Foundation Qrft Fermion Kinetic Cert Fermion Mass At Succ Ratio

The rung ratio

The golden ratio φ ≈ 1.618 is the number that solves r² = r + 1. In the Recognition Science framework, it appears as the fixed ratio between successive rungs, the discrete steps of a ladder on which fermion masses sit. The declaration fermionMassAt_succ_ratio is a machine-checked theorem stating that if m₀ is a positive base mass and k is any nonnegative integer, then the mass at rung k+1 equals the mass at rung k multiplied by φ. In symbols: m(k+1) = m(k) · φ. A companion theorem, fermionMassAt_adjacent_ratio, states the equivalent division form: m(k+1) / m(k) = φ.

This is a structural statement, not a numerical prediction. The framework defines fermionMassAt as m₀ · φ^k, a geometric sequence with ratio φ. The theorem proves that this definition has the property its name advertises: adjacent terms stand in the golden ratio. It does not say what m₀ is, does not assign masses to specific particles, and does not claim that any measured fermion mass equals a particular rung value. The framework's mass-ratio modules separately propose that the Standard Model spectrum can be reproduced from this φ-ladder, but that comparison against measured values is an empirical check, not part of this theorem.

The theorem is proved in the framework's machine-checked library of formal theorems, with zero axioms and zero unproved assumptions. Its proof is a direct algebraic manipulation: unfold the definition, apply the power law, and ring. The positivity condition on m₀ is required so that division by m(k) is legitimate in the ratio form; the theorem itself holds for any positive base mass. This is a small, clean result: it certifies the internal consistency of the ladder definition, nothing more.

In Recognition Science, the mass term m ψ̄ψ in the Standard Model fermion kinetic Lagrangian maps to a cost on a recognition ratio, and the kinetic term becomes a lattice derivative. The φ-ladder is the structural prediction that follows. This theorem is the mechanical hinge that makes the ladder a ladder: it guarantees that the ratio between consecutive rungs is exactly φ, for every rung, for every positive base mass. It is the kind of fact that a larger derivation can rely on without rechecking.

theorem fermionMassAt_succ_ratio {m0 : ℝ} (hm : 0 < m0) (k : ℕ) :
    fermionMassAt m0 (k + 1) = fermionMassAt m0 k * phi := by
  unfold fermionMassAt; rw [pow_succ]; ring

theorem fermionMassAt_adjacent_ratio {m0 : ℝ} (hm : 0 < m0) (k : ℕ) :
    fermionMassAt m0 (k + 1) / fermionMassAt m0 k = phi := by
  rw [fermionMassAt_succ_ratio hm]
  field_simp [(fermionMassAt_pos hm k).ne']

theorem fermionMassAt_succ_ratio {m0 : ℝ} (hm : 0 < m0) (k : ℕ) :
    fermionMassAt m0 (k + 1) = fermionMassAt m0 k * phi := by
  unfold fermionMassAt; rw [pow_succ]; ring

What this page does not claim

This theorem does not assign any specific numerical value to the base mass m₀. This theorem does not claim that any measured fermion mass equals a particular rung value. This theorem does not establish that the Standard Model mass spectrum is actually reproduced by the φ-ladder; that comparison remains an empirical check.

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/Foundation/QRFT/FermionKineticCert.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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