Mathematical Research Program · March 2026

The Latent

Investigating the hidden finite-dimensional representations that govern observable structure — from the distribution of prime numbers to portfolio risk, gravitational dynamics, and the architecture of neural networks.

Dr. Tamás Nagy  ·  ORCID

24 Papers Lean 4 Verified 5 Domains Open Access (CC-BY 4.0)

Research Program

The central question is always the same: what latent object, once shown to exist, forces the visible pattern into place?

Number Theory 3 papers

The Riemann Hypothesis

Machine-verified proof chain: MH → Latent → RH. Two independent paths, Lean 4 formalization (zero sorry), 15 numerical tests — all pass.

Quantitative Finance 9 papers

Portfolio Risk & Option Pricing

Exact Portfolio VaR without Monte Carlo. Spectral lognormal distribution. Machine-verified Black–Scholes. Basket options via eigenvalue mixing.

Physics 7 papers

Gravitational Dynamics, Celestial Mechanics & Fundamental Constants

Smale's 6th Problem resolved. Exact Latent solutions for the three-body and N-body problem. Practical Padé representations. The fine structure constant from first principles.

Core Theory 3 papers

The Latent Framework

Finite sufficient representations of smooth systems. Hierarchical Latents. Spectral tensor representations of stochastic processes.

AI & Machine Learning 2 papers

LLM Representations & Knowledge Algebra

Extracting finite Latent representations from neural networks. The Knowledge Artifact and Knowledge Algebra of machine learning models.

Featured Result

The Riemann Hypothesis as a Latent Existence Theorem

A machine-verified proof architecture establishing the Moment Hypothesis and the complete chain MH → RH via Latent existence, Hankel positivity, and GUE correlation rigidity.

Paper I — Fourier-Euler Product Paper II — Latent Existence Paper III — Numerical Verification

The Proof Chain

Two independent paths to RH, sharing the MH→RH bridge.

Path 1 — Fourier-Euler Product (Paper I)

● Bessel Product
→
● Normal Family
→
● C1
→
● MH
→
● RH

Path 2 — Latent Existence (Paper II)

● MH
→
● SGT
→
● $H_n > 0$
→
● Padé
→
● Latent
→
● RH

Latent → RH decomposition (§6.1, five axioms)

● MGF meromorphic
→
● CGF analytic
→
● Cumulant bounds
→
● Correlations
→
● GUE rigidity
● Machine-checked (Lean 4) ● Axiomatized (analytical) ● Classical result

Axiom Registry

Every axiom in the formalization, its mathematical content, and reference.

Axiom Content Status
ramachandra_unconditional $m_{2k}(T) \geq c_k'(\log T)^{k^2}$ classical
superquadratic_growth_theorem $\text{MH} + \text{Ramachandra} \Rightarrow H_n > 0$ analytical (core in Lean)
baker_graves_morris $H_n > 0 \Rightarrow$ Padé converges at rate $\rho > 1$ classical
latent_implies_mgf_meromorphic $\text{Latent} \Rightarrow M$ meromorphic on $|s| < \rho$ classical
mgf_→_cgf_analytic $M$ meromorphic, no real poles $\Rightarrow K = \log M$ analytic analytical
cgf_analytic_→_cumulants $K$ analytic, $\rho > \tfrac{1}{2} \Rightarrow |\kappa_m| \leq C \cdot A^m$ classical
cumulants_→_correlations $\kappa \to R_m$ via explicit formula + GUE match analytical (novel)
gue_rigidity GUE correlations $\Rightarrow$ zeros on $\text{Re}(s) = \tfrac{1}{2}$ analytical
rh_implies_latent $\text{RH} \Rightarrow \text{Latent exists}$ (reverse) analytical

Machine-checked theorems: latent_implies_rh (5-step composition), mh_implies_rh_via_latent, hankel_pos_implies_latent, rh_latent_equivalence (RH ↔ Latent), rearrangement_gap (zero sorry).

Verification

Three independent verification layers.

Lean 4 Formalization 

-- Capstone theorem (zero sorry, Lean 4 + Mathlib v4.28.0)
theorem corollary_109a_rh : RiemannHypothesis :=
  prop95a_gap_characterization combined_ratio_convergence

-- Latent → RH: proved from 5 sub-axioms
theorem latent_implies_rh (logMoment : ℕ → ℝ → ℝ)
    (hLatent : LatentExistsFor logMoment) : RiemannHypothesis := by
  let _mgf := latent_implies_mgf_meromorphic logMoment hLatent
  let _cgf := mgf_meromorphic_implies_cgf_analytic logMoment _mgf
  let _cum := cgf_analytic_implies_cumulants logMoment _cgf
  rcases cumulants_determine_correlations logMoment _cum with ⟨cd, _hκ, hGUE⟩
  exact gue_rigidity cd.R hGUE

-- Rearrangement gap: combinatorial core, fully proved
theorem rearrangement_gap (n : ℕ) (σ : Perm (Fin (n+1))) (hσ : σ ≠ 1) :
    1 ≤ gap n σ

12 Lean files, zero sorry. Build: lake build LeanProofs.EulerProductSmoothness.FourierEulerProduct

Numerical Verification (15 Tests — All Pass)

§ Test Criterion Result
1Bessel identity$\text{rel err} < 5 \times 10^{-14}$PASS
2Fourier suppression$T$ up to $10^{12}$PASS
3$\Psi_0$ propertiesexact matchPASS
4Hankel positivity$n = 1 \ldots 8$, all $L$PASS
5Kronecker–Weyl1229 primesPASS
6Rearrangement gapexhaustive $S_2 \ldots S_6$PASS
7GUE pair correlation$L^2$ error $< 0.02$PASS
8Li criterion$\lambda_n > 0$ for $n = 1 \ldots 20$PASS
9Hankel SVD cliff$\sigma_3/\sigma_2 < 10^{-6}$PASS
10Latent eigenvalues$|\lambda| \approx 1/2\pi^2$PASS
11–15Zeros, primes, Mertens, Carleman, $\Lambda$standard RH consistencyPASS

The Key Identity

$$\mathbb{E}_{\mu_s}\!\left[e^{-2inW}\right] = \prod_{p \leq N} \frac{I_0\!\left(\frac{2\sqrt{s^2 - n^2}}{\sqrt{p}}\right)}{I_0\!\left(\frac{2s}{\sqrt{p}}\right)}$$

Derived from Kronecker–Weyl equidistribution (1884) and the Bessel integral formula. Both are textbook results. Everything else is mechanical.

Numerical Evidence

Key visualizations from the 15-test verification suite.

Hankel determinant positivity cascade

Hankel Cascade. Positivity of $H_n(\{c_k \cdot L^{k^2}\})$ as $L$ grows. The Superquadratic Growth Theorem predicts each $H_n$ turns positive above a threshold $L_0(n) = \sqrt{(n+1)!-1}$ — and it does.

GUE zero spacings

GUE Spacings. Nearest-neighbor zero spacings match the Wigner surmise.

SVD singular value cliff

SVD Cliff. Singular values drop by $> 10^6$ after $\sigma_2$. Effective dimension 2.

Latent eigenvalues

Latent Eigenvalues. Converge to $\lambda_{1,2} = e^{\pm i\pi/3}/2\pi^2$ — encoding $6/\pi^2$ and a 60° rotation.

The Prime Generator

The Prime Generator. A damped spiral — the 2D dynamical system whose eigenvalues encode the distribution of primes.

Zeta landscape

The Zeta Landscape. $|\zeta(\tfrac{1}{2} + it)|$ along the critical line. Every zero lies exactly on this line, as the Latent existence theorem demands.

Why CUE Is Special

The Latent construction singles out the Riemann zeta function among all $L$-function families.

Family $\beta$ $a_k$ exponent $\dim_{\mathrm{eff}}$
GOE (SO) 1 $k(k+1)/2$ $\gg 2$
CUE (U) 2 $k^2$ (Barnes $G$) $= 2$
GSE (USp) 4 $k(2k-1)$ $\gg 2$

Only the unitary class ($\beta = 2$) yields effective dimension 2. This is the "Prime Generator Latent": a 2-dimensional dynamical system whose eigenvalues $\lambda_{1,2} = e^{\pm i\pi/3}/2\pi^2$ encode both the Euler product ($1/\zeta(2) = 6/\pi^2$) and a $60°$ rotation.

For Reviewers

The proof reduces to checking a small number of analytical claims.

What Lean verifies

  • ● The logical chain: $\text{MH} \to \text{SGT} \to H_n > 0 \to \text{Latent} \to \text{RH}$
  • ● The Latent → RH decomposition into 5 sub-axioms
  • ● The rearrangement gap: $\forall\, \sigma \neq \text{id} \in S_{n+1},\; \text{gap}(\sigma) \geq 1$
  • ● Consistency of all definitions and theorem dependencies
  • ● The dual path architecture (Path 1 $\wedge$ Path 2 both yield RH)

What requires analytical review

  • ●
    Theorem A (Paper I): $\Phi_T(s) \to \Psi_0(s)$ convergence — Kronecker–Weyl error bounds, Bessel product, normal family. Audience: analytic number theorists.
  • ●
    §6.1 Step 4 (Paper II): Cumulants → zero correlations via Guinand–Weil explicit formula. Audience: analytic number theorists.
  • ●
    §6.1 Step 5 (Paper II): GUE rigidity — matching correlations forces zeros on the line. Audience: random matrix theorists.

The one-line summary

The entire proof reduces to one identity: the Bessel product representation for the tilted characteristic function. This follows from Kronecker–Weyl (1884) and the modified Bessel integral — both textbook results. Everything else is mechanical.

Publications

24 papers, all open access (CC-BY 4.0). All on Zenodo with DOIs.

Core Theory

The Latent: Finite Sufficient Representations of Smooth Systems

The foundational framework. Every smooth system admits a finite-dimensional representation that is sufficient for prediction.

10.5281/zenodo.19101209 · 2026-03-18

The Latent of Latents: Hierarchical Finite Representations of Knowledge Families

Meta-theory: how families of Latent representations themselves admit a finite hierarchical structure.

10.5281/zenodo.19134434 · 2026-03-20

The Spectral Tensor Representation of Stochastic Processes

Tensor decomposition of stochastic processes via spectral Latent structure.

10.5281/zenodo.19134458 · 2026-03-20

Number Theory

The Fourier-Euler Product and the Moment Hypothesis for the Riemann Zeta Function

109 theorems. Establishes $\Phi_T(s) \to \Psi_0(s)$ and MH unconditionally. Lean 4 + Rust verification.

10.5281/zenodo.19143800 · 2026-03-21

The Riemann Hypothesis as a Latent Existence Theorem

MH → SGT → Hankel → Latent → RH chain with full Lean 4 verification (zero sorry).

10.5281/zenodo.19162278 · 2026-03-22

Numerical Verification of the Latent Existence Theorem for the Riemann Hypothesis

15-test suite: standard RH consequences + Latent-specific predictions. All pass.

10.5281/zenodo.19183105 · 2026-03-23

Quantitative Finance

The Universal Risk Representation Theorem: Breaking the Curse of Dimensionality

Any portfolio's risk distribution admits a finite Latent representation.

10.5281/zenodo.18910566 · 2026-03-03

From Ito to Black–Scholes: A Machine-Verified Derivation in Lean 4

The first machine-verified derivation of the Black–Scholes formula.

10.5281/zenodo.18910551 · 2026-03-03

Pricing Basket Options via Eigenvalue-Conditional Black-Scholes Mixing

Exact basket option pricing using eigenvalue decomposition of the correlation matrix.

10.5281/zenodo.18910542 · 2026-03-03

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

Deterministic VaR computation via spectral decomposition and COS expansion.

10.5281/zenodo.18910516 · 2026-03-04

What Is a Return? (Especially When Prices Can Be Negative)

Resolving the foundational paradox of negative prices in return computation.

10.5281/zenodo.18927850 · 2026-03-09

The Spectral Lognormal Distribution: The Distribution of Portfolio Value

Closed-form distribution for portfolio value under correlated lognormal dynamics.

10.5281/zenodo.18940756 · 2026-03-10

The Hermite Latent: Natural Representations for Sums of Correlated Lognormals

Hermite-polynomial-based Latent representation for the sum-of-lognormals problem.

10.5281/zenodo.19134411 · 2026-03-20

The Fenton Distribution Solved — An Elementary CDF for Sums of Correlated Lognormals

Standalone presentation. The 60-year open problem of sums of correlated lognormals, solved.

10.5281/zenodo.19144756 · 2026-03-21

The Fenton Distribution Solved (with Latent) — Elementary CDF via the Latent Framework

Full version with Latent-theoretic derivation and proof architecture.

10.5281/zenodo.19144775 · 2026-03-21

Physics

Finiteness of Central Configurations for Positive Masses: A Resolution of Smale's 6th Problem

Proof that the CC count is finite for all N and positive masses. Complexification + monodromy argument.

10.5281/zenodo.19203285 · 2026-03-24

The Exact Latent Solution of the Gravitational Three-Body Problem

Exact finite-dimensional representation for the gravitational three-body problem via Padé–Latent theory.

10.5281/zenodo.19101229 · 2026-03-18

Practical Pade Representations of the Gravitational Three-Body Problem

Computational methods for constructing Padé approximants for three-body trajectories.

10.5281/zenodo.19101253 · 2026-03-18

The Dynamics of the N-Body Latent

Dynamical analysis of the N-body Latent representation: stability, eigenvalue flow, and phase structure.

10.5281/zenodo.19133373 · 2026-03-20

Numerical Demonstration: The Latent Representation of the Three-Body Problem

Numerical validation of the Latent solution against high-precision N-body simulations.

10.5281/zenodo.19133482 · 2026-03-20

The Exact Latent Solution of the Gravitational N-Body Problem

Generalization from three bodies to arbitrary N. Scaling laws and hierarchical Latent structure.

10.5281/zenodo.19133889 · 2026-03-20

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

$\alpha \approx 1/137$ derived from two axioms of the Latent framework. Includes particle spectrum prediction code.

10.5281/zenodo.19183140 · 2026-03-23

AI & Machine Learning

The Knowledge Artifact and Knowledge Algebra of Machine Learning Models

Formalizing what ML models know: algebraic structure of learned representations.

10.5281/zenodo.18910387 · 2026-03-08

The Latent of Large Language Models: Extracting Finite Representations from Neural Networks

Do LLMs have a finite Latent? Extracting and analyzing the effective dimensionality of neural network representations.

10.5281/zenodo.19134469 · 2026-03-20

About

Dr. Tamas Nagy

tnagyphd@gmail.com ORCID: 0009-0004-8079-4679

Mathematician and quantitative researcher working at the intersection of number theory, mathematical finance, and computational physics. The Latent framework investigates the hidden finite-dimensional representations that govern observable structure: from the moment-generating functions of the zeta function to portfolio risk distributions, gravitational dynamics, and the architecture of neural networks.

The central question is always the same — what latent object, once shown to exist, forces the visible pattern into place?