wick_hinge_areaSq_closed_forms_complete
plain-language theorem explainer
For every complex parameter z, all twenty triangular hinge areas-squared of a single causal 4-simplex admit explicit closed forms: 3/16 on every all-spacelike class and z/4 - 1/16 on every class that carries a timelike edge, covering both the 4+1 and 3+2 causal types. Gravity and discrete-QG workers cite this as the B3 areas-squared completeness certificate in the Seven-Gaps Wick lane. The proof is a three-way term conjunction of the already-proved 4+1 spacelike, 4+1 timelike, and 3+2 closed-form lemmas.
Claim. For every $z \in \mathbb{C}$, the squared areas of all twenty triangular hinges of a single causal 4-simplex are given by the closed forms $3/16$ on every all-spacelike hinge class and $z/4 - 1/16$ on every hinge class that includes a timelike edge. This holds for both the $4{+}1$ edge assignment and the $3{+}2$ edge assignment.
background
Lane B3 of the QG Seven-Gaps finishing charter packages hinge-data Wick continuation over both causal 4-simplex types. A causal 4-simplex has five vertices; each unordered complementary triple determines a triangular hinge. The complex edge lengths are assigned by the canonical upper-half-plane arc at the physical point $a=1$, $\alpha=1$, via the two edge maps (4+1 and 3+2). The squared hinge area is the complex Cayley-Menger quantity evaluated on those three edges.
The module is deliberately a hinge-DATA continuation only: dihedral cosines and areas-squared of the hinges of a single simplex of each type. It is not an action-level continuation. The Regge action needs deficit angles summed over a genuine multi-simplex interior-hinge complex, which remains open (C12 prerequisite) and is untouched here.
Upstream, the 4+1 all-hinges and 3+2 hinges modules already supply the per-type area-squared identities and the interior cut-avoidance fact that the same two closed forms stay off the branch cut on the open arc (with the documented Lorentzian endpoint contact at $-5/16$ allowed).
proof idea
Term-mode proof: a single triple of conjunctions. The first conjunct is the 4+1 all-spacelike closed form (four hinges equal to $3/16$). The second is the 4+1 timelike closed form (six hinges equal to $z/4 - 1/16$). The third is the full 3+2 closed-form package (one all-spacelike hinge at $3/16$ and nine mixed hinges at $z/4 - 1/16$). No new algebra is done here; the theorem only conjoins the three already-proved per-type certificates into one completeness statement over both causal types and all twenty hinges.
why it matters
This is the areas-squared half of the B3 completeness package in the Seven-Gaps Wick campaign. Together with the companion cosine-continuation completeness theorem in the same module, it certifies that every hinge datum of a single causal 4-simplex of either type continues with explicit closed forms along the canonical arc.
Its sole recorded consumer is the documentation theorem that the action-level continuation flag remains open: this module proves hinge-data continuation and deliberately does not flip causalSimplex4DStatus.action_level_continuation_open. No FullTheoryLedger flag changes. The honest scope is kinematic hinge data only; the continued Regge action itself still requires a three-or-more-pent interior-hinge complex (C12 lane).
In the broader RS gravity stack this sits under discrete Regge/TT hinge analysis for 4D causal simplices, not under the T0-T8 forcing chain or the mass ladder. It closes the B3 areas-squared deliverable while memorializing that action-level Wick continuation is still the open gap.
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