Foundation Qrft Fermion Kinetic Cert

Fermion kinetic certificate

The standard model's fermion kinetic Lagrangian, written ψ̄(iγ^μ ∂_μ − m)ψ, describes how spin-1/2 particles propagate and acquire mass. The first term, ψ̄ γ^μ ∂_μ ψ, is the kinetic energy of a moving fermion; the second, m ψ̄ψ, is the rest-mass energy. In the Recognition Science framework, this familiar object is reinterpreted: the mass term becomes a cost on the fermion's recognition ratio, and the kinetic term becomes a derivative on a recognition lattice. The framework's machine-checked library of formal theorems packages this mapping into a certificate called FermionKineticCert.

The certificate establishes three structural facts. First, it records the standard model's count of 15 Weyl fermions per generation, derived as 5 electroweak sectors times 3 color copies. Second, it defines the fermion mass at a ladder rung k as m_k = m_0 · φ^k, where φ is the golden ratio, roughly 1.618. Third, it proves that adjacent rungs of this mass ladder are separated exactly by the golden ratio: m_{k+1} / m_k = φ. These are not empirical fits; they are structural predictions from the recognition φ-ladder already established in the mass-ratio modules.

In plain terms, the module says: if fermion masses sit on a ladder whose steps are the golden ratio, then positivity and the ratio condition follow as theorems. The certificate assembles these results into a single structure with no unproved assumptions, checked by the machine. The practical consequence is that the standard model's fermion sector, including its particle count and mass spacing, is expressible as a consequence of the recognition cost function rather than as a list of free parameters.

This is a structural claim, not a numerical prediction of specific masses. The base mass m_0 remains a free parameter; what is forced is the ratio between adjacent rungs. The certificate does not derive the fine-structure constant, does not prove the Riemann Hypothesis, and does not claim that the golden ratio explains every mass ratio in the particle data book. It establishes a clean, checkable relationship between the recognition framework and the standard model's fermion kinetic sector.

structure FermionKineticCert where
  fermions_per_gen : fermionsPerGeneration = 5 * 3
  mass_pos : ∀ {m0 : ℝ}, 0 < m0 → ∀ k, 0 < fermionMassAt m0 k
  mass_ratio : ∀ {m0 : ℝ}, 0 < m0 → ∀ k,
    fermionMassAt m0 (k + 1) / fermionMassAt m0 k = phi

/-- Structural derivation: 5 EW sectors × 3 colors = 15. -/
theorem fermionsPerGeneration_val : fermionsPerGeneration = 5 * 3 := by
  unfold fermionsPerGeneration; rfl

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']

/-- Fermion kinetic sector certificate. -/
def fermionKineticCert : FermionKineticCert where
  fermions_per_gen := fermionsPerGeneration_val
  mass_pos := @fermionMassAt_pos
  mass_ratio := @fermionMassAt_adjacent_ratio

What this page does not claim

This module does not derive specific fermion masses from first principles. This module does not prove the Riemann Hypothesis or derive the fine-structure constant. This module does not claim that every standard model mass ratio equals the golden ratio.

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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