thalens.org — Mathematical Research

Phylogenetic Tree Reconstruction via the Latent Framework

Thu, 23 Apr 2026 00:00:00 +0000

Phylogenetic inference turns molecular sequences into historical relationships. Classical likelihood-based methods excel in practice, yet the information geometry of alignment data relative to tree topology is rarely summarized in coordinates comparable across studies.

Dimension-Independent Finiteness of Central Configurations for Positive Masses

Thu, 23 Apr 2026 00:00:00 +0000

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**.

The Structured Latent Basis: Feature Engineering as Basis Selection

Wed, 22 Apr 2026 00:00:00 +0000

We introduce the Structured Latent Basis (SLB) framework, a perspective on supervised learning that unifies feature engineering and modeling as a single problem: selecting a mathematical basis in which the target function is linear.

Formal Koide Structure: Mass Bounds, Generation Counting, and Neutrino Predictions from the Z_N Ansatz

Wed, 22 Apr 2026 00:00:00 +0000

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 $Q = 2/3$.

The Yang-Mills Mass Gap via Gauge Absorption and Perelman W-Entropy

Wed, 22 Apr 2026 00:00:00 +0000

**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

The Smooth-Step Spectral Method: Unifying Smooth and Threshold Structure in Tabular Regression

Tue, 21 Apr 2026 00:00:00 +0000

We introduce the Smooth-Step Spectral Method (S³M), a structured basis regression approach that unifies smooth spectral features with learned threshold features for tabular data.

An Unconditional BGST$\to$R$_2$ Fourier Transfer via Poisson-Kernel Deconvolution

Tue, 21 Apr 2026 00:00:00 +0000

Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh [BGST23] proved the first unconditional asymptotic for Montgomery's pair-correlation function $F(\alpha, T)$, with error $O(1/\sqrt{\log T})$ uniformly for $\alpha \in [0,1]$.

The PDE Tensor Algebra: Structural Decomposition and Exact Recombination of Differential Equations

Tue, 21 Apr 2026 00:00:00 +0000

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.

Machine-Verified Derivation Chain for the Fine Structure Constant via the E8 → Standard Model Breaking Pattern

Tue, 21 Apr 2026 00:00:00 +0000

We present a machine-verified derivation chain connecting the E₈ Lie algebra to the electromagnetic fine structure constant $\alpha \approx 1/137$.

A refuted-and-vindicated pre-registration test of a spectral error model on a superconducting processor

Tue, 21 Apr 2026 00:00:00 +0000

We pre-register and test a spectral-error-mitigation prediction for the two-qubit gate fidelity of Quantum Inspire's Tuna-9 9-qubit transmon processor and execute it in four cryptographically timestamped stages.

The Latent: Finite Sufficient Representations of Smooth Systems

Mon, 20 Apr 2026 00:00:00 +0000

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.

A Lean 4 Formalization of the Birch and Swinnerton-Dyer Conjecture and Its Surrounding Theory

Sun, 19 Apr 2026 00:00:00 +0000

The Birch and Swinnerton-Dyer conjecture predicts that for an elliptic curve $E/\mathbb{Q}$ the Mordell-Weil rank equals the order of vanishing of the Hasse-Weil $L$-function at $s = 1$, and that the leading Taylor coefficient is expressed by an expl

ε-Removal for Moments of the Riemann Zeta Function via Cumulant Generating Function Analysis, Subject to Grade-2 Dominance

Sun, 19 Apr 2026 00:00:00 +0000

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 $r_\star \geq 1/A$;

An Exact Algebraic Bifurcation in the Triangle-Plus-Center Central Configuration

Sat, 18 Apr 2026 00:00:00 +0000

We prove that the triangle-plus-center central configuration of the planar four-body problem with masses $(1, 1, 1, \mu)$ undergoes an exact stability transition at the critical mass ratio $$\mu^* = \frac{81 + 64\sqrt{3}}{249},$$ the unique positiv

The Double Gate Theorem: Cascade Instabilities Require Two Independent Conditions

Sat, 18 Apr 2026 00:00:00 +0000

We prove that gravitational cascade instabilities (such as the N-body problem's runaway collisions) and debris cascade instabilities (such as the Kessler syndrome in orbital mechanics) share a common mathematical structure: both require the simultane

Integer Basics: Foundational Properties of Integer Arithmetic

Sat, 18 Apr 2026 00:00:00 +0000

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.

No Non-Trivial Collatz Cycle Has Twenty-Three or Fewer Odd Steps: Cycle Elimination, Descent Analysis, and Residue-Class Structure

Sat, 18 Apr 2026 00:00:00 +0000

We develop a systematic method for eliminating non-trivial cycles of the Collatz map $T(n) = n/2$ if $n$ is even, $T(n) = 3n+1$ if $n$ is odd, and complement it with a stopping time analysis toward the full conjecture.

Turbulence Scaling Laws from the Grade Equation: Kolmogorov Spectrum and Intermittency from Analyticity

Sat, 18 Apr 2026 00:00:00 +0000

We derive the Kolmogorov energy spectrum $E(k) \sim \varepsilon^{2/3} k^{-5/3}$ and anomalous intermittency corrections to the structure function exponents $\zeta_p$ from the Grade Equation — a universal structural decomposition theorem for analytic dynamical systems. The derivation proceeds in three steps.

Spectral Interest Rate Pricing: COS-Based Derivatives from Yield Curve Coefficients

Fri, 17 Apr 2026 00:00:00 +0000

We present the first formally verified interest rate derivatives pricing engine. The Nagy spectral yield curve represents yields as a finite cosine series $y(\tau) = A_0 + \sum_{k=1}^K A_k \cos(k\pi\tau/\tau_{\max})$ with each mode following independent Ornstein-Uhlenbeck dynamics.

Critical Exponents for Three-Dimensional Percolation: A Crossover Approach via Long-Range Renormalization

Fri, 17 Apr 2026 00:00:00 +0000

$D_\infty$ and the Goldbach Convergence Hierarchy

Fri, 17 Apr 2026 00:00:00 +0000

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.

The Goldbach Conjecture as a Latent Positivity Theorem: Twenty-Five Paths, the Convergence Theorem, and the Five-Layer Strategy

Fri, 17 Apr 2026 00:00:00 +0000

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

The Quantum Compressibility Threshold

Fri, 17 Apr 2026 00:00:00 +0000

We prove a structural dichotomy for open quantum systems.

Constraint-Forced Absorption: A General Mechanism for Singularity Prevention in Gauge-Invariant PDE Systems

Thu, 16 Apr 2026 00:00:00 +0000

We identify a general mechanism — **constraint-forced absorption (CFA)** — in PDE systems where a differential identity constrains the dynamics. In the $(D, C, P)$ framework (dissipation, coupling, constraint): **Theorem** (Constraint-Forced Absorption).

Residual Stream Denoising in Large Language Models: Gradient-Free Quality Improvement via Functional Sensitivity Analysis

Mon, 13 Apr 2026 00:00:00 +0000

An Eigenvalue-Conditioned Copula with Positive Tail Dependence: A Machine-Verified Alternative to the Gaussian Copula

Sat, 11 Apr 2026 00:00:00 +0000

The Gaussian copula's failure to capture tail dependence was a central factor in the 2008 credit crisis: CDO tranche losses far exceeded model predictions because the model assigned near-zero probability to simultaneous defaults.

Proof Circuits

Sat, 11 Apr 2026 00:00:00 +0000

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.

Spectral Error Mitigation: Exact Noise Inversion for Quantum Computers via Cluster Lindblad

Sat, 11 Apr 2026 00:00:00 +0000

Allosteric Regulation — Spectral Communication in Proteins via the Latent Framework