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:

1. 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.
2. 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.
3. 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!