The Trivial-Multiplicity Obstruction: A Representation-Theoretic Barrier for the Determinant

We establish a representation-theoretic obstruction to orbit containment of the determinant family, combining algebraic geometry and computational verification without unproved conjectures. 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 Uniform Generation, showing that Im is generated in degree 3 for all m ≥ 3, proved via an explicit seed construction and row-profile distance argument verified computationally for m = 3, 4, 5. The Trivial-Multiplicity Obstruction compares two coordinate rings under the row action of GL₃ on ℂm² induced by the inclusion GL₉ ⊂ 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, the row action of GL₃ preserves no nonzero polynomial supported on the first-row variables alone within any GLm²-translate of detm, giving k(d) = 0 for every d ≥ 1 and every m. Complete reducibility makes the inequality 0 ≥ 3 a contradiction under any hypothetical orbit containment, establishing that f𝒰(n) does not lie in the border closure of the determinant family. Since f𝒰(n) ∈ VNP as a p-projection of the permanent, this gives a separation between VP and VNP in the border complexity framework. 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 and orbital stratification within NPC.