product_form_crossing_value_mixed

Lane B2 of the QG Seven-Gaps campaign continues all ten triangular hinges of the (3,2) causal 4-simplex along the canonical upper-half-plane arc $z_{\mathrm{arc}}$ of the complex-first Wick action, at the physical point $a=1$, $\alpha=1$. Mixed hinges (six opposite pairs with one lower and one upper vertex) have asymmetric closed cofactors $C_{pp}=8z-4$ (lower) and $C_{qq}=6z-2$ (upper), with $C_{pq}=-1$ and area-squared $z/4-1/16$.

The product-form transcription of the hinge cosine uses a single complex square root of the diagonal-cofactor product $C_{pp}C_{qq}$. That transcription is valid only when the product lies in the slit plane. The mixed-class crossing value is the exact complex number that product takes at the interior parameter $t^*$ where $\mathrm{Re},z=5/12$.

Sibling closed forms and 5x5 minor checks pin the cofactor polynomials; the arc identity $z_{\mathrm{arc}}(t)=\exp(i\pi(1-t))$ and the cosine identity at $t^*$ supply the real and imaginary parts used below.

First rewrite $z_{\mathrm{arc}}(t^)$ via the exponential form of the arc and the known cosine at $t^$, obtaining $\mathrm{Re},z=5/12$ plus an imaginary sine term. Substitute into each linear factor: $8z-4$ becomes $-2/3$ plus $8i\sin(\pi(1-t^))$, and $6z-2$ becomes $1/2$ plus $6i\sin(\pi(1-t^))$.

Multiply those two complex linear forms; the cross terms cancel against $i^2=-1$ via a linear combination, leaving the real expression $-1/3-48\sin^2(\pi(1-t^))$. The identity $\sin^2=1-\cos^2$ with the known cosine at $t^$ yields $\sin^2=119/144$. Cast that rational into $\mathbb{C}$ and simplify by norm_num to $-40$.

This is the pure algebraic crossing identity that the mixed-class B2 kill certificate wraps. The parent theorem product_form_crossing_threeTwo_mixed packages three facts: $t^*$ lies in $(0,1)$, the actual CM cofactors of a witness mixed hinge multiply to $-40$, and $-40\notin\mathrm{slitPlane}$. The present lemma discharges the numerical product step of that package.

In the executed Wick-arc trace (RESULTS.txt §3), the mixed class is recorded with $\mathrm{Re},z=5/12$, value $-40$, and $t^*\approx 0.6368$. Landing on the branch cut kills the single-sqrt product transcription for every mixed threeTwo hinge, forcing the split-form (two-sqrt) certificate used elsewhere in the module. Within the Seven-Gaps finishing charter this closes one of the two remaining product-form kill certificates for the (3,2) simplex.