jarlskog_structural
plain-language theorem explainer
The structural Jarlskog invariant assembles the RS-derived Wolfenstein A, Cabibbo λ, and Berry CP phase into the unnormalized measure J = A² λ⁶ sin δ of quark-sector CP violation. Cosmology and SM certificates cite it as the CP source for the sign of baryon asymmetry and for the small-but-nonzero hierarchy. It is a direct product of three upstream structural parameters (torsion-gap A, φ-ladder λ, Berry-phase δ = π/2).
Claim. Define the structural Jarlskog invariant by $J_{\mathrm{struct}} = A^2 \lambda^6 \sin\delta$, where $A$ is the Wolfenstein $A$ from the torsion-gap ratio $\Delta\tau_{23}/\Delta\tau_{12}=6/11$, $\lambda=(\varphi-1)^2/\varphi=\varphi^{-3}$ is the structural Cabibbo parameter, and $\delta$ is the Berry-phase CP angle (equal to $\pi/2$).
background
The Jarlskog invariant $J_{CP}$ is the unique rephasing-invariant measure of CP violation in the quark sector. In the Standard Model one has $J=\mathrm{Im}(V_{us}V_{cb}V_{ub}^V_{cs}^)\approx 3.08\times 10^{-5}$. In the Wolfenstein parametrization this reduces to $J=A^2\lambda^6\eta\approx A^2\lambda^6\sin\delta$.
This module builds $J$ from RS geometry: torsion gaps and flip-count asymmetry fix the CKM angles, while the Berry phase on the Gray-code path fixes the CP angle. Upstream, the structural $A$ is the absolute torsion-gap ratio $\Delta\tau_{23}/\Delta\tau_{12}=6/11$; the structural $\lambda$ is $(\varphi-1)^2/\varphi=\varphi^{-3}$ from 1–2 mixing and flip ratio; the CP angle is $\delta=\gamma(\mathrm{gen1})-\gamma(\mathrm{gen2})=\pi/2$, so $\sin\delta=1$ (maximal per cycle).
The local setting is the Q₃-cube derivation of CKM and CP phase (CKMFromCube, CPPhaseDerivation), feeding the unnormalized structural product used for sign and hierarchy statements rather than a fitted numerical match.
proof idea
Pure definition, not a proved theorem. The body is the three-factor product of the upstream structural reals: square of the torsion-gap Wolfenstein $A$, sixth power of the φ-ladder Cabibbo $\lambda$, and $\sin$ of the Berry CP angle. No tactics or lemmas; evaluation is by unfolding those three defs.
why it matters
This is the module’s primary structural object: every positivity, smallness, and certificate result in JarlskogInvariant unfolds or cites it (jarlskog_positive, cp_violation_exists, cp_small_but_nonzero, JarlskogCert). Downstream cosmology treats it as the CP source: cp_asymmetry_parameter is definitionally equal to it; cp_source_positive and eta_B_structural := J/g_★ inherit the sign; derivation_chain_complete and BaryonAsymmetryCert list $J>0$ among the Sakharov ingredients that force a positive baryon asymmetry.
Framework landmarks in play: three generations from $D=3$ (T8), Gray-code chirality and flip counts $[4,2,2]$, torsion gaps ${0,11,17}$, and the Berry phase that sets $\delta=\pi/2$. The φ-suppression in $\lambda^6=(\varphi^{-3})^6$ is what makes $J$ small without fine-tuning. Magnitude matching to $3.08\times 10^{-5}$ and washout constants remain outside this def; only the structural skeleton and its sign are fixed here.
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