Foundation Qrft Yukawa Coupling From Jcost Yukawa At Rung8

The electron rung

In the standard model of particle physics, the mass of a fermion such as the electron or top quark is not a free number but a product: the dimensionless Yukawa coupling y_f times the Higgs vacuum expectation value v = 246 GeV. The coupling measures how strongly a fermion interacts with the Higgs field; heavier fermions have larger couplings, and the electron, with its tiny mass, has a coupling of about 2.9 × 10⁻⁶. This hierarchy, spanning many orders of magnitude, is one of the standard model's great unexplained patterns.

Recognition Science (RS) proposes a structural origin for that pattern. The framework models each fermion as sitting at a specific rung on a ladder of discrete recognition events, and defines the Yukawa coupling as recognition coherence: how well the fermion's rung aligns with the framework's fundamental cost of recognition. In symbols, the coupling at rung r is

y_f(r) = 1 − J(φ^r−8),

where φ is the golden ratio and J is the framework's cost function, which measures the price of a recognition event. The framework proves, in a machine-checked library of formal theorems, that this coupling never exceeds 1, and that at rung 8, the electron's rung, the cost J(φ⁰) = J(1) = 0, so the coupling is exactly 1. That is the content of the declaration yukawaAt_rung8: it establishes, as a theorem, that the electron's coupling equals unity in RS units.

This is not a statement about the measured electron Yukawa coupling, which is tiny. It is a statement about the framework's internal scaling: rung 8 is the reference point where the recognition cost vanishes, so the coupling is maximal. The framework's structural predictions for other fermions, such as the top quark near unity and the bottom quark near 0.02, are comparisons against measured values, not theorems. The declaration itself does not claim that the electron's physical coupling is 1, nor that the rung assignment is unique, nor that the framework's mass ladder has been derived from first principles. It proves a mathematical identity about the cost function at a chosen reference rung.

What the declaration changes is the interpretation of the mass hierarchy: instead of arbitrary constants, the hierarchy becomes a consequence of where each fermion sits on a universal ladder. The electron's tiny mass is not a small coupling but a large distance from the reference rung. That reframing is the framework's contribution; the empirical match to the measured top and bottom couplings remains a prediction to be tested, not a proven result.

/-- 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

The electron's physical Yukawa coupling is measured to be 1. The rung assignment for the electron is derived from first principles. The framework's mass ladder has been proven to match all measured fermion masses.

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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