deltaTau23
plain-language theorem explainer
The generation-torsion gap between the second and third fermion generations equals τ(2)−τ(1). With the fixed torsion ladder {0,11,17} this is the integer 6. CKM and Q₃-geometry arguments cite it as the numerator of the structural Wolfenstein A ratio before the face-flux correction. The body is a one-line difference of the local torsion map on Fin 3.
Claim. Define the generation-torsion gap $\Delta\tau_{23}:=\tau(2)-\tau(1)\in\mathbb{N}$, where the torsion map on generations is $\tau(0)=0$, $\tau(1)=11$, $\tau(2)=17$. Equivalently $\Delta\tau_{23}=17-11$.
background
The module derives the Wolfenstein parameter $A$ from $Q_3$ face-flux geometry and generation torsion, with zero sorry and zero axioms. The headline claim is $A_{\mathrm{corrected}}=9/11\approx 0.818$, within $0.6\sigma$ of the PDG value $0.826\pm 0.013$.
Generation torsion is the representation-independent ladder $\tau(0)=0$, $\tau(1)=E_{\mathrm{passive}}=11$, $\tau(2)=W=17$ (Anchor). The local copy on $\mathrm{Fin},3$ is the same triple. Gaps are $\Delta\tau_{12}=\tau(1)-\tau(0)=11$ and $\Delta\tau_{23}=\tau(2)-\tau(1)$. Together with the Gray-code flip counts $(4,2,2)$ on the three cube axes, these integers control the structural CKM ratio and the shared “44” factor that also appears in $\alpha^{-1}$ and $\eta_B$.
proof idea
Pure definition: evaluate the local torsion map at generation indices $2$ and $1$ (as elements of $\mathrm{Fin},3$) and subtract. No tactics, no lemmas. The numerical identity $\Delta\tau_{23}=6$ is discharged later by rfl in the companion equality theorems.
why it matters
This gap is step 2 of the five-line CKMExact derivation. It is the numerator of $A_{\mathrm{structural}}=\Delta\tau_{23}/\Delta\tau_{12}$, proved equal to $6/11$. Multiplying by the face-flux ratio $3/2$ yields $A_{\mathrm{corrected}}=9/11$. Downstream, A_structural, A_structural_eq, deltaTau23_eq, and six_from_torsion all read this constant; the last packages the slogan “$6=\Delta\tau_{23}$ from the CW-filtration torsion gap between generations 2 and 3.” The same integers enter the 44-connection linking CKM, fine structure, and baryon asymmetry under one $Q_3$ chirality pattern.
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