Discrete Modular Trigonometry (DMT): An Irrational-Free Angular Calculus Framework Based on Sexagesimal Harmony Mathematics (MAS)

Description / Abstract This dossier presents the mathematical foundations and computational benchmarks of Discrete Modular Trigonometry (DMT), a revolutionary angular calculus framework derived from the Sexagesimal Harmony Mathematics (MAS) paradigm and governed by the structural invariance of the Balance of Maat. ∑Ф ≡ 0 (Mod 360) Traditional trigonometry relies on the real continuum (R) and the transcendental constant π, which inherently introduces irrational numbers (√2, √3, π), non-periodic decimal expansions, and catastrophic floating-point rounding errors in digital hardware. DMT replaces this paradigm by projecting all angular functions onto the modular torus Z_60 x Z_360, redefining trigonometric projections as exact, rational fasorial ratios (100\% integer-based arithmetic). Key Scientific Contributions & Innovations: Irrational-Free Trigonometric Operators: Formalization of the Maat Discrete Cosine (Cmp_60) and Sine (Smp_60) as exact integer pairs on the ring Z_60, mapped across the 96 coprime rays (ϕ (360) = 96). Elimination of Singularities: Redefinition of the tangent operator as the Coprime Ray Ratio (Tmp_60), converting asymptotic infinities and overflow boundaries into closed modular slope invariants. O(1) Computational Efficiency: A deterministic, lookup-driven algorithm for trigonometric identities that executes in constant time without requiring Floating-Point Units (FPUs) or CORDIC iterations. Zero-Entropy Physics & Hardware: Application of DMT to quantum state simulation, fault-tolerant aerospace trajectory modeling, and loss-free signal processing. Structural Notice & Intellectual Property Protection This dossier contains the complete theoretical proofs, primary axioms, conversion matrices, and algorithmic benchmarks necessary for peer evaluation. To protect intellectual property rights and maintain system security, low-level microcode implementations, underlying reverse-engineering source routines, and lower-level derivative derivations have been intentionally omitted. When the global scientific community deeply explores and integrates the concepts established by this disruptive mathematical paradigm, the theoretical and practical possibilities opened will unlock a civilizational-scale shift in computation, theoretical physics, and space exploration.