IndisputableMonolith.Gravity.Analysis.ReggeExactFlatHessianBlochTorusBridge4DAudit
Audit shell for the 4D discrete-torus bridge that glues the exact midpoint Bloch limit into the Regge flat-Hessian residual. It re-exports and typechecks the centered (unconditional) and uncentered (R2-gated) Tendsto statements along mesh scale t(j)=2π/torusSide(j)→0. Gravity analysts cite it to confirm the bridge module compiles cleanly without new proof obligations.
claimModule-level audit of the 4D bridge: the discrete torus family is composed with the banked continuous-scale limit of the exact midpoint Bloch quotient along $t(j)=2\pi/L_j\to 0$ in the punctured neighborhood of the origin; centered form unconditional, uncentered form under residual hypothesis $R_2$.
background
In the Regge gravity analysis stack, the flat Hessian of the discrete action is probed on a family of 4D tori whose side length $L_j$ sets the mesh scale $t(j)=2\pi/L_j$. The continuous-scale object is the exact midpoint Bloch quotient; its Tendsto as $t\to 0$ away from the origin is already banked upstream.
The parent bridge module composes that continuous Tendsto along the discrete mesh sequence $t(j)\to 0$. The centered residual bridge needs no extra hypothesis. The uncentered form, which matches the typed discrete-torus residual family, is gated by a residual hypothesis conventionally labeled $R_2$.
This audit module sits one import above that bridge. It does not introduce new geometric definitions; it only re-checks that the glued Tendsto statements typecheck and remain available to downstream gravity residual arguments.
proof idea
No independent proof content. The module imports the 4D Bloch-torus bridge and serves as a compile-time audit surface: it re-exports or mentions the centered unconditional bridge and the uncentered $R_2$-conditional form so that breakage in the parent Tendsto composition is caught at this boundary. Argument structure is entirely inherited from the imported bridge's composition of tendsto_exactMidpointBloch_m2_div along $t(j)\to\mathcal{N}_{\neq 0}(0)$.
why it matters in Recognition Science
Keeps the 4D Regge flat-Hessian residual pipeline honest. The parent bridge feeds discrete-torus residual comparisons used in gravity analysis; without a clean audit edge, silent drift in the midpoint Bloch gluing would corrupt those residual claims. In the broader Recognition gravity stack this protects the discrete-to-continuum handoff on the torus family, which is a prerequisite for any later comparison of Regge curvature defects against continuum limits. No downstream consumers are recorded yet; the module is a boundary check rather than a lemma source.
scope and limits
- Does not prove the midpoint Bloch Tendsto; that lives upstream.
- Does not discharge the residual hypothesis R2 for the uncentered form.
- Does not treat non-torus meshes or dimensions other than 4D.
- Does not supply numerical bounds on the rate of t(j)→0.
- Does not connect to continuum Einstein-Hilbert variational identities.