wick_continuation_threeTwo_hinges
plain-language theorem explainer
For every hinge of the (3,2) causal 4-simplex at the physical point a=α=1, the split-form Wick continuation is branch-regular on the open arc (0,1), and the cosine path is continuous on [0,1] ending at the Euclidean value −1/4. Gravity and QG workers cite this as the B2 all-hinge certificate for threeTwo. The proof is exhaustive Fin-5 case analysis on opposite vertex pairs, wiring pre-proved pair lemmas and a symmetry lemma.
Claim. For all distinct vertices $p,q\in\{0,1,2,3,4\}$, the split-form edge continuation of the $(3,2)$ causal 4-simplex at $a=\alpha=1$ is branch-regular on the open arc $(0,1)$ for the hinge opposite $\{p,q\}$, and the associated cosine path is continuous on the closed interval $[0,1]$ with value $-(1/4)$ at the Euclidean endpoint $t=1$.
background
Lane B2 of the QG Seven-Gaps campaign treats the complex-first Wick continuation of the (3,2) causal 4-simplex along the canonical upper-half-plane arc. Vertices split as lower slice ${0,1,2}$ and upper slice ${3,4}$; the six cross edges are the timelike ones. Each unordered opposite pair ${p,q}$ labels a triangular hinge, in three classes: one spacelike hinge $(0,1,2)$ opposite $(3,4)$; six mixed hinges (one lower, one upper vertex); three upper-pair hinges opposite a lower edge.
Branch regularity means the split-form continuation stays off the arccos branch cut on the open interior. The cosine path is the continuous split-form cosine of the hinge angle along the arc. At the Euclidean endpoint the target value is the regular 4-simplex cosine $-(1/4)$. For the spacelike hinge the Lorentzian endpoint $t=0$ lands on the cut at $-(11/8)$ (classical boost angle); that contact is allowed, so its branch certificate is interior-only, while the endpoint value remains exact via the cut-free rational form $(5-6z)/(6z-2)$.
Closed cofactor forms are kernel-checked by 5×5 minors: spacelike $C_{pp}=C_{qq}=6z-2$, $C_{pq}=5-6z$; mixed asymmetric $C_{pp}=8z-4$, $C_{qq}=6z-2$, $C_{pq}=-1$; upper-pair $C_{pp}=C_{qq}=8z-4$, $C_{pq}=3-4z$.
proof idea
Term-mode proof by double fin_cases on $p,q:\mathrm{Fin},5$. The five diagonal cells $p=q$ are discharged by absurd rfl. Each of the twenty off-diagonal ordered pairs is a one-line package of a pre-proved pair lemma:
- Direct orientation:
branchRegular32_pair01…pair04,pair12…pair14,pair23,pair24, plusbranchRegular_threeTwo_spacelikefor the spacelike hinge, each paired with the matchingboundary32_*continuity-and-endpoint facts (taking.1and.2.2). - Reverse orientation:
branchRegularOn_symmon the corresponding direct certificate, andboundary32_symmon the boundary pair.
No new analytic work happens here; the theorem is the exhaustive glue of the ten unordered hinges in both orientations.
why it matters
This is the B2 headline: full hinge coverage for the (3,2) type at the physical point. Downstream, wick_hinge_data_continuation_complete (B3) conjoins it with the fourOne all-hinge certificate to get branch regularity and Euclidean endpoint $-(1/4)$ for every hinge of both causal 4-simplex types. The sibling product-form kill product_form_crossing_threeTwo_upper uses the same continuation data to show that at $t^*=2/3$ the upper-pair cofactor product hits $-48$ on the csqrt cut, memorializing why the single-sqrt product transcription fails on this class.
In the Seven-Gaps finishing charter this closes the threeTwo half of the complex-first Wick hinge-data lane. It sits under the broader Recognition gravity stack (simplicial ledger hinges, Clifford/8-tick structure upstream) without itself invoking the T0–T8 forcing chain or the mass ladder. The honest spacelike cut-contact disclosure keeps the Lorentzian endpoint classical rather than papered over.
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