IndisputableMonolith.Gravity.SevenGaps.WickActionEuclidSchlaefliAudit
Audit shell for the Euclidean Schläfli variation of the Wick-continued gravitational action at the real endpoint. It packages the certificate that the real part of the Wick action path is differentiable in the continuation parameter at that endpoint. Gravity and Regge-calculus readers cite it when checking the frozen euclidSchlaefli slot of WickActionContinuationCert. Structure is import-and-audit of the Wave C4 R4 inhabitant, not a fresh derivation.
claimAudit of the claim that there exists a real number $dS$ such that the map $\beta \mapsto \mathrm{Re}\bigl(S_{\mathrm{Wick}}(\beta,1)\bigr)$ is differentiable at the Wick endpoint parameter $\alpha$, i.e. $\exists\, dS,\; \mathrm{HasDerivAt}\bigl(\beta\mapsto \mathrm{Re}(S_{\mathrm{Wick}}(\beta,1)),\, dS,\, \alpha\bigr)$.
background
SevenGaps Wave C4 targets the Wick continuation of the gravitational action and the Euclidean identities that must hold at the real endpoint. The parent design note (D-gap6-r1) freezes a certificate field euclidSchlaefli whose mathematical content is existence of a derivative of the real part of the Wick action path in the continuation parameter.
Classically, the Schläfli identity relates the variation of polyhedral (or simplicial) volume to edge lengths and dihedral angles; in Regge and continuum gravity it controls the on-shell variation of the Euclidean action. Here the same role is played after analytic continuation: one evaluates the real part of the continued action along a path $\beta \mapsto S_{\mathrm{Wick}}(\beta,1)$ and asks for differentiability at the Euclidean endpoint labeled $\alpha$.
The audited module WickActionEuclidSchlaefli is the inhabitant of that frozen field. This audit module sits one import above it and exposes the claim for certificate checking inside the SevenGaps gravity stack.
proof idea
Definition and audit module, not a standalone proof development. It imports WickActionEuclidSchlaefli and re-exports or checks the packaged witness that HasDerivAt holds for $\beta \mapsto (\mathrm{wickActionPath},\beta,1).\mathrm{re}$ at the endpoint $\alpha$. Any real work (construction of the derivative value $dS$, justification of the path, Euclidean Schläfli reduction) lives in the imported module; the audit layer only confirms the certificate shape matches the frozen WickActionContinuationCert.euclidSchlaefli slot.
why it matters in Recognition Science
Closes the audit face of SevenGaps gap-6 / Wave C4 R4: Euclidean Schläfli and $\alpha$-variation at the Wick endpoint. Downstream certificate consumers that read WickActionContinuationCert.euclidSchlaefli depend on this slot being inhabited and shape-checked; without the audit, the frozen field is an unchecked Prop hole.
In the broader Recognition gravity program the result ties the continued action's real variation to the classical Schläfli structure expected in Euclidean signature, which is a prerequisite for matching on-shell gravitational dynamics after Wick rotation. No further used_by edges are recorded at this module boundary; its value is local certificate hygiene for the SevenGaps stack rather than a new physical identity.
scope and limits
- Does not derive the Schläfli identity from first principles; only audits the frozen certificate field.
- Does not construct the explicit derivative value dS inside this module.
- Does not address Lorentzian signature or off-endpoint continuation parameters.
- Does not prove uniqueness of the Wick path or of the endpoint parameter α.
- Does not connect to the T0–T8 forcing chain or to the α fine-structure band.