Modular Tomography of Twin Primes: A Deterministic Lattice Framework from 1 to Infinity

DOI: 10.5281/zenodo.18229787Title: Modular Tomography of Twin Primes: A Deterministic Lattice Framework from 1 to InfinityAuthor: Miles Enoch Tracy contact: milestracy@yahoo.com ORCID: https://orcid.org/0009-0006-7781-8259 Description**: This work presents a self‑contained, deterministic framework for twin prime distribution based on modular arithmetic and visual computation. Beginning with the number 1 as the sole generator, we construct explicit residue rails modulo M=210, generate complete execution tables showing every twin prime ≤1,000,000, and derive the exact amplification factor R(M)=φ(M)/|T(M)|=3.2. We prove that the Hardy‑Littlewood twin‑prime constant C₂ emerges as the infinite‑modulus limit of R(M), making the heuristic rigorous. Polar mapping of twin primes reveals 15 interleaved spiral arms whose swirl is deterministic Moiré interference, not randomness. Fourier analysis of the modular indicator function yields coefficients whose squared magnitude equals C₂/π², linking discrete modular constraints to analytic number theory. The framework predicts local deviations (e.g., class 61 count 535 vs. expected 544.6) via phase‑product corrections derived from inclusion‑exclusion, and generalizes to all prime k‑tuples. All equations use only Unicode characters (superscripts, subscripts, standard symbols), no LaTeX, and every claim is computationally verifiable with provided Python scripts. The only special number is 1; all patterns emerge from modular cancellation waves. This structure is falsifiable, scales to arbitrary bounds, and serves as a pedagogical tool accessible from elementary school through graduate research.