edge_dual_ratio
The edge dual ratio is the rational constant 1/24 that quantifies the coupling strength between face-centered generation-2 states and vertex generation-3 states through dual-lattice transitions in the cubic voxel model. CKM matrix derivations in particle physics cite this value when obtaining the geometric prediction for V_cb. The declaration is a direct abbreviation that assigns the reciprocal of the total edge transition slots without further reduction steps.
claimThe edge-dual ratio is the rational number $1/24$, which encodes the coupling between face-centered states (generation 2) and vertex states (generation 3) arising from dual-lattice transitions in the cubic voxel topology.
background
This definition sits inside the module that supplies the geometric foundation for mixing matrices by imposing cubic voxel topology constraints on the CKM and PMNS parameters. The ratio counts one transition slot per 24 available lattice positions, where a face is treated as dual to an edge. Upstream results supply the fine-structure constant alpha as the electromagnetic scale, the vacuum as the gauge-bond configuration with all bonds at rung zero, and parity patterns that label ledger states.
proof idea
The declaration is a direct definition that assigns the rational 1/24. Downstream theorems unfold it via simp and norm_num to recover V_cb_geom as the inverse of twice the cube edge count in three dimensions.
why it matters in Recognition Science
The constant supplies the geometric input for V_cb in CKMElementScoreCardCert and V_cb_match, which verify agreement with experiment inside one sigma. It realizes the cubic topology constraints that force mixing angles from the Recognition Science chain (T8 forcing D=3) and feeds the Jarlskog witness positivity result. The construction closes the link between lattice geometry and observed quark mixing without additional parameters.
scope and limits
- Does not derive the slot count 24 from the J-cost functional or phi-ladder.
- Does not include radiative corrections or loop effects.
- Does not extend to PMNS neutrino mixing angles.
- Does not prove the underlying cubic voxel structure.
Lean usage
def V_cb_geom : ℚ := edge_dual_ratioformal statement (Lean)
29def edge_dual_ratio : ℚ := 1 / 24
proof body
Definition body.
30
31/-- The parity-split fine structure leakage.
32 Represents the coupling between non-adjacent generations (1 and 3)
33 mediated by the vacuum polarization field alpha. -/