pentHingeCosPath
plain-language theorem explainer
Defines the shared three-pent hinge cosine along the Wick arc as the complex dihedral cosine on chart pair (3,4) of the (3,2) causal 4-simplex edge continuation at lattice scale a=1. Gravity/Regge analysts cite it as the single cosine path that, under structural collapse, represents all three hinge angles. The body is a one-line composition of the split cofactor cosine with the complex edge continuation.
Claim. For real parameters $\alpha$ and $t$, let $x(t)$ be the complex squared-edge 10-tuple of the $(3,2)$ causal 4-simplex continued along the Wick arc at lattice scale $a=1$. Then $\mathrm{pentHingeCosPath}(\alpha,t)$ is the complex dihedral cosine of $x(t)$ at the hinge opposite vertex pair $(3,4)$.
background
Module Wave C4 R2 freezes the schema for the 4d Wick action continuation and the complex arccos lift carccos. Arc convention is binding: $t=0$ is Lorentzian, $t=1$ Euclidean. The complex angle is $\mathrm{carccos},w:=-I\log(w+I\sqrt{1-w^2})$ with the repo half-power square root applied only to $1-w^2$.
Causal 4-simplices in 4d CDT come in two types; threeTwo has three vertices on slice $t$ and two on $t+1$. The complex edge map continuationEdgesC sends timelike edges along the Wick arc $Z$ and holds spacelike edges at $a^2$. The split cofactor cosine dihedralCosSplitC is the complex Cayley–Menger dihedral cosine at the hinge opposite a vertex pair $(p,q)$.
Structural collapse on the three-pent object identifies the shared hinge with the same-slice all-spacelike triangle ${0,1,2}$. All three dihedral cosine paths are definitionally the chart pair $(3,4)$ of the threeTwo continuation, so the deficit sum is $3\cdot\theta(t)$ and hinge area-squared is the constant real $3/16$ along the arc.
proof idea
One-line definitional wrapper. Instantiate the complex edge continuation at type threeTwo, scale $a=1$, parameters $\alpha,t$; feed that 10-tuple into the split cofactor dihedral cosine at vertex pair $(3,4)$. No tactics, no lemmas beyond the two composed defs.
why it matters
This is the cosine path that the entire interior-hinge Wick package rides on. Downstream cert assembly uses it for: continuity of the path and of $\mathrm{carccos}\circ$ path on $(0,1]$; the off-arccos-cut predicate along that interval; the pointwise Lorentzian cut value $\mathrm{carccos}(\mathrm{path}(1,0))=\pi-I,\mathrm{arcosh}(11/8)$; the decisive mismatch between principal value and one-sided cut limit; and continuity of the Wick action path (three times the complex angle). It feeds WickActionContinuationCertV2 at $\alpha=1$.
In the Seven Gaps gravity campaign it encodes the structural-collapse reduction that turns three hinge angles into one path, which is the design freeze for gap-6 Lorentzian action work. It does not itself close the terminal wick_action_continuation_4d (still demands the frozen V1 family) and does not flip any action-level Bool. Framework context: Regge-style deficit angles on causal simplices under Wick rotation toward a Euclidean action comparison; not a T0–T8 forcing step, but infrastructure for the gravity side of the RS ledger.
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