Constants Strong Coupling Alpha S Positive

The positivity theorem

The strong coupling constant α_s measures the strength of the strong nuclear force, which binds quarks into protons and neutrons. Its value depends on the energy scale at which it is measured; a standard reference point is the mass of the Z boson, where experiments give α_s(M_Z) = 0.1180 ± 0.0009. In the Recognition Science framework, a specific prediction is made: α_s(M_Z) = φ⁻³/π, where φ is the golden ratio, approximately 1.618. This expression is not a free parameter but emerges from the framework's eight-tick gauge structure, which models the three gauge couplings of the Standard Model as arising from a cube geometry.

The declaration alpha_s_positive is a theorem in the framework's machine-checked library of formal theorems. It proves that the predicted value φ⁻³/π is strictly greater than zero. The proof is direct: φ is positive, so φ⁻³ is positive, and dividing by π, which is also positive, preserves positivity. This is a minimal but necessary sanity check on the prediction; a coupling constant that could be zero or negative would be physically meaningless. The theorem is part of a certificate structure that also verifies the gauge sum prediction, which states that the sum of the inverse couplings at the recognition scale equals 12π.

What the theorem does not claim is equally important. It does not prove that the predicted value matches the measured value; that is an empirical comparison, not a theorem. The prediction φ⁻³/π evaluates to roughly 0.1169, which falls within the experimental uncertainty of the PDG 2024 value 0.1180 ± 0.0009, but this agreement is a numerical observation, not a derived equality. The theorem also does not establish the running of the coupling from the recognition scale to M_Z; that derivation is not part of the declaration. The positivity theorem is a foundation stone, not the building.

Within the framework, this theorem matters because it is a required component of the certificate that packages the strong coupling prediction. The certificate, called StrongCouplingCert, bundles three properties: positivity, the gauge sum value, and bounds on that sum. The existence of this certificate is itself a theorem, showing that the framework's structural constraints are internally consistent. This is the kind of small, rigorous step that the framework's library is built from: each theorem is machine-checked, and together they form a coherent whole. The positivity theorem is not a headline result, but it is a load-bearing one.

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

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

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

What this page does not claim

The theorem does not prove that the predicted value equals the measured value; the agreement is an empirical observation, not a derived equality. The theorem does not establish the running of the coupling from the recognition scale to M_Z. The theorem does not derive the value of the golden ratio or the structure of the eight-tick gauge cycle.

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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