zArc_zero
plain-language theorem explainer
At the Lorentzian endpoint of the physical Wick arc the complex edge parameter equals $-1$. Anyone working the unit-lattice 4-simplex hinge continuation cites this base value before evaluating cofactors or dihedral cosines. The proof unfolds the physical specialization and applies the general arc-endpoint lemma at unit scale.
Claim. Let $z_{\mathrm{arc}}(t)=\exp(i\pi(1-t))$ be the physical Wick path (unit lattice spacing, unit scale factor). Then $z_{\mathrm{arc}}(0)=-1$, the Lorentzian endpoint of the upper-half-plane arc.
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 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 arc is the unit specialization $a=1$, $\alpha=1$, matching the repo physical causal pentachoron. Upstream, the general endpoint lemma states that at $t=0$ the arc evaluates to the real negative value $-(\alpha a^2)$ cast into $\mathbb{C}$, via $\exp(i\pi)=-1$. This theorem is that specialization at the physical point.
Honest scope of the module: hinge-data continuation only; full action-level continuation of an interior-hinge complex remains the open C12 ledger gap.
proof idea
Term-mode, three steps. Unfold the physical arc definition to arcZ 1 1 0. Rewrite by the general Lorentzian-endpoint lemma, which gives $-(\alpha a^2)$ at $t=0$. With $a=\alpha=1$, norm_num reduces the right-hand side to $-1$.
why it matters
Base evaluation for every Lorentzian-endpoint identity on the physical path. Downstream, the split-form hinge cosine at $t=0$ rewrites through this fact to obtain $-(3/8)$, and the endpoint cofactor identity uses it to show the diagonal cofactor equals $-8$ (on the square-root branch cut, disclosed as allowed endpoint contact).
Boundary-continuation theorems for the four-one and three-two causal classes (spacelike, timelike, and mixed hinge pairs) all evaluate their Lorentzian endpoints by substituting this value into the Möbius or cofactor formulae along the arc. Without the physical endpoint pinned, the closed-interval continuity statements and the Euclidean-to-Lorentzian matching at the two ends of the Wick path cannot close.
Framework role: C11 panel-locked lane of the QG Seven-Gaps campaign; supports path-selected branch certificates on the open arc interior while leaving the FullTheoryLedger action-level gap untouched.
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