Constants Strong Coupling Gauge Sum Value

The gauge sum identity

The strong nuclear force binds quarks into protons and neutrons. Its strength is not constant; it changes with energy. At the mass of the Z boson, the carrier of the weak force, experiments measure it at about 0.1180 plus or minus 0.0009 (PDG 2024). A separate quantity, the reciprocal of the fine-structure constant for electromagnetism, is about 137.036. These numbers look arbitrary. The Recognition Science (RS) framework attempts to show they are not arbitrary, but are forced by a discrete structure of eight ticks, a ledger, a discrete record of events that reality keeps.

The framework's library, a machine-checked collection of formal theorems, contains a definition called gauge_sum_prediction. It adds the reciprocals of the three gauge couplings at the recognition scale, the scale where the framework's structure applies. The definition sets this sum equal to the number of edges of a three-dimensional cube, which is 12, multiplied by pi. The theorem gauge_sum_value proves this equality: gauge_sum_prediction = 12 * Real.pi. The proof is a direct computation, unfolding the definition of cube_edges and simplifying. A second theorem, gauge_sum_bounds, proves the sum lies strictly between 36 and 48, which follows from pi being between 3 and 4.

This result is a structural identity. It states that the sum of the reciprocals of the three couplings equals a number built from the geometry of a cube. It does not, by itself, give the value of any individual coupling. The strong coupling prediction is a separate definition, alpha_s_prediction, set to phi to the power of negative 3 divided by pi, where phi is the golden ratio. The framework has a certificate structure, StrongCouplingCert, that packages the positivity of the strong coupling prediction, the gauge sum identity, and the bounds into one object. The theorem strong_coupling_cert_exists proves this certificate is nonempty.

In plain terms, the declaration establishes a clean numerical relationship among three force strengths, a relationship that has a geometric flavor. It is a theorem about a definition, not a measurement. The framework's claim is that this relationship is forced by its structure, not that it has measured the couplings. The value of the strong coupling at the Z mass, 0.1180, is an external measurement used for comparison, not a result derived here. The identity is a piece of the framework's internal consistency, a check that its structure produces numbers in the right range.

The consequence is a concrete, checkable statement: the reciprocal sum of the three gauge couplings equals 12 pi. This is a fact a reader can verify from the definitions alone. It shows the framework's geometry produces a specific number, not a vague range. The bounds place the sum between 36 and 48, a narrow window. This is the kind of exact statement that distinguishes a formal derivation from a hand-waving analogy.

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]

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

This declaration does not derive the value of the strong coupling constant at the Z mass. It does not prove the measured value of 0.1180 from the framework. It does not establish that the three couplings unify at a single energy scale.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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