arcZ
plain-language theorem explainer
Defines the canonical upper-half-plane Wick arc on a timelike squared edge: $z(t)=\alpha a^2\exp(i\pi(1-t))$ for real $a,\alpha,t$. Lorentzian endpoint at $t=0$ is $-\alpha a^2$; Euclidean at $t=1$ is $+\alpha a^2$; interior stays in $\mathrm{Im}>0$. Cited by the complex edge-tuple continuation and endpoint lemmas in the C11 hinge-data Wick lane. Pure definitional abbreviation of that complex exponential path.
Claim. For real parameters $a$, $\alpha$, and $t$, the complex value $z(t) = \alpha a^{2}\,\exp\bigl(i\pi(1-t)\bigr)$ is the upper-half-plane continuation arc on a timelike squared edge length scale $a$ with scale factor $\alpha$.
background
Module C11 formalizes a complex-first 4D Lorentzian Wick continuation of Regge hinge data (complex Cayley–Menger areas-squared and cofactor dihedral cosines) for causal 4-simplex classes. Scope is hinge-data only: dihedral cosines and triangular hinge areas of a single causal 4-simplex; full action-level continuation on an interior-hinge complex is deferred to C12 and remains open in the ledger.
The arc (S1) is the path on each timelike squared edge $z(t)=\alpha a^{2}\exp(i\pi(1-t))$, $t\in[0,1]$, with $z(0)=-\alpha a^{2}$ (Lorentzian) and $z(1)=+\alpha a^{2}$ (Euclidean), matching the repo sign convention for Lorentzian versus Euclidean squared-edge tuples and the executed numeric gate. Interior points lie strictly in the open upper half-plane.
Here $a$ is the real edge-length scale and $\alpha$ is a real scale factor on the squared edge (not the fine-structure constant from other modules). Spacelike edges stay at the real value $a^{2}$ under the companion edge-tuple map.
proof idea
Definitional: cast the real product $\alpha a^{2}$ to $\mathbb{C}$, multiply by $\exp\bigl((\pi(1-t))\cdot i\bigr)$. No lemmas; endpoints and positivity are proved downstream by unfolding and standard complex exponential identities ($\exp(i\pi)=-1$, $\exp(0)=1$).
why it matters
This is the S1 model arc for the entire C11 Wick lane. It feeds the complex squared-edge tuple (timelike edges follow the arc, spacelike stay $a^{2}$), the endpoint theorems (Lorentzian $t=0$, Euclidean $t=1$), the unit physical-point specialization $z_{\mathrm{arc}}(t)=\mathrm{arcZ}(1,1,t)$, and the hinge area-squared constructions built on continued edges.
Together those objects support the path-selected boundary continuation with a branch certificate on the open arc interior. The module does not close the FullTheoryLedger gap on 4D action-level Wick continuation; that remains open. Framework role is geometric QG scaffolding (Regge hinge data), not the T0–T8 forcing chain or RCL uniqueness.
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