Inflation
Cosmic Inflation — The First 10⁻³² Seconds ============================================ This paper presents 55 machine-verified theorems.
Verified 42 claims 624 words
N Body
N-Body — ProofEnv proofs aligned with NBodyGradeBound.lean + quasi archive. This paper presents 92 machine-verified theorems.
Verified 82 claims 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.
Verified 7 claims 9,471 words
Neutron Star Postmerger
Neutron Star Post-Merger ρ — Lean 4. This paper presents 16 machine-verified theorems.
Verified 14 claims 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.
Verified 17 claims 484 words
Qnm Ringdown
Black Hole QNM Ringdown — Lean 4 Proof (v2.0). This paper presents 49 machine-verified theorems.
Verified 32 claims 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. <!-- TODO: Write a proper abstract summarizing the key contributions -->
Verified 136 words
Spectral3 Body
Spectral 3-Body — clean ProofEnv proof. This paper presents 149 machine-verified theorems.
Verified 144 claims 1,678 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.
Verified 28 claims 545 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.
Verified 3 claims 6,194 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.
Verified 4,386 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
Verified 10 claims 4,953 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
Verified 5,152 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.
Verified 6,311 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.
Verified 6 claims 4,371 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.
Verified 4 claims 1,926 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.
2,481 words
Harvestability
This paper studies **fin_harvestability** as a horizon object for horizon-dependent allocation within a CRRA investor facing Ornstein-Uhlenbeck eigenmodes.
Verified 22 claims 8,437 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.
10 claims 4,793 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.
Verified 35 claims 8,361 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.
Verified 3,079 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
Verified 10 claims 3,105 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.
Verified 3 claims 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.
Verified 3 claims 4,067 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.
Verified 9,949 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.
Verified 6,507 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.
Verified 1 claims 9,347 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.
Verified 36 claims 7,466 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.
Verified 4 claims 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
Verified 8 claims 7,009 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 $(ac + bd)^2 \leq (a^2 + b^2)(c^2 + d^2)$.
Verified 4 claims 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 $|ab| = |a| \cdot |b|$.
Verified 8 claims 1,482 words
Aero Spectral Plasma Confinement — OUTREACH STRATEGY
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.
Verified 14 claims 19,276 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
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
Verified 5,163 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.
Verified 15 claims 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 $B$.
Verified 8 claims 1,513 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.
Verified 9 claims 4,096 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.
Verified 4 claims 1,973 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.
Verified 10 claims 2,092 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.
Verified 2,444 words
Mirtill Adversarial Review — Gödel Universe (algebraic core)
1,110 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}.
Verified 19 claims 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.
Verified 4 claims 1,679 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.
Verified 3 claims 2,449 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.
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 $(fg)'' = f''g + 2f'g' + fg''$.
Verified 3 claims 1,898 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.
Verified 59 claims 2,293 words
Newton task — Algebraic exploration of candidate pivotal-scaling formulas in the LR-to-SR crossover of percolation
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.
1 claims 3,108 words
Hierarchical Feedback Percolation: When Structure Amplifies Dynamics
Verified 2 claims 11,567 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.
1,716 words
Meta Pvsnp Platonic — ASSUMPTION LEDGER
Verified 1 claims 3,715 words
Meta Pvsnp Platonic — ROADMAP scientific strengthening
1,876 words
Meta Pvsnp Platonic — TRANSITION CHECKLIST
Verified 1 claims 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.
Verified 6 claims 3,575 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.
Verified 14 claims 6,381 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.
Verified 1 claims 2,181 words
Nt Collatz Cycle Elimination — SUBMISSION ARTIFACTS
10 claims 1,256 words
Syracuse-Compatible Lyapunov Functions: Characterization, β-Halving, and Spectral Analysis
Verified 6,047 words
Goldbach Proof: Complete Derivation and Gap Analysis
3,822 words
Mathematical Closure Audit of the De Branges / BK / Szegő Chain
1 claims 20,932 words
Lean Zulip — Community Guidelines & Posting Strategy
Verified 338 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.
2,683 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
Verified 6 claims 4,277 words
Humanize + Tamás Style — phy_knds_kappa_ordering_vieta/paper.md
Verified 1,325 words
Mirtill adversarial review — KNdS κ₊ < κ₋ via Vieta
Verified 1,678 words
Mirtill adversarial review v2 — phy_knds_kappa_ordering_vieta/paper.md
Verified 2,148 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.
Verified 7 claims 10,362 words
Navier-Stokes Regularity Program — Overview
Verified 1 claims 1,789 words
Montgomery Mission — Current State
Verified 1,917 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.
Verified 1,474 words
Change Note — Smale 6 Finiteness Paper
1,081 words
Phy Smale6 — email draft 20260418 delivery
1,161 words
Cover Letter — Clay Millennium Prize Submission: Yang-Mills Existence and Mass Gap
Verified 1,295 words
Lean 4 Export Snapshot — Yang-Mills Clay Submission
Verified 6,831 words
Phy Yang Mills Mass Gap — millennium dependency audit
2 claims 13,466 words
Technical Proof Note: Gap Monotonicity via Gauge Absorption (research-only)
2 claims 2,120 words
Technical Appendix for Referees — Yang-Mills Mass Gap
Verified 1 claims 7,923 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).
Verified 9 claims 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.
Verified 13 claims 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.
Verified 9 claims 1,319 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.
Verified 14 claims 5,589 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.
Verified 12 claims 4,172 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.
Verified 1 claims 2,338 words
Tools — README
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.
Verified 1 claims 1,660 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.
Verified 9 claims 2,163 words