IndisputableMonolith.StandardModel.CKM_Cabibbo_Exact_RS
Module packaging the Recognition Science treatment of the exact Cabibbo angle in the CKM matrix. It defines a domain cost, a positive canonical threshold, and an inhabited certificate that the Cabibbo sector matches the RS prediction. Phenomenologists comparing RS quark mixing to data would cite the certificate. Structure is mostly definitional, with nonnegativity and positivity lemmas.
claimThe module introduces a domain cost $C$, proves $C \ge 0$ and records its on-diagonal value, fixes a canonical threshold $\tau > 0$, and supplies an inhabited certificate that the Cabibbo angle of the CKM matrix is exact in Recognition Science units.
background
The CKM matrix encodes charged-current mixing among the three quark generations. Its leading $2\times 2$ block is historically the Cabibbo angle. Recognition Science derives mixing angles from the same $J$-cost and $\varphi$-ladder structure that forces the gauge and mass spectra along the T5--T8 chain (unique $J$, golden fixed point, eight-tick octave, $D=3$).
This module lives in the StandardModel layer. It imports RS constants (tick $\tau_0$, $\varphi$) and the cost calculus. Sibling definitions introduce a nonnegative domain cost, evaluate that cost at equality, and fix a strictly positive canonical threshold against which Cabibbo exactness is certified.
Notation follows the rest of the monolith: costs are real-valued, thresholds are positive reals, and certificates are Prop-level packages with inhabited witnesses.
proof idea
Definition-and-certificate module rather than a deep derivation. The domain cost is introduced as a real functional, shown nonnegative, and evaluated on the diagonal. A separate definition fixes the canonical threshold and proves it is positive. The Cabibbo exactness certificate is a structure packaging those ingredients; an inhabitation lemma supplies a concrete witness. No long tactic scripts: the argument is assembly of local positivity facts around the certificate type.
why it matters in Recognition Science
Places an exact Cabibbo claim inside the RS Standard Model layer, so later phenomenology can cite a single certificate rather than re-deriving the angle from $J$ and $\varphi$. The module currently has no recorded downstream dependents in the graph, so it is a leaf packaging step for quark-mixing claims. It sits beside mass-ladder and coupling work that already use $\varphi$-powers and the eight-tick structure; the Cabibbo certificate is the natural mixing counterpart. Open follow-ons include full $3\times 3$ CKM phases and numerical comparison to the PDG Cabibbo band.
scope and limits
- Does not derive the full $3\times 3$ CKM matrix or CP phase.
- Does not claim a numerical PDG fit or error bar.
- Does not re-prove $J$-uniqueness or $\varphi$ forcing (imports Constants/Cost).
- Does not treat PMNS lepton mixing.
- Does not assert experimental units; values stay in RS-native normalization.