Strong Coupling alpha_s from the Gauge Sum

1. Setting

Strong Coupling alpha_s from the Gauge Sum is anchored in Constants.StrongCoupling. The page is not a loose explainer: it is a public map from the Recognition Science forcing chain into one Lean-checked declaration bundle. The primary anchor determines what is proved, and the surrounding declarations show how the result is used.

2. Equations

(E1)

$$ J(x)=\frac12(x+x^{-1})-1,\qquad \varphi^2=\varphi+1 $$

Shared constant-forcing backbone.

3. Prediction or structural target

• alpha_s(M_Z): predicted RS gauge-sum prediction (dimensionless); empirical 0.1180(9). Source: PDG 2024

This page is currently a structural derivation. Where the claim has direct empirical content, the prediction table gives the measurable target; otherwise the claim is a formal bridge inside the Lean canon.

4. Formal anchor

The primary anchor is Constants.StrongCoupling..alpha_s_prediction.

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

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

/-! ## Structural Constraints

The three gauge couplings at the recognition scale satisfy:

5. What is inside the Lean module

Key theorems:

• alpha_s_positive
• gauge_sum_value
• gauge_sum_bounds
• strong_coupling_cert_exists

Key definitions:

• alpha_s_prediction
• gauge_sum_prediction
• StrongCouplingCert

6. Derivation chain

• alpha_s_prediction - alpha_s prediction
• alpha_s_positive - alpha_s positive
• gauge_sum_bounds - Gauge sum bounds
• strong_coupling_cert_exists - Cert exists

7. Falsifier

A precision measurement outside the stated RS interval, after checking SI calibration and systematic error, refutes this constant-level derivation.

8. Where this derivation stops

Below this page the chain reduces to the RS forcing sequence: J-cost uniqueness, phi forcing, the eight-tick cycle, and the D=3 recognition substrate. If any upstream theorem changes, this page must be versioned rather than patched silently. The published URL is stable, but the version field is the contract.

9. Reading note

The minimal way to audit this page is to open the first Lean anchor and then walk the supporting declarations listed above. If the primary theorem is a module-level anchor, the key theorems section names the internal declarations that carry the mathematical load. This keeps the public derivation readable without severing it from the proof object.

10. Audit path

To audit strong-coupling-from-rs, start with the primary Lean anchor Constants.StrongCoupling.alpha_s_prediction. Then inspect the theorem names listed in the module-content section. The page is intentionally built so the public explanation is not a substitute for the proof object; it is a map into it. The mathematical dependency is the same in every case: reciprocal cost fixes J, J fixes the phi-ladder, the eight-tick cycle fixes the recognition clock, and the domain theorem listed above supplies the last step. If that last step is empirical, the falsifier section names what observation would break it. If that last step is formal, a Lean-checkable counterexample is the relevant failure mode.