Physics Lepton Universality3 From Jcost

Lepton universality and the cost ledger

Lepton universality is the physics observation that the three charged leptons, the electron, the muon, and the tau, couple to the weak nuclear force with the same strength. The Standard Model predicts this equality because the weak coupling constant does not depend on lepton flavor. Measurements of tau decay and Z boson branching ratios confirm the prediction to about a tenth of a percent. The ratio g_tau/g_mu, where g is the weak coupling for each lepton, is measured at 1.0011, within 0.1 percent of 1.

Recognition Science (RS) models this equality through a ledger, a discrete record of recognition events, and its forced cost, the price the ledger must pay to register a distinction. The framework proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. From this cost, RS derives a phi-power ladder for particle masses, where phi is the golden ratio. In this ladder, the ratio of couplings between lepton generations is predicted to be phi^(n_tau - n_mu)/phi^(n_tau - n_mu), which equals 1 exactly. The framework models the small measured deviation as a correction from rung mismatch, estimated as delta = J(phi)^2 per rung difference, which gives 1.014 versus the measured 1.0011.

The module in question, Lepton_Universality3_FromJCost, does not prove the full universality claim. Its Lean code proves three general facts about the cost function: it vanishes when the two masses are equal, it is nonnegative for positive inputs, and the golden ratio minus 3/2 is positive. These are universal properties of J, not specific to leptons. The module defines domainCost as J(m/e) without defining m and e in lepton terms, so the universality result remains a research note, not a theorem. The machine-checked library of formal theorems proves only the general cost facts.

What the module does establish is that the cost function has the right shape for a universality argument. If the ratio m/e equals 1, the cost is zero, meaning no distinction is registered. The nonnegativity ensures the cost is a genuine price, never a negative reward. The positivity of phi - 3/2 is a technical threshold used in the ladder. These facts are the groundwork, but they do not by themselves force lepton universality. The bridge from cost to lepton couplings remains open.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The module does not prove lepton universality; it proves general cost properties only. The framework does not derive the measured 1.0011 deviation; it models it as a correction. The phi-power ladder is a model, not a theorem, for lepton couplings.

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/Physics/Lepton_Universality3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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