This work places on record a proposed explanatory framework for the repeated appearance of the transcendental constants π, φ, and e in particle mass correspondences documented across prior ENSO-related studies. The central claim advanced here is one of geometric necessity: under simultaneous constraints of metric closure, recursive self-similarity, and stable mapping between logarithmic and linear structure, the constants π, φ, and e arise as unavoidable invariants rather than adjustable or selected parameters. Within this view, particle masses occupy discrete allowed positions determined by geometric and scaling constraints, forming a sparse ladder in logarithmic space. The paper does not assert completeness, finality, or exclusivity of interpretation. It does not propose new particles, modify the Standard Model, or introduce fitted parameters. Its purpose is narrower and procedural: to articulate a mechanism-class consistent with, and synthesizing, a large body of previously published results in which π-, φ-, and e-based constructions repeatedly yield statistically significant agreement with experimental particle data. All quantities, transformations, and tests are specified explicitly. An accompanying script reproduces the ladder construction and statistical analysis described in the text. Negative results and sparsity are treated as structural features rather than noise. This document is released as a timestamped synthesis note. It is intended to establish priority for the geometric-necessity interpretation underlying earlier empirical findings, while leaving detailed mathematical development, physical modelling, and critical evaluation to subsequent work.