product_form_crossing_threeTwo_upper
plain-language theorem explainer
At Wick-arc parameter t = 2/3, the product of the two diagonal Cayley-Menger cofactors for every upper-pair hinge of the (3,2) causal 4-simplex equals -48 and sits on the complex square-root branch cut. Cite this for the B2 product-form kill certificate on the upper-pair class. The proof rewrites to closed cofactor forms, applies a precomputed value lemma, then excludes slit-plane membership by real arithmetic.
Claim. The arc parameter $t^*=2/3$ lies in $(0,1)$, the product of complex Cayley-Menger cofactors $C_{1,1}\cdot C_{2,2}$ of the Wick-continued squared-edge data for the $(3,2)$ causal 4-simplex at the physical point $(a,\alpha)=(1,1)$ equals $-48$, and $-48\notin\mathbb{C}\setminus(-\infty,0]$ (the complex slit plane).
background
Lane B2 of the QG Seven-Gaps campaign treats complex-first Wick continuation of all ten triangular hinges of the threeTwo causal 4-simplex (three vertices on slice $t$, two on $t+1$). Opposite vertex pairs classify hinges: spacelike (one pair), mixed (six), and upper-pair (three pairs inside the lower triple). Upper-pair hinges have closed cofactors $C_{pp}=C_{qq}=8z-4$ and $C_{pq}=3-4z$.
The complex Cayley-Menger cofactor $C_{r,c}$ is the signed minor of the bordered $6\times 6$ matrix built from the ten complex squared edge lengths. Continuation along the canonical upper-half-plane arc supplies those lengths at physical $a=1$, $\alpha=1$. The slit plane is the usual domain of the principal square root; landing on its complement means the product sits on the branch cut.
This certificate targets the upper-pair class at the exact interior point $t^*=2/3$ (where $\mathrm{Re},z=1/2$), matching the executed arc-trace table.
proof idea
Term-mode refine splits the three conjuncts. The open-interval claim $2/3\in(0,1)$ is norm_num. The product identity rewrites the continued edges via the physical threeTwo specialization, replaces the two diagonal cofactors by their closed forms (cof32_d1, cof32_d2), and finishes with the precomputed lemma that the upper-pair product equals $-48$. The slit-plane non-membership assumes membership, unfolds the slit-plane criterion, simplifies, and obtains a real-arithmetic contradiction by linarith (negative reals lie off the slit plane).
why it matters
This is one of the two product-form kill certificates demanded by the B2 finishing charter (RESULTS.txt §3). It shows that the single-sqrt product transcription of the upper-pair threeTwo hinges fails at an interior arc point: the diagonal-cofactor product is exactly $-48$ on the csqrt cut. Downstream, wick_product_form_kills_memorialized packages this certificate with the mixed-class kill (value $-40$) into one kernel statement that the product form stays killed and the split-sqrt form is the repaired convention. Within the broader Recognition gravity stack this closes a negative gate on the (3,2) simplex Wick data rather than a forcing-chain landmark (T0–T8); it is bookkeeping that keeps the complex-first hinge calculus honest before any continuum or effective-action claims.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.