IndisputableMonolith.Gravity.Analysis.SRSTTFirstVariation4DAudit
Audit layer for the Euclidean weak-field TT directional first variation of the closed 4D midpoint Bloch symbol. It packages honesty and completeness checks on the cross-term variation and on the torus-normalized continuum transport via the banked Einstein–Hilbert convergence. Gravity analysts cite it when verifying that the first-variation face is fully discharged before polarization. Structure is import-and-audit: it re-exports and stress-tests the parent analysis module rather than proving new identities.
claimAudit of the TT directional first variation of the closed 4D midpoint Bloch symbol $S_{\mathrm{mid}}$ in the Euclidean weak-field sector: the genuine cross-term $\delta_{\mathrm{dir}} S_{\mathrm{mid}}$, its torus-normalized continuum face, and transport along the banked limit $S_{\mathrm{RS}}\to S_{\mathrm{EH}}^{4\mathrm{d,cl}}$ on the combinations $H+K$ and $H-K$, together with polarization.
background
Recognition Science gravity analysis works in a Euclidean weak-field transverse-traceless (TT) sector. The closed 4D midpoint Bloch symbol is the discrete action face whose continuum limit is tied to the Einstein–Hilbert action on a closed 4-manifold. The parent module derives the genuine directional (cross-term) first variation of that symbol, then moves the torus-normalized continuum face across the banked convergence $S_{\mathrm{RS}}\to S_{\mathrm{EH}}^{4\mathrm{d,cl}}$ evaluated on $H+K$ and $H-K$, finishing with polarization.
This audit module sits one layer above that derivation. Its job is not to recompute the variation, but to record and enforce the honesty constraints that the parent doc-comment flags as binding: what is proved versus what is assumed, and whether the continuum transport and polarization steps are fully wired.
Notation: $H$ and $K$ are the linearized metric combinations appearing in the weak-field TT decomposition; the Bloch symbol is the lattice-side generating object before continuum identification.
proof idea
This is an audit module, not a primary proof module. It imports the full TT first-variation development and organizes status, honesty, and completeness checks around the directional cross-term, the torus-normalized continuum face, the banked Tendsto transport on $H+K$ and $H-K$, and the final polarization step. No independent analytic identity is proved here; the argument structure is re-export plus audit scaffolding over the parent analysis.
why it matters in Recognition Science
In the Recognition gravity stack, continuum identification of discrete Bloch faces with Einstein–Hilbert data is only as strong as the first-variation and honesty layer beneath it. This audit module is the checkpoint that the TT directional first variation of the closed 4D midpoint symbol, and its transport through the banked $S_{\mathrm{RS}}\to S_{\mathrm{EH}}^{4\mathrm{d,cl}}$ limit, meet the binding honesty bar stated in the parent module.
No downstream consumers are wired yet in the graph (used_by is empty), so its present role is internal QA for the Gravity.Analysis chain. It does not itself advance T0–T8 forcing, the Recognition Composition Law, or the $\varphi$-ladder mass formula; it protects the 4D weak-field bridge those landmarks eventually sit on.
scope and limits
- Does not derive the TT directional first variation; that lives in the parent analysis module.
- Does not prove continuum convergence $S_{\mathrm{RS}}\to S_{\mathrm{EH}}$; it only audits use of the banked limit.
- Does not treat non-TT, Lorentzian, or strong-field sectors.
- Does not feed named downstream theorems yet; the use graph is empty.
- Does not establish dimension forcing, eight-tick structure, or $\varphi$-ladder mass claims.