Constants Strong Coupling Gauge Sum Prediction

The gauge sum

The ledger, a discrete record of recognition events, forces an eight-tick cycle in the Recognition Science framework. From that cycle the framework models the three gauge couplings, the strengths of the electromagnetic, weak, and strong forces, as quantities set by the geometry of a cube. The declaration gauge_sum_prediction states that the sum of their inverses equals the number of edges of a three-dimensional cube times π, which is 12π. The machine-checked library of formal theorems proves this identity and bounds the sum between 36 and 48, a narrow window that follows from π lying between 3 and 4.

The number 12π is not a measured value. It is a definitional choice, a model of how the three couplings might relate at a high energy scale. The framework's library proves the identity as a matter of arithmetic, not as a physical law. The sum's value is a structural constraint, the framework's analog of gauge coupling unification, the idea that the three forces merge into one at extreme energies. The claim is that if the framework's geometric picture holds, then the inverse couplings must add to 12π; it does not say which coupling carries which share.

The framework also defines a prediction for the strong coupling at the Z boson mass, roughly 91 GeV: alpha_s_prediction equals φ⁻³/π, where φ is the golden ratio. The library proves this quantity is positive, a modest but real theorem. The structural prediction stated in the framework's documentation is that the strong coupling should equal φ⁻ᵏ for some integer k, with k fixed by the energy running from the recognition scale down to the Z mass. This is a hypothesis with a named falsifier: the measured value of α_s(M_Z) = 0.1180 ± 0.0009 from the Particle Data Group's 2024 review. The framework's expression lands near that window, but the framework does not prove the running, the integer k, or the match.

In plain terms, the gauge sum is a clean, checkable piece of the framework's internal logic, a geometric constraint with a fixed numerical value. What it does not do is derive the strong coupling constant from first principles. The exact value of α_s, the fine details of its energy dependence, and the bridge from the recognition scale to laboratory energies all remain open targets. The sum is a promise about structure, not a receipt for a measurement.

noncomputable def gauge_sum_prediction : ℝ :=
  (cube_edges 3 : ℝ) * Real.pi

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]

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

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

What this page does not claim

This answer does not claim that the strong coupling constant is derived from the framework. This answer does not claim that the gauge sum prediction matches any measured value. This answer does not claim that the integer k in the strong coupling prediction is known or proved.

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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