Variational Backbone and Regime Closures VII: Elimination Duality and Structural Invariants No-escape shape, criticality equivalence, and κ-classification
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2026
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| _version_ | 1866901743529361408 |
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| author | Yi, Yunbeom |
| author_facet | Yi, Yunbeom |
| contents | <p>What is new is not that Schur complements or variational formulations exist in isolation,<br>but that a single fixed variational primitive, together with a reading protocol and controlled<br>eliminations, yields both the closed-elimination and open-elimination corridors and leaves a<br>small, order-independent structural core. Parts V–VI establish two complementary reductions<br>of the same fixed primitive functional: a closed-elimination Schur map producing open-sector<br>mass data (Part V) and an open-elimination dual-Schur map producing a closed-curvature<br>effective operator (Part VI). This paper studies what remains invariant when both eliminations<br>are allowed under the same reading protocol. To make the order-independence concrete,<br>we also include a one-page finite-dimensional 2 × 2 illustrative block example in which<br>both reductions, the determinant identity, and the criticality equivalence are explicit. We<br>present a central “no-escape” ultraviolet shape example: within the declared Laplace-type<br>corridor, open elimination forces a resolvent response of order −2, hence a universal Fourier<br>scaling ∼ |k|<br>−2<br>, independent of cards and rails. We prove a criticality equivalence theorem<br>(gap closing on the open side ⇐⇒ gap closing on the closed side ⇐⇒ a null mode of the<br>full block Hessian) under explicit operator-theoretic gates. Finally we define a candidate<br>minimal set of order-independent structural invariants (within the declared corridor)—(index,<br>gap, log-determinant within a determinant class)—and classify corridor-realizations by the<br>invariant coupling κ = η/√<br>αβ.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18227389 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | Variational Backbone and Regime Closures VII: Elimination Duality and Structural Invariants No-escape shape, criticality equivalence, and κ-classification Yi, Yunbeom Functional analysis Operator Theory Partial differential equations Mathematical physics Schur complement dual Schur complement block operator matrix Laplace-type operator principal symbol pseudodifferential operator resolvent asymptotics ultraviolet scaling spectral gap / criticality Morse index determinant class zeta-regularized determinant (logdet) <p>What is new is not that Schur complements or variational formulations exist in isolation,<br>but that a single fixed variational primitive, together with a reading protocol and controlled<br>eliminations, yields both the closed-elimination and open-elimination corridors and leaves a<br>small, order-independent structural core. Parts V–VI establish two complementary reductions<br>of the same fixed primitive functional: a closed-elimination Schur map producing open-sector<br>mass data (Part V) and an open-elimination dual-Schur map producing a closed-curvature<br>effective operator (Part VI). This paper studies what remains invariant when both eliminations<br>are allowed under the same reading protocol. To make the order-independence concrete,<br>we also include a one-page finite-dimensional 2 × 2 illustrative block example in which<br>both reductions, the determinant identity, and the criticality equivalence are explicit. We<br>present a central “no-escape” ultraviolet shape example: within the declared Laplace-type<br>corridor, open elimination forces a resolvent response of order −2, hence a universal Fourier<br>scaling ∼ |k|<br>−2<br>, independent of cards and rails. We prove a criticality equivalence theorem<br>(gap closing on the open side ⇐⇒ gap closing on the closed side ⇐⇒ a null mode of the<br>full block Hessian) under explicit operator-theoretic gates. Finally we define a candidate<br>minimal set of order-independent structural invariants (within the declared corridor)—(index,<br>gap, log-determinant within a determinant class)—and classify corridor-realizations by the<br>invariant coupling κ = η/√<br>αβ.</p> |
| title | Variational Backbone and Regime Closures VII: Elimination Duality and Structural Invariants No-escape shape, criticality equivalence, and κ-classification |
| topic | Functional analysis Operator Theory Partial differential equations Mathematical physics Schur complement dual Schur complement block operator matrix Laplace-type operator principal symbol pseudodifferential operator resolvent asymptotics ultraviolet scaling spectral gap / criticality Morse index determinant class zeta-regularized determinant (logdet) |
| url | https://doi.org/10.5281/zenodo.18227389 |