Three Gaps, One Index: the m-uniform rigidity of a lossy magnetic pair is topological, not spectral

In one fully specified, machine-checked special case — the charge-k magnetic flux family on an m × m torus of lossy (rank-deficient) links — we disambiguate three distinct quantities that the almost-commuting-operator and topological-protection literatures all call “the gap,” and determine which survive uniformly in the system size m. The local gaps — the commutator norm ‖[U,V]‖, the magnetic Hamiltonian's spectral gap, and the spectral-localizer gap they control — all vanish like the flux-per-plaquette 2πk/m² → 0. Two other gaps are m-uniform: the core gap (the rank/isometry structure of the links, independent of the magnetic data) and the Bott branch gap (well-definedness of the index). The consequence is a sharp negative with a constructive moral: the m-uniform operator-norm separation c(k) between charge and flatness — the Plateau quantity — cannot be sourced from any local spectral gap, and is therefore irreducibly topological (a Voiculescu distance-to-commuting that stays ≈ √2 while every local gap collapses), transported into the non-invertible setting by the rank-structure core gap, not by a spectral gap. This locates exactly where a proof of the Plateau Conjecture must, and must not, look. We then make the positive locale concrete: reading the rank loss as the ideal of an extension reduces the remaining unitary-to-contraction passage to a single off-diagonal Schur block, which numerics (Exp. E, and an adversarial worst-case Exp. F) show is core-gap-controlled and m-uniform — recasting “extend a dimension-free theorem to contractions” as “bound one block.” A direct minimization of the distance to flatness itself (Exp. G) independently fails to collapse the distance, converging up to the same √2 as the index-jump radius — the conjecture's most direct numerical test, passed. The √2 above is the codimension-1 value (d = r+1; the tested case d = 3, r = 2). For codim = d−r ≥ 2, a trivial flat competitor supported on an r-plane orthogonal to the rotating core sits at operator distance exactly 1 from the flux family at every m, so distOpFlat ≤ 1 < √2 there. The separation is therefore codimension-dependent: c(k) = √2 at codimension 1, c(k) = 1 at codimension ≥ 2 — both m- and k-uniform. The Plateau Conjecture (c(k) > 0) holds across 2 ≤ r ≤ d, with uniform floor 1; what is codimension-1-specific is the mechanism — the dimension-free unitary (Voiculescu/Exel–Loring) bound transports only at codimension 1, while for codimension ≥ 2 the binding competitor is the orthogonal one the unitary bound cannot see. House style. Every claim in this note compiles in Lean ([THEOREM]), cites the literature ([CITED]), carries numerical evidence with its script ([EVIDENCE]), or wears a [CONJECTURE] tag. Numerics (experiments A–G, full detail in §9) are the evidence base and are referenced throughout by letter.