The Latent Framework

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.

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 $r_\star \geq 1/A$;

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.

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

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.

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,

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).

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

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.

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.

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.

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 $\rho$.

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

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

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

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

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

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

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).

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

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.

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

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*.

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.

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.

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.

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.

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

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.

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.