Coarse information and finite rank obstructions for flat lattice Hodge forms

On a finite periodic lattice with unit spacing, a flat Hodge form cannot have a positive coercivity constant uniform under refinement when supplemented only by linear observations in a fixed finite-dimensional real space and linear fine-space repairs of uniformly bounded rank. We construct a real physical Fourier band, prove a necessary information budget and a separate bounded-rank sector obstruction, and explain why a classical singular-Schur condition prevents reversing that budget. Constructed derivative repairs and differentiable observations with locally upper-quadratic penalties admit tangent obstructions, including an exact regular nonlinear-sector argument. For nonnegative weights, a subquadratic directional spatial mean forces output-rank divergence. For signed weights, the spatial mean of the pointwise positive part gives a sufficient condition; ordinary signed means do not, as an actual fixed-output counterexample with every mean zero shows. An attributed classical square cancellation delimits the sufficient criterion; a finite-sum extension covering every eligible pair is displayed analytically. A full volume coexact packet also gives a quantitative rank budget under spatial means of the pointwise positive total coefficient on its two occupied field components. A derived integer cutoff yields positive output rank density on every sufficiently large lattice, while exact trace averaging improves a sufficient low frequency constant. Formal endpoints are separately pinned, and additional analytic arguments are identified. These are finite flat statements, with no interacting or continuum Yang–Mills mass-gap claim or new general spectral-principle claim. The integrated continuation also proves a critical inverse-moment sufficient condition for global spectral observations, together with limits on spatial localization. Rectangular means have a general upper bound with exponent d/4 and the target exponent d/2 for an explicitly restricted scaled-cube class. Two constructive counterexamples exclude a single-global-threshold algorithm and a universal exact cube decomposition, while preserving successful rectangular partitions for those same profiles. The general critical rectangular question remains open. Self-contained local and supplied-cube proofs are included, along with an actual physical fixed-map obstruction and its separately scoped exact finite penalty threshold. The present integrated revision adds positive matrix- gradient estimates, finite-rank signed spectral repair and an exact critical-penalty obstruction for prescribed rectangular means. Its leading penalty coefficient persists on actual flat coexact fields; the complete ambient scalar optimum is proved and accompanied by paired exact rational replay. These supporting results do not settle the general critical rectangular rank- order question. The spatial continuation derives an actual coexact point-support obstruction in dimension three, an arbitrary-geometry fixed-coefficient-gap rectangular construction, and local detours showing that even sharp minimum-based rectangle certificates can require unboundedly many cells where one mean suffices. An isospectral nonlocal graph example delimits spectral-only conversion. Complete analytical proofs and separately scoped exact finite reproductions are retained; the full critical rectangular question stays open.