Riesz Stability and the Infinite-Rank Barrier for Multipole Gram Operators

This preprint develops a functional-analytic framework for passing from finite-depth Graph-Schur certification of multipole Gram matrices to infinite-rank Riesz stability on coefficient space. The manuscript studies countable multipole response families in a separable observable-dual Hilbert space and distinguishes Bessel sequences, frames, Riesz sequences, and Riesz bases. The central point is that finite-dimensional invertibility of every truncated Gram matrix does not by itself imply bounded invertibility of the limiting Gram operator on ℓ². The main results include a normalized block-Schur stability theorem with an explicit inverse bound, a Hilbert-Schmidt compactness/Fredholm alternative, barrier results for vanishing diagonal gaps and degenerating finite-depth margins, and geometric sufficient criteria for bounded positive weights on the torus and sphere. The paper also states a conditional infinite-rank BBGKY interface under a named Riesz-stability hypothesis, tail-decay assumptions, and a scale-safe Gronwall closure. All unconditional results are functional-analytic statements about bounded Gram operators, Riesz sequences, compact perturbations, Schur certificates, and weighted Hilbert-space realizations. The BBGKY consequences are conditional interface results. No unregularized Coulomb closure, trace-norm propagation-of-chaos theorem, or automatic infinite-rank passage from finite Graph-Schur ladders is claimed. Version v1.1 is the assumption-provenance and final-scope polish version. It clarifies the bounded-diagonal operator layer, the conditional Gronwall layer, the provenance of structural assumptions, and the role of finite-dimensional diagnostics.