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 if it is supplemented only by linear observations in a fixed finite-dimensional real space and linear fine-space repairs of uniformly bounded rank. The linear observations may vary with refinement, and their scalar penalty need only vanish at zero. We prove a separate obstruction on kernels of bounded-rank linear constraints using a constructed real Fourier band, derive a necessary dimension budget and explain why it is not sufficient through a classical singular-Schur criterion and a cross-coupling example. Formal evidence is separately pinned; analytic statements and source specializations are identified. No interacting, continuum or Yang-Mills mass-gap claim is made.