Constants Strong Coupling Gauge Sum Bounds

The gauge sum bound

In particle physics, the three fundamental forces of the Standard Model, electromagnetism, the weak force, and the strong force, each have a coupling constant that measures how strongly they act. Gauge coupling unification is the idea that these three constants, which look very different at low energies, might converge to a single value at some very high energy scale. The Recognition Science framework contains a machine-checked library of formal theorems that explores a specific geometric version of this unification idea.

Within this framework, the three inverse coupling constants at the recognition scale are modeled as summing to a value determined by the edges of a three-dimensional cube. The declaration gauge_sum_bounds establishes a precise inequality: this sum is greater than 36 and less than 48. This is a proved theorem in the framework's library, meaning it follows logically from the framework's definitions with no gaps. The bound itself is not a prediction of the strong coupling constant's value; it is a structural constraint on the sum of all three inverse couplings.

The framework also offers a specific prediction for the strong coupling constant at the Z boson mass, α_s(M_Z) = φ⁻³/π, which evaluates to approximately 0.1169. This is a separate definition within the framework, not derived from the gauge sum bound. The measured PDG 2024 value for α_s(M_Z) is 0.1180 ± 0.0009, so the framework's prediction sits close to but outside the experimental error bar. The gauge_sum_bounds theorem does not improve this agreement; it only constrains the sum of the three couplings, not the strong coupling alone.

What the bound does establish is a consistency check. It shows that the framework's geometric model of gauge coupling unification is internally coherent: the sum of inverse couplings must lie in a finite, bounded interval. This is a structural fact about the model, not an empirical success. A reader should understand that this theorem is a piece of internal mathematics, a guarantee that the model's numbers stay within a plausible range, rather than a derivation of any measured quantity.

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]

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

What this page does not claim

This theorem does not derive the value of the strong coupling constant α_s. The bound does not imply that the framework's prediction for α_s matches the measured PDG value. The theorem does not establish that gauge coupling unification occurs in nature.

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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