Constants Strong Coupling Strong Coupling Cert Exists

The strong coupling certificate

The strong coupling constant, written α_s, measures how powerfully quarks and gluons interact. At the energy scale of the Z boson, experiments give α_s = 0.1180 ± 0.0009. The standard model does not predict this number; it is an input measured from collider data.

Recognition Science, a framework that derives physical structure from a ledger of recognition events, offers a geometric picture. It models the three gauge couplings as emerging from an 8-tick structure and cube geometry. The framework's library, a machine-checked collection of formal theorems, contains a declaration named strong_coupling_cert_exists. This theorem establishes that a certificate exists: a formal object bundling three proved facts about the strong coupling prediction.

The certificate's first fact is that the framework's predicted value, phi^(-3)/pi, is positive. The second is that the sum of inverse gauge couplings equals 12π. The third bounds that sum strictly between 36 and 48. These are exact statements proved in the library, with no unproved assumptions.

What the certificate does not do is derive the measured value 0.1180. The framework predicts a specific number, but the certificate only proves its positivity and its place in a gauge sum. The match to experiment remains a separate empirical check, not a theorem. The certificate is a structural statement, not a numerical derivation.

noncomputable def alpha_s_prediction : ℝ := phi ^ (-(3 : ℤ)) / Real.pi

theorem strong_coupling_cert_exists : Nonempty StrongCouplingCert :=
  ⟨{ positive := alpha_s_positive
     gauge_structure := gauge_sum_value
     gauge_bounded := gauge_sum_bounds }⟩

theorem alpha_s_positive : 0 < alpha_s_prediction := by
  unfold alpha_s_prediction
  exact div_pos (zpow_pos phi_pos _) Real.pi_pos

theorem gauge_sum_value : gauge_sum_prediction = 12 * Real.pi := by
  unfold gauge_sum_prediction cube_edges
  simp [D]

theorem gauge_sum_bounds :
    (36 : ℝ) < gauge_sum_prediction ∧ gauge_sum_prediction < (48 : ℝ) := by
  rw [gauge_sum_value]
  constructor <;> nlinarith [Real.pi_gt_three, Real.pi_lt_four]

What this page does not claim

The certificate does not prove that the predicted strong coupling value equals the measured 0.1180. The certificate does not derive the strong coupling constant from first principles; it only proves structural properties of a specific prediction. The certificate does not establish that the framework's gauge structure is the unique explanation for the strong 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/Constants/StrongCoupling.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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