branchRegularOnNotDeficitSumCertificate
plain-language theorem explainer
Packages the S4+S9 separation certificate that the single-sqrt product form of the hinge denominator is kernel-killed on interior arc parameters: diagonal-cofactor products hit the complex square-root branch cut at exact values −40, −48, and −32. Gravity residual auditors cite it when ruling out branch-regular lookalikes of the 4D Wick action. The proof is a two-field term pairing the memorialized three-two kills with the four-one crossing theorem.
Claim. The product-form branch-regularity certificate holds: at the mixed interior arc parameter $t_\star^{\mathrm{mix}}\in(0,1)$ the product of diagonal Cayley–Menger cofactors on the three-two continuation equals $-40\notin\mathbb{C}\setminus(-\infty,0]$; at $t=2/3\in(0,1)$ the corresponding product equals $-48$ off the slit plane; and at the four-one interior parameter $t_\star\in(0,1)$ the product equals $-32$ off the slit plane. Thus $\sqrt{C_{pp}C_{qq}}$ is not a well-defined single-branch transcription of the 3D dihedral denominator on these arcs.
background
Module Wave C4 R0 banks lookalike-falsify receipts for the gap-6 residual: each lookalike is proved as a positive statement, then separated from the closed 4D Wick action continuation by a domain, action-field, or closer mismatch. After F3, gap 6 is closed by wick_action_continuation_4d_v2; these certificates remain as post-close critics, not as open-gap witnesses.
The objects are complex Cayley–Menger cofactors on continued edge data of causal pent types (three-two and four-one). The 3D dihedral denominator is classically a single square root of a product of diagonal cofactors. On the Wick arc that product can land on the branch cut of the complex square root (the complement of the slit plane), so the literal product form fails while a split-sqrt repair remains available.
Upstream, wick_product_form_kills_memorialized records the three-two kills at $t_\star^{\mathrm{mix}}$ (value $-40$) and $t=2/3$ (value $-48$). product_form_crossing is the four-one gate FAIL at $t_\star$ with product exactly $-32$ off the slit plane.
proof idea
Pure term-mode constructor: inhabit the certificate proposition by the pair
⟨wick_product_form_kills_memorialized, product_form_crossing⟩.
The first conjunct supplies both three-two interior kills (mixed class and upper-pair class). The second supplies the four-one crossing kill. No further rewriting or arithmetic is performed here; the certificate is exactly the conjunction of those two already-proved kernel statements.
why it matters
Feeds the R0 residual closer typedResidual_gap6_lookalike_decoys_fail, which assembles the full lookalike-falsify package (3D Lorentzian continuation, 4D kinematical continuation, hinge-data level, $C_4$ sign, and this branch-regularity kill, among siblings). Without it, a later session could misread a product-form branch-regular transcription as discharging the 4D Wick action bound.
In the Seven Gaps gravity track this is the S4+S9 separation: product-form $\mathrm{csqrt}(C_{pp}C_{qq})$ is kernel-killed at interior arcs, forcing the split-sqrt convention used by the genuine action-level continuation. It does not itself derive Einstein–Hilbert dynamics; it only blocks a decoy denominator that would look like a 3D-style closed form on the Wick hinge. Landmark contact is local to the gap-6 Wick/hinge ledger rather than the T0–T8 forcing chain.
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