Preprint

This research provides a geometric proof of the separation between P and NP complexity classes by analyzing the orbit closures of the Determinant and the Permanent polynomials. The study introduces the "Staircase Obstruction Lemma," which identifies a critical multiplicity gap in the representation space of SL_m.  

Key aspects of the work include:

Barrier Bypass: The method utilizes non-constructive algebraic properties of the SL_m symmetry group to bypass the "Natural Proofs" barrier.  

Combinatorial Analysis: Using Littlewood-Richardson rules and plethysm, the author demonstrates that the staircase weight \lambda_n exhibits asymmetry that the Determinant's structure cannot generate.  

Border Complexity: The proof leverages the lower semi-continuity of the multiplicity function to show that the Permanent cannot be approximated by the Determinant even at the topological boundary of the orbit.  

Verification: The computational verification for n=3 and n=4 confirms the existence of the identified algebraic obstruction.  

The results confirm that P \neq NP is a direct consequence of the algebraic rigidity of representation structures in characteristic 0.