Foundation Qrft Yukawa Coupling From Jcost

Mass from recognition cost

The Standard Model of particle physics assigns every fermion, such as an electron or a top quark, a mass through a dimensionless number called its Yukawa coupling, a knob that sets how strongly the particle interacts with the Higgs field. The measured values span five orders of magnitude, from the electron's tiny 0.00001 to the top quark's near-unity 0.99, and the Standard Model offers no explanation for why each fermion has its particular value. Recognition Science proposes that the hierarchy is not random but forced by a single mathematical object, the cost, a function that measures how expensive it is for reality to recognize a given scale.

The cost function J(x) is defined by five plain conditions: it is symmetric under x to 1/x, vanishes at x equals 1, obeys a forced composition law, is calibrated, and is continuous. A proved theorem in the framework's machine-checked library of formal theorems shows these conditions force J(x) = (x + 1/x)/2 - 1. From this, the framework derives a ladder of scales based on the golden ratio phi, approximately 1.618. Each fermion sits at a specific rung of this ladder, and its Yukawa coupling is defined as y_f = 1 - J(phi^(rung_f - 8)). At rung 8, the electron rung, J(1) equals 0, so the coupling is exactly 1, and the framework proves the coupling is always bounded above by 1.

The framework's module YukawaCouplingFromJCost establishes this mapping formally. It defines the coupling yukawaAt r for any integer rung, proves the coupling never exceeds 1, and proves it equals exactly 1 at rung 8. The mass hierarchy then falls out of the ladder structure: a heavier fermion sits at a higher rung, where phi raised to a larger power makes J(x) smaller, so the coupling y_f = 1 - J(x) approaches 1. A lighter fermion at a lower rung has a larger J(x) and thus a smaller coupling. The docstring gives structural predictions: the top quark at rung 27 has y_t approximately 1, the bottom quark at rung 19 has y_b approximately 0.02, and the tau lepton at rung 17 has y_tau approximately 0.01, with the bottom value agreeing with the measured PDG value of 0.018.

What changes with this result is the status of the mass hierarchy. In the Standard Model, the Yukawa couplings are free input parameters, nineteen arbitrary numbers fitted to experiment. In Recognition Science, they are outputs of a single forced function, with each fermion's rung position determining its coupling. The framework does not yet prove which rung each fermion occupies; that assignment remains a structural prediction, and the comparison against measured PDG values is an empirical check, not a theorem. But the module does prove the backbone: the coupling formula itself, its upper bound, and its unity at the electron rung.

structure YukawaCert where
  bounded_above : ∀ r : ℤ, yukawaAt r ≤ 1
  rung8_unity : yukawaAt 8 = 1

/-- Yukawa coupling for a fermion at φ-ladder rung `r` (relative to rung 8). -/
def yukawaAt (r : ℤ) : ℝ := 1 - Jcost (phi ^ (r - 8))

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

This answer does not claim the framework proves which rung each fermion occupies. This answer does not claim the framework derives the fine-structure constant alpha. This answer does not claim the measured PDG values are theorems; they are empirical comparisons.

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)

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