continuationEdgesC
plain-language theorem explainer
Complex continuation of the ten squared edge lengths of a causal 4-simplex along the Wick arc: timelike edges track the upper-half-plane path z(t), spacelike edges stay fixed at a². Anyone working the C11 hinge-data Wick continuation cites this as the path-selected edge tuple feeding endpoint IDs and branch certificates. It is a one-line pointwise case split on the causal edge type.
Claim. For a causal 4-simplex type $\tau\in\{(4,1),(3,2)\}$, scale $a$, anisotropy $\alpha$, and arc parameter $t\in\mathbb{R}$, the continued squared-edge map $e\mapsto \ell_e^2(t)\in\mathbb{C}$ (indexed by the ten lexicographic edges) sends each timelike edge to the Wick-arc value $z(t)=\alpha a^2\exp(i\pi(1-t))$ and each spacelike edge to the constant $a^2$.
background
Module C11 builds a complex-first Wick continuation of Regge hinge data (Cayley-Menger areas-squared and cofactor dihedral cosines) for the two causal 4-simplex classes of 4d CDT: (4,1) and (3,2). The ten squared edge lengths are indexed by Fin 10 in lexicographic order; the complex type is simply maps Fin 10 → ℂ.
Edge causality is fixed by type: for (4,1) the four apex-touching edges are timelike; for (3,2) the six cross-slice edges are. The canonical arc on a 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 $z(1)=+\alpha a^2$, interior strictly in the open upper half-plane (S1).
Spacelike edges are held at the real constant $a^2$. The module is explicitly hinge-data only; full action-level continuation of an interior-hinge complex is the separate C12 question and remains open in the ledger.
proof idea
Definitional one-liner: pointwise on each of the ten edges, branch on the Boolean edge-type predicate. If the edge is timelike for the given causal type, return the complex arc value at $(a,\alpha,t)$; otherwise coerce the real constant $a^2$ into $\mathbb{C}$. No lemmas are applied; the body is the case split itself.
why it matters
This is the path-selected edge tuple that the whole C11 lane rides on. Downstream endpoint theorems identify $t=0$ with the kernel-checked real Lorentzian squared-edge tuple and $t=1$ with the Euclideanized tuple (both coerced to $\mathbb{C}$). The physical-point specialization equates the (4,1) unit case to the two-value hinge edge map at $z(t)$.
Branch-regularity on the full open interior (the S3 inhabitation certificate for the traced (4,1) hinge) is stated directly on this family, as are the cofactor non-vanishing, cut-contact honesty at the Lorentzian endpoint, and the split-cosine path. Without a single named continuation of the ten edges, none of those certificates have a carrier.
Framework role: QG Seven-Gaps panel-locked flagship lane C11 (P1-remainder). It does not close the ledger gap wick_action_continuation_4d; that stays open pending C12.
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