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 the necessary dimension budget and a bounded-rank linear-sector obstruction, and explain the fixed-map cross-coupling limitation through a classical singular-Schur criterion. We construct componentwise derivative repairs and prove a differentiable-observation tangent obstruction. An analytic implicit-function argument extends the obstruction to exact nonlinear regular sectors; counterexamples delimit its regularity and quadratic-energy hypotheses. The operator and full-space tangent results have separately pinned Lean evidence; the exact regular-sector extension remains analytic. These are finite flat statements, with no interacting or continuum mass-gap claim.