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Abstract

We study concentrated signed potential forms in the derivative-energy space H_0^1(a,b). An explicit finite matrix retains the complete projected primitive Gram and original spline trace, with bounded norm error independent of center separation, boundary distance and rank loss. If the smallest node or boundary gap h resolves the width ε, every spectral pair instead follows the exact primitive-weighted Dirichlet Green matrix with relative error O(log(2+h/ε)/√(h/ε)). This holds for a fixed number of zero-mean profiles with uniform envelopes and primitive L^2 norms bounded below, arbitrary resolved nested geometry and boundary approach. An exact trace factorization and a small shear pay trace entries that can exceed secondary spectral scales. Integrable primitives improve the relative error to O(√(ε/h)). Neutral fixed-order stencils give vanishing residual spectrum and critical scale d^p/ε^(p+1/2); their explicit rational realizations also distinguish finite, vanishing and escaping secondary channels. Unresolved Cauchy collision and cancellation examples explain why the resolution condition cannot simply be removed. All conclusions concern fixed-order bounded-interval operators.