The Discrete Algebraic Structure of the Fermion Mass Spectrum and its Pythagorean Geometric Origin

This paper presents a systematic numerical analysis of the charged fermion mass spectrum across three generations in the Standard Model. We report the discovery of a precise network of algebraic relations among quark and lepton masses, characterized by simple rational numbers, basic constants,and Pythagorean number patterns. Key findings include: (1) Square-root mass ratios exhibit high accuracy simple fractions, e.g., √mc/√mu ≈97/4 with an error below 10−5; (2) Extended Koide-type relations satisfy sum rules Kδu + Kδd = 9/10 and Ku + K∆ ≈ 17/10; (3) Leptons and quarks are connected via a fixed proportion 8/9, leading to a unified equation; (4) Square-root mass-difference ratios show integer/semi-integer patterns: R(ℓ) ≈ 4, R(u) ≈ 12, R(d) ≈ 34/5, with R(u)/R(ℓ) ≈ 3 precisely matching the quark color number; (5) All key numbers (3,4,5,7,8,9,12,17,19,24,34,35,51,97...) satisfy the difference-of-squares relation a2 − b2 and systematically map to primitive Pythagorean triples. We propose that each fermion sector corresponds to a characteristic triple: leptons ↔ (3,4,5), up-type quarks ↔(5,12,13), down-type quarks ↔ (7,24,25), with Koide-related structures ↔ (8,15,17). The statistical significance of these relations far exceeds current mass uncertainties, forming an algebraic closure that strongly suggests the mass spectrum originates from a geometric structure on a two-dimensional discrete lattice, potentially related to the arithmetic of Gaussian integers Z[i]. Quantitative constraints from empirical relations provide stringent tests for beyond Standard Model theories. All computational codes are publicly available for verification. This work provides explicit empirical foundations and constructive directions for flavor theories based on discrete or modular symmetries.