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This note demonstrates VP ≠ VNP using sparsity and invertibility in mathematical structures, suggesting foundational implications.
A self-contained note deriving VP ≠ VNP from two elementary structural properties - sparsity of the canonical SAT polynomial f𝒰(n) and invertibility of GLn² - uniformly for all n ≥ 3. The argument requires no Rigidity Theorem, no Castelnuovo–Mumford regularity, and no case-by-case computation: an explicit unipotent transformation gives k(d) ≥ 1 for the SAT orbit closure, while a torus-weight observation gives k(d) = 0 for the determinant. The canonical object f𝒰(n) and seven-step proof structure follow TMO_OIGNPC_v6 (same author); all other ingredients are independent.
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Gilberto Ismael Ramirez (2026) studied this question.
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