continuationEdgesC_physical
plain-language theorem explainer
At unit spacelike length and unit arc scale, the complex Wick continuation of the (4,1) causal 4-simplex squared-edge 10-tuple equals the two-value hinge assignment with spacelike edges fixed at 1 and timelike edges equal to the arc value z_arc(t). Branch-regularity, Möbius cosine collapse, and cut-contact theorems on the fourOne arc all rewrite through this identity. The proof is a pointwise case split on the timelike edge mask.
Claim. For every real $t$, the complex squared-edge 10-tuple obtained by Wick-continuing the $(4,1)$ causal 4-simplex at spacelike length $a=1$ and arc scale $\alpha=1$ equals the two-value edge map that assigns $1$ to every spacelike edge and $z_{\mathrm{arc}}(t)$ to every timelike edge.
background
Module C11 formalizes complex-first 4D Wick continuation of Regge hinge data (Cayley-Menger areas-squared and cofactor dihedral cosines) for the causal 4-simplex classes of CausalSimplex4D. Scope is hinge data only; full action-level continuation remains open (C12 / ledger gap).
The arc on the timelike squared length is $z(t)=\alpha a^2\exp(i\pi(1-t))$ for $t\in[0,1]$, with Lorentzian endpoint $z(0)=-\alpha a^2$ and Euclidean endpoint $z(1)=+\alpha a^2$. The complex continuation of the edge tuple sends timelike edges (those connecting the two CDT slices) along this arc and holds spacelike edges at $a^2$. For type $(4,1)$ the timelike mask is edges ${3,6,8,9}$ (those touching the apex).
At the physical point $a=\alpha=1$ the arc collapses to a pure phase $z_{\mathrm{arc}}(t)$, and the continuation is expected to match the two-value hinge edge assignment (spacelike $1$, timelike $z$) used throughout the closed-form cofactor and cosine calculations.
proof idea
Term-mode proof by functional extensionality on the ten edges. Unfold the three definitions (continuation at $(4,1)$ with $a=\alpha=1$, the two-value hinge edge map, and the unit arc). Case-split on whether the edge is timelike for type $(4,1)$: both sides take the arc value on the true branch and the constant $1$ on the false branch (the false branch finishes by norm_num). No external lemmas are required.
why it matters
This is the bridge from the general parametric continuation to the specialized two-value hinge calculus used on the unit arc. Downstream rewrites include: the full-interior branch-regularity certificate for the traced fourOne hinge; the Möbius collapse of the split-form cosine to $(1-2z)/(6z-2)$; interior off-cut statements for hinge area-squared; endpoint cofactor cut-contact disclosure; and the product-form crossing that kills the single-sqrt 3D denominator transcription. The same identity is reused in WickFourOneAllHinges for spacelike- and timelike-pair branch certificates and path equalities.
Within the Seven-Gaps campaign it is a small but load-bearing normalization step on lane C11 (panel-locked hinge-data continuation). It does not close the open ledger gap on genuine action-level 4D Wick continuation.
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