The Trivial-Multiplicity Obstruction and VP ≠ VNP
Year: 2026
Type: preprint
Abstract: We establish VP ≠ VNP via a self-contained representation-theoretic obstruction requiring no unproved conjecture. The central object is the canonical complexity polynomial f𝒰(3) = perm(M𝒰(3)), defined by the Strong Complete Instance Access Property (CIAP⁺) applied to irredundant 2-SAT instances. Three geometric results are proved: the Exact Stabilizer Formula dim(StabGL₉(f𝒰(3))) = 31; the Rigidity Theorem ℂ[𝒱𝒰]d ≅ s(d)⊕3 as GL₃-modules for all d ≥ 3; and Regularity Stabilization reg(Im) = 3 for all polynomial m ≥ 3. The Trivial-Multiplicity Obstruction then compares two coordinate rings under the row action of GL₃ ⊂ GLm². For the cylindrical SAT variety, the Rigidity Theorem gives multiplicity k(d) = 3 copies of s(d,0,0) at every degree d ≥ 3. For the determinant, det(Xm×m) is a highest-weight vector of weight (1,1,1) — verified directly: every raising operator Eij sends det to a matrix with two equal rows, hence to zero — so det ∈ s(1,1,1)(ℂ³) and k(d)(det) = 0 for every d ≥ 1. Complete reducibility then makes the inequality 0 ≥ 3 a contradiction under any hypothetical orbit containment, for every polynomial m. Since f𝒰(n) ∈ VNP as a p-projection of the permanent, this gives f𝒰(n) ∉ VP and therefore VP ≠ VNP. This paper is a self-contained extraction of the central result of the full technical report (https://doi.org/10.5281/zenodo.20950127). The full report contains the entanglement spectrum, orbital stratification within NPC, and the conditional separation P ≠ NP.
Author: n, n
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7166302769
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