continuationEdgesC_physical32
plain-language theorem explainer
At the physical point (spacelike squared length 1), the complex-continued edge tuple of a (3,2) causal 4-simplex along the Wick arc equals the two-value hinge assignment with 1 on spacelike edges and zArc(t) on timelike edges. Every branch-regularity and product-form certificate for the ten triangular hinges of the threeTwo simplex cites this identification. The proof is a pointwise case split on the timelike flag after unfolding both sides.
Claim. For every real parameter $t$, the complex edge-length continuation of a $(3,2)$ causal $4$-simplex at the physical point (spacelike squared length $1$, continuation scale $1$) equals the two-value edge map that assigns $1$ to every spacelike edge and $z_{\mathrm{arc}}(t)$ to every timelike cross edge.
background
Lane B2 of the QG Seven-Gaps campaign treats the all-hinge complex-first Wick continuation of the $(3,2)$ causal $4$-simplex at the physical point $a=1$, $\alpha=1$, along the canonical upper-half-plane arc $z_{\mathrm{arc}}$ of the complex-first Wick module. A $(3,2)$ simplex has three vertices on slice $t$ and two on slice $t+1$; its six cross edges are the timelike ones (indices ${2,3,5,6,7,8}$ in the Fin-10 edge order).
The generic continuation map builds a complex edge $10$-tuple by keeping spacelike edges real and sending timelike edges along the arc. The specialized two-value map for the threeTwo type does the same with a single complex parameter $z$: spacelike edges get $1$, timelike edges get $z$. The bordered Cayley-Menger matrix of that assignment is the complex analogue of the real $(3,2)$ pent matrix with $p=1$ and $q=z$.
Hinge classes are fixed by the opposite vertex pair: the pure spacelike hinge $(0,1,2)$, six mixed lower-upper pairs, and three upper-pair hinges inside the lower triple. All later closed-form cofactors and cosine paths are written against the two-value edge map.
proof idea
Term-mode proof by extensionality on the Fin-10 edge index. Unfold both the generic physical-point continuation and the specialized two-value hinge map (and the arc). Case on whether the edge is timelike for type threeTwo. On the timelike branch both sides reduce to the same if-true arm, so the rewrites match. On the spacelike branch both sides reduce to the if-false arm; norm_num discharges the residual real equality (both constants are $1$). No external lemmas beyond the edge-type predicate and the two definitions.
why it matters
This is the dictionary that lets every threeTwo hinge certificate work in the specialized $z$-language while still quoting the physical-point continuation used by the Wick arc trace. Downstream it is applied by the three class branch certificates (spacelike, mixed-pair, upper-pair), the three split-cosine path identities, and the two product-form kill certificates that place diagonal-cofactor products on the complex square-root cut at interior arc parameters.
Without the equality, cofactor closed forms such as $C_{pp}=8z-4$, $C_{qq}=6z-2$, $C_{pq}=-1$ (mixed) or $C_{pp}=C_{qq}=6z-2$, $C_{pq}=5-6z$ (spacelike) would not transfer to the continuation path that the finishing charter actually continues. In the broader RS gravity stack this closes the edge-data step of lane B2 before the arccos-branch and product-form negative results are stated. It does not itself touch T0-T8 or the RCL; it is pure CDT simplex geometry feeding the Wick-regularity gate.
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