wick_boundary_continuation_fourOne_hinge
plain-language theorem explainer
The split-form complex dihedral cosine of the traced four-one timelike hinge is continuous on [0,1] and joins the Lorentzian endpoint value -(3/8) at t=0 to the Euclidean regular-4-simplex value -(1/4) at t=1. Workers on the C11 Wick hinge-data lane cite this as the S4 boundary-continuation receipt. The proof is a one-line packing of path continuity with the two endpoint evaluations.
Claim. The split-form complex dihedral cosine path of the traced four-one timelike hinge is continuous on the closed interval $[0,1]$, takes the value $-(3/8)$ at the Lorentzian endpoint $t=0$, and takes the value $-(1/4)$ at the Euclidean regular 4-simplex endpoint $t=1$.
background
Module C11 formalizes a complex-first 4D Wick continuation of Regge hinge data (complex Cayley-Menger areas-squared and cofactor dihedral cosines) for causal 4-simplex classes. The continuation path on the timelike squared edge is the 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 $z(1)=+(\alpha a^2)$. Interior points lie in the open upper half-plane.
The object continued here is the split-form complex dihedral cosine of one traced four-one timelike hinge, not a full simplicial action. Split-sqrt denominators are mandatory: the product of diagonal cofactors hits the branch cut of a single complex square root on the open arc, so the 3D product-form transcription fails and the denominator is defined split. The Lorentzian endpoint value carries a documented sign factor relative to the real product-form formula; unrestricted equality with that real formula is not claimed.
This declaration is the S4 boundary-continuation receipt: continuity of that cosine path on the closed unit interval together with its two endpoint values.
proof idea
Term-mode one-line packing. The proof is the triple constructor of the three conjuncts: continuity of the hinge cosine path on the closed unit interval, its value at $t=0$, and its value at $t=1$. No further algebraic work occurs at this site; the three named lemmas already discharge continuity and the two endpoint identities.
why it matters
S4 is the boundary-continuation receipt for the C11 flagship lane (panel-locked QG Seven-Gaps campaign). Downstream, the realized theorem packages the full path-selected continuation with a proved branch certificate on the open arc interior, and the product-form crossing theorem records the interior FAIL event: at an interior parameter the product of diagonal cofactors equals $-32$, which lies on the complex square-root branch cut and kills the literal single-sqrt product transcription of the 3D dihedral denominator. That negative certificate is why the split denominator is definitional.
Honest scope remains hinge-data only. The FullTheoryLedger gap for action-level 4D Wick continuation stays open; no ledger flag is touched. The result sits inside the gravity/Regge hinge analysis feeding Recognition Science continuum limits, not inside the T0-T8 forcing chain itself.
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