Core Theory — thalens.org

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.

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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,

Verified 17 claims 10,585 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.

Verified 135 claims 2,289 words

Latent Core

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

Verified 111 claims 1,506 words

Latent Optimization

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

Verified 120 claims 1,492 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.

Verified 16 claims 401 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.

Verified 8 claims 2,149 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.

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 $\rho$.

Verified 4,876 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

6 claims 22,583 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.

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.

10 claims 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.

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).

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.

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.

1 claims 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.

Verified 2 claims 3,355 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.

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.

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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.

Verified 1,849 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.

Verified 6 claims 8,801 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.

4,317 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.

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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).

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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.

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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.

Verified 6 claims 3,480 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.

8 claims 4,797 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 $G(z; \mathbf{v}_0)$.

4 claims 3,837 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 $N_P$.

7 claims 3,167 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.

1,849 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.

Verified 1 claims 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 $\mathcal{L}$.

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.

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.

Verified 1 claims 2,651 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).

2 claims 2,885 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.

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Bio Protein Folding Dynamics — HANDOFF

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LatentFold — Business Plan & Product Strategy

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The Thought Line: Universal Smoothing and the Latent Bridge

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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.

Verified 11 claims 1,896 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.

Verified 11 claims 1,870 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 $\varepsilon$.

Verified 28 claims 1,604 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.

1,854 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.

Verified 33 claims 2,207 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.

Verified 1 claims 2,242 words

Publication Strategy — QNM Latent

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Phy Qnm Latent — email frei draft

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Phy Qnm Latent — email gergely draft

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Phy Qnm Latent — email isi draft

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Phy Qnm Latent — email racz draft

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Reproducibility Package — Quantum USRT Spectral Dichotomy

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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.

Verified 5,108 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.

13 claims 1,994 words

Unconditional Results: Deriving Latent Conditions from First Principles

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