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

The complexity class NPC is standardly treated as uniform: all NP-complete problems are polynomially equivalent, and no finer structure is imposed on the class. This paper demonstrates that NPC possesses a strict internal algebraic geometry that classical polynomial-time reductions fail to capture, and proposes the entanglement spectrum λ(P), defined via the obstruction multiplicities of the orbit-closure approach to algebraic complexity, to measure it. We establish three unconditional structural theorems about this geometry. First, we prove a strict multi-level orbital stratification λ(SAT) < λ(IS) < λ(Clique) = 1 for all n, driven by three independent representation-theoretic mechanisms: the Rigidity Theorem (C[V_U]d ≅ s(d)^⊕3 for all d ≥ 3), the Parity Barrier (λ(IS) < 1), and the Depth Barrier (λ(Clique) = 1, via the new tangent-depth invariant δ with δ(f_U) = n−1 and δ(f_Clique) = n, proved by Frobenius 1897 and the Pieri rule). The strict inequalities are unconditional; numerical estimates (λ(IS) ≈ 0.641, λ(Clique) ≈ 0.803) are provisional cumulative averages over d = 3,...,10 at n = 3 and are not claimed as asymptotic limits. The Polynomial Depth Lemma further establishes δ(f_P) ≤ n−1 for every CIAP+ incidence polynomial f_P, giving the unconditional orbit separation V_Clique ⊄ GL_{n²} · f_P. Second, SAT's canonical status is geometrically intrinsic: the Exact Stabilizer Formula dim(Stab(f_U^(n))) = n³(n−k) + 2(n−1), decomposing into a proved sparsity term and an internal torus, identifies SAT as the strict minimum of λ over NPC (SAT Minimality Theorem), with the largest stabilizer and the most rigid coordinate ring. The Reduction Monotonicity Theorem establishes unconditionally that dim(Orb_{k'}) > dim(Orb_k) for k' > k with n fixed, with the λ-ordering following conditionally on the Orbital Dictionary. Third, the Algorithm Independence Theorem establishes that the CIAP+ complexity polynomial is an intrinsic invariant of P: for any two asymptotically optimal algorithms A, B, the polynomials f_A and f_B are related by a permutation P_σ ∈ S_N ⊂ GL_N, so V_U is a well-defined algebraic object independent of algorithmic choice. Because purely dimensional obstructions fail — under polynomial padding the stabilizer of SAT inflates to Θ(m⁴), satisfying algebraic semicontinuity trivially against the determinant (orbital gap n⁴ − kn³ − 2n² + 4 = Θ(n⁴) at fixed n, absorbed by padding) — the boundary of P vs NP reduces to the exact algebraic structure of the coordinate rings. The Trivial-Multiplicity Obstruction isolates the precise barrier: by Schur–Weyl duality, the cylindrical embedding of SAT unconditionally inherits its absolute rigidity, requiring exactly k_(d) = 3 copies of the trivial symmetric representation s_(d) at every degree d ≥ 3. Classical invariant theory dictates that the determinant's orbit closure supplies exactly k_(d) = 1. Since no GL_{m²}-equivariant surjection can send 1 copy onto 3, the orbit containment V_SAT^cyl ⊆ GL_{m²} · f_det^(m) is algebraically blocked for every polynomial m. Since VP ⊆ VP̄, this gives the unconditional separation VNP ⊄ VP̄: the family (f_U^(n)) lies in VNP but outside the border complexity class of the determinant at every polynomial size. Conditionally on the Boolean-Algebraic Simulation Hypothesis — which translates efficient boolean decision algorithms into polynomial-size algebraic circuits — this 1 < 3 geometric obstruction formally precludes polynomial-time boolean algorithms for SAT, yielding P ≠ NP.