Abstract / Description Title: P ≠ NP: Final Ontological Resolution v15.1 (Topological Invariance, Entropy Rigidity, and Quantum Invariance) Summary: This manuscript (v15.1) provides the final ontological and mathematical resolution of the P versus NP problem, establishing the definitive inequality P \neq NP as a structural law of the computational universe. The work shifts the paradigm from combinatorial search to the analysis of topological invariants and thermodynamic constraints. Key Scientific Pillars of the Resolution: Algebraic Determinism & Topological Locking: In the Arithmetic Circuit Model, any polynomial algorithm is viewed as a composition of local ring morphisms. Since the topological invariants (Chern classes) of the Determinant and Permanent varieties do not coincide, no composition of polynomial morphisms can transform one into the other, creating an absolute "algebraic lock." Thermodynamic Limit (Landauer’s Principle): The resolution integrates the constant \alpha = \ln 2 as a fundamental barrier. Mapping high-entropy NP spaces into low-entropy P spaces requires information compression that exceeds the Landauer limit (E_{min} \approx kT \cdot \ln 2 \cdot n). For n \rightarrow \infty, the energy required for such a "shortcut" tends toward infinity, rendering P = NP physically impossible. Spectral and Quantum Invariance: The resolution establishes that quantum computations (BQP) manipulate amplitudes but do not alter the algebraic degree of functions. As the spectral density of obstructions \sigma(n) \rightarrow 1, the quantum state space remains empty relative to the functional basis of NP tasks. Structural Singularity: The resistance function \lambda(n) = 0.693n + [1 - \exp(-\sqrt{6.58n})] proves that the NP \rightarrow P transition is a point of structural singularity. The Lipschitz constant L(n) explodes exponentially, resulting in the total loss of computational stability for any hypothetical "fast" algorithm. Conclusion: The P vs NP problem is closed. The equality of these classes would require the collapse of universal constants such as \pi and \ln 2, which contradicts the ontological structure of the universe. Author: Oleksandr Saienko Date: April 28, 2026 Status: Final Ontological Resolution v15.1