Foundation Qrft Yukawa Coupling From Jcost Yukawa Cert

The Yukawa certificate

In the Standard Model of particle physics, fermion masses arise through the Yukawa coupling, a dimensionless number that measures how strongly a fermion interacts with the Higgs field. The top quark couples with a strength near one; the electron couples far more weakly. Recognition Science (RS) offers a structural account of these couplings: each fermion sits at a rung on a ladder, and its coupling is a function of the recognition cost at that rung.

The central object is the recognition cost, a discrete record of how much it costs to recognize a given scale. The framework's proved cost function J(x) = (x + 1/x)/2 - 1 assigns a nonnegative cost to any positive scale x, with cost zero at x = 1. The Yukawa coupling for a fermion at rung r is defined as y_f = 1 - J(φ^(r-8)), where φ is the golden ratio and r is measured relative to rung 8, the electron rung. At rung 8, the argument is φ^0 = 1, so the cost is zero and the coupling is exactly one.

The machine-checked declaration YukawaCert establishes two facts. First, for every integer rung r, the coupling is at most one: yukawaAt r ≤ 1. Second, at rung 8 the coupling equals one exactly: yukawaAt 8 = 1. These are proved in the framework's library of formal theorems with no unproved assumptions. The certificate is a compact package: two theorems bundled into one structure, ready for other proofs to consume.

What YukawaCert does not claim is just as important. It does not claim that any particular fermion sits at any particular rung; the rung assignments for the top, bottom, and tau are structural predictions in the docstring, not part of the certificate. It does not claim that the electron's coupling is measured to be one; the electron's actual Yukawa coupling is small, and the rung-8 equality is a normalization convention, not an empirical match. It does not claim that the coupling values agree with experiment; the certificate is purely a formal statement about the defined function, and any comparison against measured PDG values is a separate empirical check.

What the certificate does buy is a clean foundation: any later RS proof that needs the Yukawa coupling to be bounded and normalized can cite YukawaCert rather than re-deriving those facts. The certificate is the load-bearing wall that lets the framework's mass-to-recognition mapping stand on proved ground.

/-- At rung 8 (the electron rung), J(φ^0) = J(1) = 0, so coupling = 1. -/
theorem yukawaAt_rung8 : yukawaAt 8 = 1 := by
  unfold yukawaAt
  norm_num [Jcost_unit0]

theorem yukawaAt_bounded_above (r : ℤ) :
    yukawaAt r ≤ 1 := by
  unfold yukawaAt
  have hpos : 0 < phi ^ (r - 8) := zpow_pos Constants.phi_pos _
  linarith [Jcost_nonneg hpos]

def yukawaCert : YukawaCert where
  bounded_above := yukawaAt_bounded_above
  rung8_unity := yukawaAt_rung8

What this page does not claim

The certificate does not assert any specific rung assignment for any fermion. The certificate does not claim the electron's measured coupling equals one. The certificate does not claim agreement with any measured PDG value.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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