hingeCosPath_eq_moebius
plain-language theorem explainer
On the Wick arc the split-form dihedral cosine of the traced fourOne hinge equals the cut-free rational map $(1-2z)/(6z-2)$ for every real parameter $t$, endpoints included. Cite this when proving continuity of the cosine path, endpoint values, or the S3 branch-regularity certificate. The proof unfolds the split cosine and denominator, inserts the physical cofactors, and multiplies the two square roots back to the non-vanishing cofactor.
Claim. For every real $t$, the split-form cosine of the traced fourOne timelike hinge along the Wick arc equals $\frac{1 - 2 z(t)}{6 z(t) - 2}$, where $z(t)$ is the complex arc value at $t$. Equivalently, the two split square roots in the denominator multiply back to the cofactor $6z-2$, so the branch cuts cancel and the cosine collapses to a Möbius rational function of $z$.
background
Module C11 formalizes a complex-first 4D Wick continuation of Regge hinge data (Cayley-Menger areas-squared and cofactor dihedral cosines) for causal 4-simplex classes. The path on the timelike squared edge is the upper-half-plane arc $z(t)=\alpha a^2\exp(i\pi(1-t))$ for $t\in[0,1]$, with Lorentzian endpoint $z(0)=-\alpha a^2$ and Euclidean endpoint $z(1)=+\alpha a^2$. Interior points lie in the open upper half-plane.
The split-form cosine is mandatory: the dihedral cosine is written with a product of two complex square roots in the denominator (one per cofactor), rather than a single product-form square root. That split is forced by the numeric gate so that branch cuts can be tracked separately. The cofactor that appears for the traced hinge (triangle $(0,1,4)$, opposite pair $(2,3)$, unit $a=\alpha=1$) is $6z-2$.
This identity is the algebraic branch-collapse step: once the two square roots multiply back to that cofactor, the cosine becomes the cut-free rational function $(1-2z)/(6z-2)$ everywhere on the real line in $t$.
proof idea
Term-mode proof by direct expansion. Unfold the split cosine path and its split denominator. Rewrite the edge data to the physical continuation edges. Specialize the Cayley-Menger vertex index map at opposite vertices $2$ and $3$ (indices $3$ and $4$). Substitute the three cofactor evaluations (diagonal pair and off-diagonal), then apply the identity that the product of the two complex square roots equals the cofactor itself, using non-vanishing of the denominator along the arc. The remaining algebra is the rational expression $(1-2z)/(6z-2)$.
why it matters
This is the branch-collapse lemma for the C11 hinge-data continuation. Downstream, continuity of the cosine path on the closed interval $[0,1]$ is proved by transporting continuity of the cut-free rational function across this equality. Endpoint theorems evaluate the same rational map at $z(0)$ and $z(1)$ to get Lorentzian value $-3/8$ and Euclidean value $-1/4$. The Lorentzian sign-factor theorem compares the split form to the real product-form value and records the documented factor $-1$. The S3 inhabitation certificate (branch-regularity on the full open interior) uses the collapsed form to control imaginary parts off the sqrt and arccos cuts.
Framework role: panel-locked QG Seven-Gaps lane C11 (hinge-data Wick path). It does not close the FullTheoryLedger gap on action-level continuation; that remains the separate C12 question. No T0–T8 forcing step is claimed here; the result is local to Regge hinge geometry on the arc.
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