wick_product_form_kills_memorialized
plain-language theorem explainer
At two interior arc parameters on the three-two causal 4-simplex, the product of a pair of diagonal Cayley-Menger cofactors lands exactly on the complex square-root branch cut: value -40 at the mixed-class star and -48 at t=2/3. Anyone citing the B3 kill certificates or the Gap-6 lookalike receipt needs this packaging. The proof is a one-line pair of the two already-proved product-form crossing lemmas.
Claim. The following two kill events hold simultaneously. (i) The mixed-class interior parameter $t_\star^{\mathrm{mixed}}$ lies in $(0,1)$, the product of the $(1,1)$ and $(4,4)$ complex Cayley-Menger cofactors of the three-two continuation edge matrix at that parameter equals $-40$, and $-40\notin\mathbb{C}\setminus(-\infty,0]$. (ii) The upper-pair parameter $2/3$ lies in $(0,1)$, the product of the $(1,1)$ and $(2,2)$ cofactors at $t=2/3$ equals $-48$, and $-48\notin\mathbb{C}\setminus(-\infty,0]$.
background
Lane B3 of the QG Seven-Gaps campaign packages hinge-data Wick continuation over both causal 4-simplex types (four-one and three-two) and all twenty triangular hinges. The module certifies continuation of dihedral cosines and areas-squared along the canonical upper-half-plane arc at the physical point $a=1$, $\alpha=1$. It is deliberately not an action-level continuation: summed deficit angles over multi-pent interior hinges remain open.
The objects here are complex Cayley-Menger cofactors of the continued edge matrix of a three-two causal pent. The single-sqrt product transcription of a cosine uses a product of two such cofactors under one outer square root. When that product lands on the slit plane (the non-positive reals), the principal branch csqrt is discontinuous or undefined in the usual sense, so the product form is killed.
Upstream B2 work already isolates the two exact crossings: mixed class at the interior star parameter, and upper-pair class at the rational interior point $t=2/3$. This theorem only memorializes both FAIL events as one kernel conjunction.
proof idea
Term-mode proof: a single pair constructor. The left conjunct is the mixed-class product-form crossing lemma (interior $t_\star^{\mathrm{mixed}}$, cofactor product $-40$ on the slit). The right conjunct is the upper-pair product-form crossing lemma (parameter $2/3$, cofactor product $-48$ on the slit). No new algebra is performed; the theorem is the packaged conjunction of those two certificates.
why it matters
In the Recognition gravity stack this is the B3 kill-certificate memorial: it freezes the fact that the single-sqrt product transcription of hinge cosines is dead on the three-two type at two concrete interior points, forcing the split-sqrt repaired convention used by the cosine-continuation theorems in the same module.
Downstream, branchRegularOnNotDeficitSumCertificate in the Gap-6 lookalike receipt consumes this packaging together with the generic product-form crossing fact, so Gap-6 can distinguish branch-regular hinge data from a genuine deficit-sum (action-level) certificate. The sibling documentation theorem action_level_still_open records that this module does not flip the kinematical-layer flag: multi-pent interior-hinge Regge action continuation stays open (C12 prerequisite).
Framework context: the ambient geometry is the $D=3$ eight-tick causal simplex setting forced by the T7/T8 chain; the present result is local complex analysis of Cayley-Menger data, not a new forcing step.
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