The Topological Phase Transition of Algorithmic Complexity: From Non-Hermitian Symmetry Breaking to Zeta Function Collapse

Embedding NP-complete combinatorial constraints into a continuous Morse manifold subjects the discrete configuration space to geometric relaxation. Generic perturbations, dictated by the Thom transversality theorem, ensure the non-degeneracy of critical points within the Riemannian manifold. The localized constraints enforce a structural fragmentation of the sub-level sets. The proliferation of index-1 saddle points establishes a rigid topological lower bound for the first Betti number through the evaluation of strong Morse inequalities. This geometric bound scales as β₁ ≥ O(e(γ₁ - γ₀)N) near the satisfiability threshold. Physical traversal of these macroscopic topological voids necessitates continuous information erasure. The associated thermodynamic cost dictates the introduction of an imaginary dissipation phase Γ into the effective Hamiltonian Heff = H₀ + iΓ. The non-Hermitian operator iΓ physically annihilates the Time-Reversal Symmetry of the state space. The spectral density of the critical Hessian matrix undergoes a spontaneous transition from the Gaussian Orthogonal Ensemble to the Gaussian Unitary Ensemble, governed by the Bohigas-Giannoni-Schmit conjecture. The spectral determinant of the Ihara Zeta function translates the discrete combinatorial topology into a continuous spectral operator. This determinant exhibits asymptotic statistical isomorphism to the Montgomery-Odlyzko distribution governing the zeros of the Riemann ζ function in the thermodynamic limit. Finite-size scaling analysis restricts the Lee-Yang zero trajectories associated with the partition function. The scaling of the lowest-lying zero's imaginary part Im(z₀) ~ O(N-y) establishes an absolute cryptographic horizon lower bound N ≥ Nc = ε-1/y. Constraint lengths exceeding Nc force the algorithmic pseudo-random state to undergo a phase transition into the resolvable analytic symmetry of the GUE attractor. The scaling exponent y exhibits strict dependence on non-mean-field topological connectivity.