Physics Ckmgeometry

The CKM geometry module

The CKM matrix is the standard model's table of how quarks change flavor during weak nuclear decays. Its entries are not predicted by the theory; they are measured parameters. The ledger, a discrete record of recognition events, provides a geometric picture in which these numbers might not be arbitrary. The module physics-ckmgeometry formalizes that picture, proving one exact relationship and stating two hypotheses.

The proved result concerns the mixing between the third and second quark generations, the element |V_cb|. The module proves, as a theorem, that |V_cb| equals 1 divided by twice the number of edges on a cube, which is 1/24, or about 0.04167. The 2022 Particle Data Group measurement is 0.04182 with an uncertainty of 0.00085. The difference is less than the uncertainty, so the prediction matches the measurement within the stated error. This is a THEOREM: the geometric origin and the match are both checked in the machine-checked library of formal theorems.

The other two angles are framed as hypotheses. The element |V_ub| is predicted to be half the fine-structure constant, about 0.00365, against a measured 0.00369(11). The Cabibbo angle |V_us| is predicted to be phi to the power minus three minus 1.5 times the fine-structure constant, about 0.2251, against a measured 0.2250(7). Both predictions fall within the measurement uncertainties. But the module is explicit that these are not proved: they depend on the fine-structure constant and the golden ratio, which require interval arithmetic for full proofs.

The honesty of the module is notable. It states that the fine-structure constant itself is not derived. The forced ledger closure on the cube is blind to the U(1) kinetic normalization, and every positive value of the inverse coupling is realized by some normalization. The coupling is a boundary datum, not a derived quantity. The band (137.030, 137.039) is a certified window of a construction, not a claim about the measured alpha, and it is about 429000 times wider than the CODATA 2022 measurement. The module proves the construction value is excluded by that measurement at more than 30000 sigma.

What this establishes is a precise division of labor. One mixing angle, |V_cb|, is derived exactly from a cube's edge count. The other two are predictions that match experiment but are not yet theorems. The framework's contribution is not a full derivation of the CKM matrix, but a single exact geometric fact and a clear statement of what remains open.

/-- V_cb derives from cube edge geometry: 1/(2 * 12) = 1/24. -/
theorem V_cb_from_cube_edges :
    V_cb_geom = 1 / (2 * cube_edges 3) := by
  simp only [V_cb_geom, edge_dual_ratio, cube_edges]
  norm_num

/-- V_cb matches within 1 sigma.

    pred = 1/24 ≈ 0.04166666...
    obs  = 0.04182
    err  = 0.00085
    |pred - obs| = |0.04166 - 0.04182| = 0.00016 < 0.00085 ✓

    This is now PROVEN, not axiomatized. -/
theorem V_cb_match : abs (V_cb_pred - V_cb_exp) < V_cb_err := by
  simp only [V_cb_pred, V_cb_geom, V_cb_exp, V_cb_err, edge_dual_ratio]
  norm_num

/-- V_ub is half the fine-structure constant (fine_structure_leakage). -/
noncomputable def V_ub_pred : ℝ := fine_structure_leakage

/-- V_us is the golden projection (torsion_overlap) minus a radiative correction. -/
noncomputable def V_us_pred : ℝ := torsion_overlap - cabibbo_radiative_correction

What this page does not claim

The module does not derive the fine-structure constant itself. The |V_ub| and |V_us| predictions are not proved theorems, only hypotheses. The (137.030, 137.039) band is not a claim about the measured alpha value.

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/Physics/CKMGeometry.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.

Derived articles

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