Physics Anomalous Moments Anomalous E Tau Universal

The universality theorem

The anomalous magnetic moment of a charged lepton, written a_l = (g-2)/2, measures how much the particle's magnetism deviates from the simple Dirac prediction. For the electron, this deviation is famously small: about 0.001159652, a value measured to parts per trillion. For the heavier tau, the same quantity is roughly 0.001177, though far less precisely known. The classical story is that these numbers differ because the electron and tau have different masses, so higher-order quantum corrections, which depend on mass, split them apart.

In Recognition Science, the framework's machine-checked library of formal theorems contains a declaration named anomalous_e_tau_universal. It proves that the framework's own correction term, the part it adds on top of the standard Schwinger term, is identical for the electron and the tau. The reason is structural: both particles carry the same electric charge, Q = -1, and in the framework's ledger, a discrete record of recognition events, that charge maps to the same integer Z = 1332. Since the framework's correction depends only on Z, and not on mass, the theorem follows by direct computation: anomalous_moment e = anomalous_moment tau.

The theorem is honest about its scope. It does not claim the full measured anomalous moments are equal. The framework's correction is universal, but the full moment is defined as the Schwinger term plus the RS correction, with higher-loop, mass-dependent effects left out of the formal statement. The pack's own empirical note concedes the measured values differ: a_e ≈ 1.16e-3 versus a_tau ≈ 1.17e-3. So the theorem establishes a symmetry in one specific correction channel, not a prediction that experiment will find identical moments.

What the theorem does is isolate a clean structural fact: within this framework, the electron and tau share a common correction because they share a common charge. That is a statement about the framework's internal logic, verified by the kernel, and it is a different kind of claim from a numerical match to experiment. The framework models the correction as universal; the mass-dependent parts are explicitly not part of the theorem.

/-- Universality: same dimless target from equal Z (φ-ladder). -/
theorem anomalous_e_tau_universal : anomalous_moment Lepton.e = anomalous_moment Lepton.tau := by
  simp [anomalous_moment, rs_correction, gap_lepton, Z_lepton]
  -- Z same ⇒ gap same

noncomputable def anomalous_moment (l : Lepton) : ℝ :=
  schwinger + rs_correction l  -- Higher loops mass-dependent, but RS universal

/-- Empirical note: RS predicts universal correction; full a differs by mass-dependent loops (PDG a_e ≈ 1.16e-3, a_τ ≈ 1.17e-3 within bands). -/
@[simp] noncomputable def pdg_a_e : ℝ := 0.00115965218073  -- Placeholder CODATA

What this page does not claim

This theorem does not claim the measured anomalous magnetic moments of the electron and tau are equal. This theorem does not predict a numerical value for either anomalous moment that matches experiment. This theorem does not derive the fine-structure constant or any other coupling constant.

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

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