lorentzian_endpoint_sign_factor
plain-language theorem explainer
At the Lorentzian endpoint of the Wick arc, the split-form complex dihedral cosine equals minus the real product-form cosine. Anyone matching complex hinge data to the real Lorentzian kernel cites this for the documented S4 sign convention. The proof rewrites the two endpoint identities and finishes by ring.
Claim. At the Lorentzian endpoint $t=0$ of the Wick arc, the split-form hinge cosine equals $(-1)$ times the real Lorentzian product-form cosine (cast to $\mathbb{C}$). Equivalently, when both diagonal cofactors are $-8$, the identity $\sqrt{w}\sqrt{w}=w$ (not $|w|$) forces a global sign factor of $-1$ relative to the real formula that uses $\sqrt{64}=+8$.
background
Module C11 builds a complex-first 4D Wick continuation of Regge hinge data (Cayley–Menger areas-squared and cofactor dihedral cosines) for causal 4-simplex classes. The arc is $z(t)=\alpha a^2\exp(i\pi(1-t))$ on $t\in[0,1]$, with $t=0$ Lorentzian and $t=1$ Euclidean, matching the repo edge conventions.
Dihedral cosines are read from a split-sqrt denominator: two complex square roots of the diagonal cofactors rather than one sqrt of their product. That split is mandatory because the product $C_{pp}C_{qq}$ hits the branch cut in the open arc interior. At the Lorentzian endpoint both cofactors equal $-8$, so the split denominator is $-8$ while the real product formula uses $+8=\sqrt{64}$.
The module is hinge-data only; full action-level continuation on an interior-hinge complex is the separate C12 question and stays open in the ledger.
proof idea
Term-mode algebraic identity. Rewrite the left side by the closed-form Lorentzian endpoint evaluation of the split path, and the right side by the defining equality for the real product-form cosine. After push_cast into $\mathbb{C}$, ring discharges the remaining scalar identity, which is exactly the factor $-1$ between split $-3/8$ and real $+3/8$.
why it matters
Documents the S4 sign convention that reconciles split-form complex hinge cosines with the real Lorentzian product formula at $t=0$ (RESULTS.txt §3 endpoint note). Downstream, the continuous closed-interval path, the Euclidean endpoint value, the product-form cut-crossing certificate, and the path-selected realization receipt all sit on the same hingeCosPath; this lemma pins the Lorentzian boundary value those results compare against.
It does not close the FullTheoryLedger gap on action-level 4D Wick continuation: that remains open pending C12. Within C11 it is the honest endpoint contact statement that the branch certificate lives on the open interior while endpoint values stay exact via the cut-free Möbius form.
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