The Geometric Origin of Particle Mass Spectrum: From Trinity Symmetry to φ-Hexagonal Vacuum

This paper presents a geometric framework for understanding the origin of the particle mass spectrum. Rather than treating particle masses as independent parameters, the work proposes that mass emerges from a structural transition in vacuum geometry governed by symmetry and constraint.

Starting from a minimal axiom of constraint, the framework shows how a small set of fundamental geometric principles leads to distinct symmetry phases. A critical transition region is identified in which stable physical structures can form, providing a natural explanation for why particle masses cluster in specific ways rather than appearing arbitrary.

The paper combines analytical derivation with computational validation. All central claims are tested using a fully reproducible Python script, demonstrating that the proposed geometric transition aligns with observed particle mass distributions without parameter tuning or curve fitting.

This work does not replace the Standard Model, but offers a complementary explanatory perspective: particle masses are interpreted as emergent features of vacuum structure rather than fundamental inputs. The results suggest that geometry plays a primary role in organizing physical reality, with dynamics emerging as a secondary consequence.

The accompanying validation script is included to allow independent verification of all results and to support transparent, reproducible analysis.

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