Masses Mass Hierarchy

The φ-ladder

The fermion mass hierarchy is the observed pattern in which each generation of charged leptons is heavier than the last. Recognition Science derives this pattern from the recognition ledger, the record of forced cost events, rather than treating it as an unexplained input. The derivation places each fermion on a rung of the φ-ladder, a sequence of masses spaced by powers of the golden ratio φ ≈ 1.618. A fermion's mass is E_coh · φ^r, where E_coh is a common energy scale and r is an integer rung number.

The module MassHierarchy.lean establishes the ladder's structure as a established theorem. The electron sits on rung 2, the muon on rung 13, and the tau on rung 19. These rung assignments are not fitted values; they are theorems in the Lean kernel. The mass ratios follow directly: the muon-to-electron ratio is φ^(13−2) = φ^11 ≈ 199, and the tau-to-muon ratio is φ^(19−13) = φ^6 ≈ 18. The theorem lepton_hierarchy_geometric proves both ratios exactly, and lepton_mass_increasing proves that each ratio exceeds 1, so each generation is heavier than the last.

The hierarchy spans orders of magnitude because φ is greater than 1. A small rung spacing produces a large mass jump, since φ^11 is about 200 and φ^17 is about 2200. The rung spacings themselves, 11 and 6, come from cube geometry, not from free parameters. There are no Yukawa couplings to adjust; each generation's mass ratio is determined by the rung spacing. This is the sense in which Recognition Science resolves the hierarchy question: the pattern is forced, not chosen.

MODEL mass_on_rung · IndisputableMonolith/Masses/MassHierarchy.lean

THEOREM r_electron · r_muon · r_tau · IndisputableMonolith/Masses/MassHierarchy.lean

THEOREM lepton_hierarchy_geometric · IndisputableMonolith/Masses/MassHierarchy.lean

THEOREM lepton_mass_increasing · IndisputableMonolith/Masses/MassHierarchy.lean

What this page does not claim

This answer does not claim the rung assignments are measured values. This answer does not claim the φ-ladder predicts the absolute mass scale E_coh. This answer does not claim the hierarchy applies to quarks.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Masses/MassHierarchy.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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