March 3, 20260 citationsOpen Access

Variational Backbone and Regime Closures VI: Closed Curvature from Open Elimination Dual Schur complement, Green-operator kernels, and channel selectivity

Key Points

Closed curvature is determined using a dual elimination method, resulting in an effective operator formulation.
The effective operator showcases coercive properties with a nonlocal kernel, enhancing mathematical robustness.
Results reveal a discrete spectrum shift and accumulation behavior, indicating significant topological features.
Findings may enable more accurate mathematical models in closed manifolds, though the findings need external validation.

Abstract

We record the dual elimination complementary to the open-sector Schur correction usedfor mass extraction: for the same quadratic backbone, eliminating the open variation yieldsa closed-curvature effective operator on Hcl of the form C − B∗A−1B. In the canonicalfirst-derivative representative, this becomes Lcl,eff = αI − η2 PRan− β∆ + V′′⋆−1PRan,which is coercive up to a smoothing (order −2) nonlocal kernel. On closed manifolds thecorrection is compact, so the closed spectrum is a discrete shift from α with accumulationat α. We record the required invertibility gate on the physical open subspace, the resultingspectral and topological features in the de Rham instance, and a basis-invariant finite-modeextraction protocol for the closed sector.

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