Stability of Almost-Flat Lossy Cocycles: an Index Obstruction Survives Rank Loss

The stability theory of almost-representations — when is a map that almost satisfies a group's relations close to one that satisfies them exactly? — has reached a sharp modern form: for higher-rank lattices, pointwise Frobenius-norm stability follows from second-cohomology vanishing, pointwise operator-norm stability fails through index obstructions, and uniform stability holds for all submultiplicative norms. Every result in this landscape assumes invertible targets: unitaries. We formulate the corresponding stability problem for lossy targets — rank-deficient partial isometries, the maps that arise when translations between heterogeneous systems destroy information — over graphs with cycle relations. A structure lemma identifies the exact objects: a family of lossy edge maps is exactly consistent if and only if it has the form Te = VjViT for orthonormal frames V at the nodes, in which case every cycle composite is a projection. Against that family we measure stability across the grid of (matrix norm) × (defect functional), with planted-obstruction, noise-floor, and invertible-column controls. Dense systems are uniformly stable with a bounded penalty for rank deficiency. On the torus with a unit-Chern flux planted in the surviving subspace, the local defect vanishes with system size while the operator-norm distance to the nearest exact family stays pinned at order one — identically at full rank, corank one, and half rank — while the Frobenius distance decays. We conjecture that the pointwise Frobenius/operator dichotomy of the invertible theory, including its index mechanism, survives rank loss. The conjecture is machine-checked down to a single estimate: a Lean 4 formalization proves that it follows — uniformly in system size, with the constant preserved — from one transported rigidity statement (a dimension-free lower bound on the distance from the flux magnetic-translation pair to every commuting contraction pair), which direct adversarial optimization over commuting contraction pairs fails to falsify: no competitor below the trivial zero-pair distance of exactly 1 was found at any size or rank-padding, so the sought constant lies in (0, 1]. The open problem this paper poses to the stability community is exactly that estimate.