Sexagesimal Harmonic Analysis on Classical Diophantine Problems: A Unified Framework for x^3 + y^3 + z^3 = n, the Erdős–Straus Conjecture, and Lander–Parkin–Selfridge Power Sums

Summary This publication contains the complete mathematical monograph presenting the application of Matemática de Armonía Sexagesimal (MAS) and the Sexagesimal Harmony Algorithm (SHA) to three classical open problems in Diophantine and Transcendental Number Theory: 1. The Sum of Three Cubes Problem (x^3 + y^3 + z^3 = n): A continuous phase-space analysis in the Sexagesimal Harmonic Space H_60 demonstrating the modulo 9 obstruction as a Maat symmetry defect and establishing bounded 1D search space reductions. 2. The Erdős–Straus Conjecture (4/n = \1/x + 1/y + 1/z): Proof of the prime reduction principle, parametric identities across residue classes modulo 840, and proof of the null asymptotic density of potential counterexamples under the action of the Harmonic Quantization Operator Q_60. 3. The Lander–Parkin–Selfridge (LPS) Conjecture : Analysis of phase coupling in equal-power sums, showing that counterexamples for k=3, 4 correspond to zero-eigenvalue phase degeneracies in the base-60 resonance matrix H_60, and deriving stability bounds for exponents k ≥ 5. Key Mathematical Constructs Introduced - Sexagesimal Harmonic Space (H_60^m): A quotient Kähler manifold C^m / Λ_60^(m) equipped with a base-60 flat Hermitian metric. - Harmonic Quantization Operator (Q_60): A discrete multi-index Fourier phase projection operator that suppresses non-resonant algebraic noise. -Maat Balance Defect ∆_Maat: A metric norm quantifying energy conservation and phase equilibrium in complex exponential and Diophantine varieties. Mathematics of sexagesimal harmony, applications to number theory (DOI: 10.5281/zenodo.22941034) Communitas scientifica, avemus aevum sexagesimale!