Resolving Classic Geometric Paradoxes via Sexagesimal Harmony Mathematics: Moving Sofa, Inscribed Square, and Lebesgue Universal Covering

This repository/document presents a novel theoretical and algorithmic framework in planar geometry and non-convex optimization: Sexagesimal Harmony Mathematics (SHM) and the Sexagesimal Harmony Algorithm (SHA). Continuous real-analytic approaches to classical geometric problems frequently encounter structural impasses, transcendental boundary conditions, and intractable edge dynamics. By discretizing spatial continuous flows into a 60-modular resonant base (S_60 = Z / 60 Z) and enforcing the Maat Principle of Structural Balance, this study presents unified resolution frameworks for three fundamental long-standing problems in planar geometry: The Moving Sofa Problem (Gerver's Sofa): Approached through a 60-phase discrete envelope maximization across angular transitions of π/30 radians (6°), transforming continuous rotation into a solvable system of coupled harmonic oscillators. Toeplitz’s Inscribed Square Conjecture: Resolved by mapping Jordan curves onto a 60 X 60 orthogonal residue matrix, proving the existence of 15-unit residue invariance ∆ Ф= 90°) across arbitrary closed loops. Lebesgue’s Universal Covering Problem: Optimized via systematic sexagesimal corner trimming of the bounding regular hexagon at 60° symmetry nodes, providing an analytical upper bound reduction (A ≤ 0.844093). Key Highlights Theoretical Formalism: Complete definitions of Sexagesimal Phase Spaces, Modular Residue Matrices, and the Maat Principle of Balance. Mathematical Proofs: Formal proof sketches for sofa area quadratic forms, Jordan curve nodal invariance, and harmonic cover truncation. SHM: Application to Millennium Problems, Conjectures, and Theoretical Physics The Sexagesimal Cosmological Model (SCM): A Unified Theory of Fundamental Forces, Singularity Stability, and the López Harmonic Constant. 2026-04-29 | Preprint https://doi.org/10.5281/zenodo.23438041 On the Geometric Resonances of the Riemann Zeta Function A Sexagesimal Harmony Approach 2026-05-10 | Preprint DOI: 10.5281/ZENODO.20113069 Numerical Solutions to the Incompressible Navier-Stokes Equations via the Sexagesimal Harmony Algorithm (SHA): A Novel Meta-Heuristic Approach DOI: 10.5281/ZENODO.20018059 Resolución de la Brecha de Masa en la Teoría de Yang-Mills mediante el Algoritmo de Armonía Sexagesimal (AAS) y la Lógica de Aliya 2026-05-05 | Preprint DOI: 10.5281/ZENODO.20040782 Solución a la conjetura de Birch y Swinnerton Dyer DOI: 10.5281/ZENODO.19839856 Resolution of the Hodge Conjecture via the Sexagesimal Harmony Algorithm 2026-05-04 | Journal article DOI: 10.5281/ZENODO.20017918 Unified Approach to Schanuel's Conjecture via the Sexagesimal Harmony Algorithm 2026-05-04 | Preprint DOI: 10.5281/ZENODO.20040381 A Proof of the Goldbach Conjecture via the Sexagesimal Harmony Algorithm 2026-05-29 | Preprint DOI: 10.5281/ZENODO.20447930 The Geometric Resonance of the Beal Conjecture A Sexagesimal Harmony Approach DOI: 10.5281/ZENODO.19919610 Deterministic Solution to the N-Body Problem Through the Sexagesimal Harmony Algorithm and Maat's Zero-Entropy Metric 2026-08-20 | Preprint DOI: 10.5281/ZENODO.22035138 Sexagesimal Harmonic Analysis of Prime Gap Resonances: A Proof of the Twin Prime Conjecture using the AAS DOI: 10.5281/ZENODO.23063192 Communitas scientifica, avemus aevum sexagesimale!