branchRegular_fourOne_hinge
plain-language theorem explainer
On the open Wick arc t ∈ (0,1), the traced fourOne hinge (opposite vertices 2,3; a=α=1) is branch-regular: both diagonal Cayley–Menger cofactors stay in the slit plane and the split dihedral cosine stays off the arccos cuts. Anyone citing the C11 hinge-data continuation or the S3 interior certificate needs this. The proof is a direct imag-part computation along the physical arc, reducing cofactors to 6z−2 and the cosine to a Möbius path with strictly negative imaginary part.
Claim. Let $x(t)$ be the complex squared-edge 10-tuple of the causal $4$-simplex of type fourOne along the physical Wick arc with $a=1$, $\alpha=1$. For opposite vertex indices $p=2$, $q=3$ and every $t\in(0,1)$, both diagonal cofactors $C_{pp}(x(t))$ and $C_{qq}(x(t))$ lie in the complex slit plane, and the split dihedral cosine $C_{pq}/(\sqrt{C_{pp}}\sqrt{C_{qq}})$ lies off the arccos branch cuts.
background
Module C11 formalizes complex-first 4D Wick continuation of Regge hinge data (Cayley–Menger areas-squared and cofactor dihedral cosines) for the two causal 4-simplex classes of CDT. The arc is $z(t)=\alpha a^2\exp(i\pi(1-t))$ for $t\in[0,1]$, with Lorentzian endpoint $t=0$ and Euclidean endpoint $t=1$; the open interior lies in the upper half-plane. Scope is hinge-data only: action-level continuation of a full interior-hinge complex is the separate C12 question and remains open on the ledger.
Branch regularity on a parameter set $s$ means: for every $t\in s$, both diagonal cofactors of the continued edge data stay in Complex.slitPlane (the continuity region of principal square root), and the split-form cosine ratio stays off the arccos cuts. The split denominator is mandatory; the product form $C_{pp}C_{qq}$ hits the cut in the interior (gate FAIL event).
The fourOne type has four vertices on one slice and one on the next. The traced hinge is triangle $(0,1,4)$ with opposite pair $(2,3)$. Upstream edge data and cofactor maps come from the causal simplex and Cayley–Menger complexification in this module.
proof idea
Fix $t\in(0,1)$. Positive imaginary part of $z_{\mathrm{arc}}(t)$ is immediate from the arc lemma. Vertex-index rewrites send cofactors at $p=2$ and $q=3$ to the physical continuation, where both equal $6z-2$. Its imaginary part is $6,\mathrm{im},z>0$, so $6z-2$ lies in the slit plane (right disjunct of the slit-plane criterion).
For the cosine: the split dihedral cosine along this path equals the hinge cosine path, which is the Möbius expression $(1-2z)/(6z-2)$. A direct real/imag expansion shows the numerator of the imaginary part of the quotient is $-2,\mathrm{im},z$. Division by $|6z-2|^2>0$ yields a strictly negative imaginary part, hence off the arccos cuts. The three conjuncts of branch regularity follow.
why it matters
This is the S3 inhabitation certificate for the full open interior of the traced fourOne timelike hinge: Lean transcription of the numeric interior margins (0.625 on this hinge; RESULTS.txt §3). It supplies the positive branch certificate that the module’s path-selected continuation theorem packages as realized on the open arc.
Downstream, product_form_crossing uses the same hinge data to prove the complementary FAIL event: at an interior $t_\star$ the product $C_{pp}C_{qq}$ equals $-32$, which lies on the sqrt cut. Together they justify why the denominator must be definitionally split rather than a single sqrt of the product (the 3D formula does not lift).
In the QG Seven-Gaps campaign this closes the C11 hinge-data lane’s interior regularity obligation. It does not touch the FullTheoryLedger gap wick_action_continuation_4d; that remains open pending C12. Framework context is Regge/CDT hinge geometry under Wick rotation, not the T0–T8 forcing chain directly.
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