Variational Backbone and Regime Closures VII (Duality): Elimination Duality and Structural Invariants From the fixed core density to route-invariant Schur data
Year: 2026
Type: article
Abstract: This synthesis paper identifies the invariant package shared by both admissible eliminationroutes of one fixed backbone. Standalone note. This paper is readable on its own; referencesto earlier parts are for orientation only. Positioning. This part is a capstone/synthesis: itcollects the route-independence consequences of allowing both eliminations under a singlefixed variational primitive. The two reduced operators produced by the two routes neednot coincide; what is invariant (within the declared gates) is the Laplace-type principalsymbol class (hence the no-escape |k|−2 UV shape) and a compact global invariant package(index, gap/criticality, LogDet) in a fixed determinant convention. Parts V–VI establishtwo complementary reductions of the same fixed primitive functional: a closed-eliminationSchur map producing open-sector mass data (Part V) and an open-elimination dual-Schurmap producing a closed-curvature effective operator (Part VI). This paper studies whatremains invariant when both eliminations are allowed under the same reading protocol.To make the order-independence concrete, we also include a one-page finite-dimensional2 × 2 illustrative block example in which both reductions, the determinant identity, andthe criticality equivalence are explicit. We present a central “no-escape” ultraviolet shapeexample: within the declared Laplace-type regime, open elimination forces a resolventresponse of order −2, hence a universal Fourier scaling ∼ |k|−2, independent of instancesand rails. We prove a criticality equivalence theorem (gap closing on the open side ⇐⇒gap closing on the closed side ⇐⇒ a null mode of the full block Hessian) under explicitoperator-theoretic gates. Finally we define a candidate minimal set of order-independentstructural invariants (within the declared regime)—(index, gap, log-determinant within adeterminant class)—and classify regime-realizations by the invariant coupling κ = η/√αβ.Why only this combination? The two eliminations normalize the open/closed quadraticbackbones in opposite directions; the ratio κ := η/√αβ is the unique dimensionless residuethat survives simultaneous rescaling.Determinant convention. Whenever a log-determinant is invoked, it is understood within adeclared determinant convention (determinant class); see Appendix B.
Author: Yi, Yunbeom
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7123643290
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