IndisputableMonolith.Gravity.Analysis.ReggeNormalizationDerived4DAudit
Audit layer for Arc 2 step 7: it packages the continuum Einstein-Hilbert second variation on a real transverse-traceless plane wave against the banked Regge midpoint dictionary in four dimensions. Anyone checking that the derived face factor matches the discrete normalization cites this module. The argument is structural: import the continuum derivation and the Regge-side measurement, then record the comparison in one place.
claimIn 4D, the phase-averaged second variation of the Einstein-Hilbert action on a real transverse-traceless cosine wave $H$ with wavevector $k$ equals $\mathrm{ehFace}(H,k)=-(1/4)\,|k|^2\,\|H\|_F^2$, and this continuum face factor is audited against the Regge midpoint dictionary normalization.
background
Arc 2 of the gravity analysis derives the continuum transverse-traceless second variation of $\int R\sqrt{g}$ from the Levi-Civita connection alone, with no input from the discrete Regge side. The continuum module fixes the convention so that the phase average of $d^2/dt^2\int R\sqrt{g}$ per unit volume on a real TT cosine wave is the face factor $\mathrm{ehFace}(H,k)=-(1/4)\cdot|k|^2\cdot|H|_F^2$.
The companion module then measures that same number through the banked Regge midpoint dictionary, obtaining Regge's normalization constant by comparison rather than by postulate. This audit module sits on top of both imports: it is the bookkeeping surface that records the match between continuum EH curvature response and discrete Regge edge weights in four spacetime dimensions.
Notation: $H$ is a real symmetric transverse-traceless polarization tensor, $|H|_F$ its Frobenius norm, and $k$ the spatial wavevector of the plane wave. The setting is pure classical GR linearized about flat space; no Recognition-ladder mass or $\phi$ constants enter here.
proof idea
Definition and audit packaging, not a deep new proof. The module imports ContinuumTTSecondVariation4D (which derives the continuum face factor from Levi-Civita) and ReggeNormalizationDerived4D (which reads the same factor off the Regge midpoint dictionary). It then exposes the equality or residual checks that confirm the two sides agree on $\mathrm{ehFace}(H,k)=-(1/4)|k|^2|H|_F^2$. Expect thin wrappers, named equalities, and status lemmas rather than a fresh curvature calculation.
why it matters in Recognition Science
Closes the second half of Arc 2 step 7: continuum EH second variation and Regge normalization must speak the same number before any discrete-to-continuum gravity claim is trusted. Downstream gravity theorems that quote a derived (not fitted) Regge face weight depend on this audit holding. In the broader Recognition stack this is classical GR infrastructure, not a T0-T8 forcing step, but it underwrites later discrete curvature and action comparisons that feed the gravity domain. No used_by edges are recorded yet; the module is a terminal audit surface for the 4D normalization arc.
scope and limits
- Does not re-derive the continuum TT second variation; that lives upstream.
- Does not treat dimensions other than 4D or non-TT polarizations.
- Does not introduce Recognition $\phi$-ladder, mass, or $\alpha$ constants.
- Does not prove nonlinear or fully discrete Regge calculus identities.
- Does not claim experimental gravity bounds; pure classical consistency audit.