Foundation Qrft Yukawa Coupling From Jcost Yukawa At Bounded Above

The coupling bound

In the Standard Model of particle physics, a fermion's mass comes from its Yukawa coupling, a dimensionless number that measures how strongly the particle interacts with the Higgs field. The top quark has a coupling near 1, while the electron's is far smaller. Recognition Science (RS) offers a structural account of this hierarchy: each fermion sits at a rung on a ladder of scaling factors, and its coupling is defined as one minus a recognition cost, the price the framework's ledger charges for that recognition event.

The declaration yukawaAt_bounded_above proves, in the framework's machine-checked library of formal theorems, that this coupling is always at most 1. The proof is short: the cost function J is nonnegative, so subtracting it from 1 cannot push the result above 1. This is a theorem, not a model choice, and it holds for every integer rung r. At rung 8, the electron rung, the cost is exactly zero, so the coupling equals 1 exactly; the framework's own theorem yukawaAt_rung8 establishes that limiting case.

What the bound does not claim is just as important. It does not say that any particular fermion's measured coupling equals the framework's value; that comparison is an empirical check, not a theorem. It does not derive the fine-structure constant. It does not say that the coupling cannot be negative, only that it cannot exceed 1. The theorem is a structural ceiling, a consistency guarantee about the framework's own definitions, not a prediction about any specific particle.

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]

/-- 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]

What this page does not claim

The theorem does not predict any measured Yukawa coupling value for a specific fermion. The theorem does not establish that the coupling is nonnegative, only that it is at most 1. The theorem does not derive the fine-structure constant or any other Standard Model coupling.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: