zArc_one
plain-language theorem explainer
At the Euclidean endpoint of the physical-point Wick arc, the complex edge parameter equals 1. Anyone evaluating hinge cosines or boundary continuations of causal 4-simplex data at the Euclidean end cites this normalization. The proof unfolds the unit-lattice arc definition, applies the general Euclidean endpoint lemma, and closes by arithmetic.
Claim. The physical-point Wick arc satisfies $z_{\mathrm{arc}}(1)=1$ in $\mathbb{C}$, where $z_{\mathrm{arc}}(t)=\exp(i\pi(1-t))$ is the unit-spacing, unit-scale specialization of the upper-half-plane continuation path $z(t)=\alpha a^{2}\exp(i\pi(1-t))$.
background
Module C11 formalizes complex-first 4D Wick continuation of Regge hinge data (Cayley-Menger areas-squared and cofactor dihedral cosines) for causal 4-simplex classes. The continuation path on the timelike squared edge is the canonical 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 $z(1)=+\alpha a^{2}$.
The physical-point specialization fixes unit lattice spacing and $\alpha=1$, so the arc collapses to $z_{\mathrm{arc}}(t)=\mathrm{arcZ}(1,1,t)=\exp(i\pi(1-t))$. Upstream, the general Euclidean endpoint lemma states $\mathrm{arcZ}(a,\alpha,1)=(\alpha a^{2}:\mathbb{R}:\mathbb{C})$. This module is hinge-data continuation only; full action-level continuation of an interior-hinge complex remains the open C12 question.
proof idea
Three-step tactic proof. Unfold the physical-point definition $z_{\mathrm{arc}}(t):=\mathrm{arcZ}(1,1,t)$. Rewrite with the general Euclidean endpoint theorem $\mathrm{arcZ}(a,\alpha,1)=(\alpha a^{2}:\mathbb{R}:\mathbb{C})$, which specializes to $1\cdot 1^{2}=1$. Finish with norm_num to identify the real embedding of $1$ in $\mathbb{C}$.
why it matters
Pins the Euclidean endpoint of every physical-point hinge path in the C11 Wick lane. Downstream, the hinge-cosine path theorem uses it to obtain the regular unit 4-simplex value $-1/4$ at $t=1$. The fourOne and threeTwo boundary-continuation theorems (spacelike, timelike, mixed, and upper pairs) all evaluate Euclidean endpoints through this identity, matching the RESULTS.txt endpoint table and the repo sign convention for euclideanSqEdges.
Within the Seven-Gaps campaign this is bookkeeping on the S1 arc, not a new geometric claim. It does not close the FullTheoryLedger gap on action-level continuation; that stays open pending C12.
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