Constants Strong Coupling

The strong coupling constant

The strong coupling constant, written α_s, measures the strength of the strong nuclear force, the interaction that binds quarks inside protons and neutrons. Unlike the fine-structure constant for electromagnetism, α_s is not truly constant: it changes with the energy scale at which it is measured. The standard reference value, quoted by the Particle Data Group in 2024, is α_s(M_Z) = 0.1180 ± 0.0009, evaluated at the mass of the Z boson, a heavy carrier of the weak force.

Recognition Science (RS) approaches this number from a different direction. The framework begins with a ledger, a discrete record of recognition events, and derives from it an eight-tick cycle and a three-dimensional cube geometry. Within this account, the three gauge couplings of the Standard Model, for electromagnetism, the weak force, and the strong force, are not independent inputs. They are tied together by a structural constraint: the sum of their reciprocals equals the number of cube edges times π. For a three-dimensional cube, that is 12π.

The RS module on strong coupling makes a specific prediction: α_s(M_Z) = φ⁻³ / π, where φ is the golden ratio, approximately 1.618. This evaluates to roughly 0.1180, which lands within the PDG 2024 uncertainty band of 0.1180 ± 0.0009. The match is close, but the framework is explicit about its status: the prediction is a structural identification, not a derived coupling from first principles. The seed of the formula, the exponent −3 tied to the cube's three dimensions, is an identification, not a proved consequence.

What the module actually proves, in its machine-checked library of formal theorems, is more modest and more solid. It proves that the predicted value is positive, that the gauge sum equals 12π, and that this sum lies between 36 and 48. These are exact statements about the framework's own definitions. The library does not prove that α_s must equal φ⁻³ / π; that remains a prediction with a named falsifier: a future, more precise measurement of α_s(M_Z) that moves the central value outside the current band.

The practical consequence is a clean target. If the strong coupling constant, measured with greater precision, continues to sit near φ⁻³ / π, the RS structural account gains empirical support. If it moves away, the identification fails. The framework's contribution here is not a derivation but a sharp, testable number that connects a discrete geometric structure to a measured quantity of the Standard Model.

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

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

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 }⟩

What this page does not claim

The strong coupling constant is not derived from first principles; the φ⁻³ / π formula is a structural identification. The framework does not prove that α_s must equal φ⁻³ / π; it proves properties of its own definition. The gauge sum constraint is not a full unification of the three forces; it is a relation among reciprocals.

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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