On the Internal Geometry of NP-Completeness: A Partial Order on NPC via Algorithmic Decomposition and Representation-Theoretic Obstructions

On the Internal Geometry of NP-Completeness This paper develops the algebraic geometry of the complexity class NPC, demonstrating that NP-complete problems possess a strict internal orbital structure that polynomial-time reductions do not capture. The central object is the complexity polynomial f𝒰(n) = perm(M𝒰(n)), the canonical receptacle of the primitive unit 𝒰 under the Strong Complete Instance Access Property (CIAP⁺). Three unconditional structural results are established: 1. The Exact Stabilizer Formula gives dim(StabGL(n²)(f𝒰(n))) = n³(n−k) + 2(n−1), decomposing into a sparsity component and an internal torus. 2. The Rigidity Theorem gives ℂ[𝒱𝒰]d ≅ s(d)⊕3 as GL₃-modules for all d ≥ 3, verified computationally for d = 3,…,10. 3. Regularity Stabilization gives reg(I(𝒱𝒰cyl,(m))) = 3 for all m ≥ 3. These results yield the Trivial-Multiplicity Obstruction: under the row action of GL₃, the determinant det(Xm×m) is a highest-weight vector of weight (1,1,1) lying in s(1,1,1)(ℂ³), so k(d)(det) = 0 for every d ≥ 1, while k(d)(𝒱𝒰cyl) = 3. No GL(m²)-equivariant surjection can map a module with k(d) = 0 onto one with k(d) = 3, blocking the orbit containment 𝒱𝒰cyl ⊆ closure(GL(m²) · fdet(m)) for every m. Since f𝒰(n) ∈ VNP, this gives the unconditional separation VP ≠ VNP. The paper additionally develops an entanglement spectrum λ(P) measuring finer orbital structure within NPC, establishes a strict stratification λ(SAT) < λ(IS) < λ(Clique) = 1 via three independent mechanisms (the Rigidity Theorem, a Parity Barrier, and a Depth Barrier via the tangent-depth invariant δ), and proves unconditional orbit separations including 𝒱Clique ⊄ GL(n²) · fP for every CIAP⁺ polynomial fP. Numerical estimates of λ at n = 3, d = 3,…,10 are reported as provisional averages, not asymptotic limits.