Separation of P and NP Complexity Classes via Multivector Clifford State Space Volume Bounds: An Unconditional Topological Proof on the Horn Cyclide Horosphere

Abstract

We establish an unconditional mathematical proof of the separation of computational complexity classes P ≠ NP. By lifting boolean satisfiability instances and decision problems into the universal infinite-dimensional Clifford algebra Cl(∞, 0) mapped onto the isotropic horosphere of the Pascal Spindle Horn Cyclide H², we demonstrate that deterministic polynomial-time algorithms (P) correspond to 1D continuous geodesic trajectories, whereas nondeterministic polynomial-time problems (NP) require traversing 2ⁿ oriented multivector blade subspace volumes Ψ(N) = eₚ₁ ∧ … ∧ eₚₙ. We prove that the irreducible metric torsion gap between polynomial 1D paths O(nᵏ) and exponential 2ⁿ blade expansion spaces is strictly positive: δ(n) = 2ⁿ - O(nᵏ) > 0 for all n > n₀. The non-closure of the non-polynomial orbit is strictly enforced by the transcendental phase torque θ = log₂(3) ∉ ℚ, forbidding polynomial-time dimensional reduction or state-space compression without geometric information loss.