Mathematics

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

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).

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

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

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).

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 $k$.

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.

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

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

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

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.

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. $

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).

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

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): $S = \sum w_i e^{Y_i}$, $Y \sim \mathcal{N}(\mu, \Sigma)$.

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

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.

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.

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

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.

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.

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.

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 $\Delta > 0$.

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.