ArXe Lepton Mass Hierarchy: Complete Ontological Derivation

Update Note — A Second, Convergent Derivation Route (ALO)

The spiral/recursive derivation below (v2.1, November 2025) remains valid and is unchanged. Since then, a second and independent derivation route — through ALO’s arity-grammar reading of the electroweak VEV — has reached the same masses with substantially better precision. Both routes are presented here because they are genuinely different arguments arriving at the same place, which is itself evidence worth keeping visible rather than silently replacing one with the other.

Route 1 (original, below): logical spirals from ternary ambiguity

m_μ/m_e = 206.664   (experimental 206.768, error 0.05%)
m_τ/m_e = 3444.3    (experimental 3477.15, error 0.94%)

Derived from the recursive formula mₙ = aⁿ·m₀ + π(aⁿ−1)/(a−1), with a=8/π, starting from the muon and iterating through the “logical spiral” generated by the ambiguous middle in ternary logic. Full derivation preserved below, unchanged.

Route 2 (new): ALO — direct arity-grammar formulas

m_μ = m_e × (3⁴ + 40π + 2/19)   = 105.66 MeV   (error 0.0001%)
m_τ = m_μ × (17 − 11/59)         = 1776.8 MeV   (error 0.0034%)

This route does not use the spiral/recursion structure at all — it reads the mass ratios directly as combinations of ArXe arity operators (3=CYC, 19=DARK, 17=SPEC, 11=REG/EM, 59=STAB) plus the π anchor, with roughly an order of magnitude better precision than Route 1.

Reading the convergence

Two structurally distinct derivations — one built from recursive geometric spirals, one built from direct arity-operator combination — land on the same two mass ratios to within their respective error bars. This is treated as corroborating evidence for the underlying claim (lepton masses are not free parameters but structured positions in the ArXe hierarchy), not as a reason to discard either route: the spiral derivation offers a geometric/dynamical picture of why a recursive relation should hold between generations, while the ALO formula offers the more precise but more “static” arity-grammar reading. Both open questions from the original document remain open — most notably, the absolute electron mass is still fixed only relative to other quantities, not derived independently in either route.

Original derivation (v2.1, November 2025) — unchanged below

[The original document continues here: the ontological foundation of the ambiguous middle in ternary logic, the genesis of the spiral form, the full step-by-step recursive mass calculation, and the closed-form verification — all as originally published.]

For the complete ALO Standard Model reconstruction this update draws on: “Arity-Logical Ontology: An Interpretive Framework for Physical Constants via Recursive n-ary Structure”