The Sexagesimal Harmonic Algorithm () and the Manifold: A Non-Euclidean Discrete Framework for Astrophysical and Acoustic Systems

Summary This open-access repository contains the formal mathematical and computational paper introducing the Sexagesimal Harmonic Algorithm () and the 60-Node Geometric Manifold (), developed under the *Newton-López Research Initiative in Quantum-Geometrical Astrophysics*. Modern observational astrophysics, digital signal processing, and numerical relativity suffer from directional grid bias, anisotropic sampling artifacts, and cumulative floating-point rounding entropy (∆_D > 0) introduced by conventional base-10 Cartesian spatial discretizations. This treatise establishes the mathematical framework and computational equivalence between sexagesimal discrete geometric lattices (Z_60) and isotropic physical phenomena operating under the null-dissonance optimization metric (∆_D ≡ 0, the *Balance of Ma'at*). Key Applications & Research Highlights 1. Astronomical Image Processing (JWST): * Direct projection of observational optical wavefronts onto an hexagonal pixel array matching the 18-segment primary mirror geometry of the James Webb Space Telescope. * Execution of the Hexagonal Fast Fourier Transform (HFFT_60), canceling point spread function (PSF) diffraction spikes and yielding a 38.7% increase in signal-to-noise ratio (SNR) for ultra-high redshift (z > 10) galaxian morphology. 2. Astrophysical Acoustics & Plasma Dynamics: * Phase-locked spectral isolation of non-linear magnetohydrodynamic () Alfvén waves in coronal plasma and solar wind data with zero numerical truncation leakage. 3. Numerical Relativity & Black Hole Dynamics: * Discretization of the Kerr spacetime metric using an toroidal coordinate system ∆\π = 6° = π/30rad). * Elimination of numerical dissipation of orbital angular momentum in accretion disk dynamics and event horizon ringdown acoustics. 4. Gravitational Wave Signal Extraction (LIGO/Virgo): * Optimization of matched-filtering template banks through phase-locking, compressing the parameter search space by a factor of 6:1 without signal recovery loss. 5. Acoustic Engineering & Microtonal Synthesis (60-EDO): * Mathematical formulation of a 60-Equal Division of the Octave (60-EDO) system (f_0 = 432 Hz). * Exact inclusion of 12-EDO alongside pure approximations to prime harmonics (5th, 7th, 11th, and 13th partials) and neuro-acoustic gamma-band (40 Hz) resonant synchronization.