gap_13

In the CKM-from-cube module, mass eigenstates on the three-generation space are labeled by Recognition torsion (CW coupling level): generation 1 has torsion 0 (ground state), generation 2 has torsion 11 (edge-dressed), and generation 3 has torsion 17 (edge plus face dressed). Off-diagonal mass-basis overlaps are suppressed by $\varphi^{-|\Delta\tau|}$, so the integer gaps $\Delta\tau_{ij}$ are the raw exponents in the mixing angles.

The local helper torsionGap i j is defined as the integer difference $\tau j-\tau i$. The present statement fixes the $(1,3)$ entry of that table. Module context ties these gaps to the Q₃ hypercube, Gray-code chirality $[4,2,2]$, and the weak-basis even-sign-flip generators that complete the CKM overlap.

One-line computational proof: native_decide evaluates the closed definition of the torsion gap at indices $0$ and $2$ and checks equality with the literal $17$. No lemmas are invoked beyond the definition $\mathrm{torsionGap},i,j:=\tau j-\tau i$ and the fixed torsion assignment ${0,11,17}$.

This is one of the three elementary gap identities (alongside the $0!\to!1$ and $1!\to!2$ gaps) that pin the mass-basis J-cost suppressions used for CKM angles. The module derives $V_{ij}$ from Q₃ geometry with $\sin^2\theta_{ij}\propto(\mathrm{flip_count_ratio})\times\varphi^{-|\Delta\tau_{ij}|}$; the exponent for the $1$–$3$ channel is exactly this gap of $17$. Downstream siblings such as the suppression exponents and the torsion-gap hierarchy consume these values. In the broader RS chain the torsions themselves come from cube dressing (edges plus faces), consistent with the eight-tick / $D=3$ forcing that makes Q₃ the spatial recognition complex. No parent theorem is recorded yet in the use graph; the declaration is infrastructure for the CKM amplitude formulas.