threeTwoCosPath_eq_upper
plain-language theorem explainer
On the Wick arc, every upper-pair hinge of the (3,2) causal 4-simplex has split dihedral cosine collapsing to the cut-free rational (3−4z)/(8z−4). Gravity and QG auditors cite this as the class-C algebraic reduction before branch and boundary work. The proof unfolds the split form, substitutes the three cofactor identities, and cancels the complex square root against itself.
Claim. Fix opposite vertices $p,q$ of the $(3,2)$ causal $4$-simplex. Suppose the complex Cayley–Menger cofactors of the hinge edge matrix satisfy $C_{pp}(z)=C_{qq}(z)=8z-4$ and $C_{pq}(z)=3-4z$ for every $z\in\mathbb{C}$. Then for every real arc parameter $t$, the split cosine path equals $(3-4z(t))/(8z(t)-4)$, where $z(t)$ is the canonical upper-half-plane Wick arc.
background
Lane B2 of the QG Seven-Gaps campaign continues all ten triangular hinges of the threeTwo causal 4-simplex (lower slice ${0,1,2}$, upper ${3,4}$) at the physical point $a=1$, $\alpha=1$, along the canonical upper-half-plane arc of the complex-first Wick action.
Hinges fall into three classes by opposite pair. Class C is the three upper-pair hinges $(0,3,4)$, $(1,3,4)$, $(2,3,4)$: both diagonal cofactors equal $8z-4$, the off-diagonal equals $3-4z$, and squared area is $z/4-1/16$. These closed forms are kernel-checked by explicit $5\times 5$ minors against the executed trace table.
The split cosine is the ratio of a numerator cofactor path to a denominator built from the two diagonal cofactors (with a complex square-root factor). Collapse means that ratio becomes an ordinary rational function of the arc coordinate, free of the branch cut of the square root.
proof idea
Term-mode, eight lines. Unfold the split cosine path and its numerator/denominator helpers. Rewrite with the physical continuation of the $(3,2)$ hinge edges, then substitute the three parametric cofactor hypotheses (both diagonals $8z-4$, off-diagonal $3-4z$). Finish by csqrt_mul_self on the nonzero class-C denominator along the arc, so the square-root factors cancel and the path equals the rational $(3-4z)/(8z-4)$.
why it matters
This is the algebraic collapse step for every class-C upper-pair hinge. It is consumed immediately by the parametric branch certificate (cofactor $8z-4$ stays in the open upper half-plane; collapsed cosine has imaginary part $-8,\mathrm{Im},z/|8z-4|^2\neq 0$ on the interior, trace margin $0.4167$, worst interior margin of the whole trace) and by the closed-interval boundary continuation (Lorentzian value $-7/12$, Euclidean $-1/4$).
In the Seven-Gaps finishing charter this discharges the class-C half of the all-hinge Wick continuation for the $(3,2)$ simplex: once the cosine is the cut-free rational, regularity and endpoint values are ordinary complex analysis on the arc. It does not touch the spacelike hinge endpoint contact or the mixed-pair asymmetric cofactors; those are separate hinge classes in the same module.
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