A Geometric and Dynamical Approach to Legendre's Conjecture: Parity Breaking via Topological Bifurcation in a Smoothed Brun Sieve

Legendre's conjecture states that there is always at least one prime between n² and (n+1)² for all n ≥ 1. Standard analytic approaches are hindered by the error term of automorphic forms, while classical sieve methods fail due to the parity problem. In this work, we introduce a geometric and dynamical paradigm to overcome these obstacles. By embedding the Legendre interval into a two-dimensional hyperbolic space, we apply a Beurling-Selberg type smooth weight function to reduce Dirichlet lattice fluctuations to order O(1). We then combine this geometric Brun sieve with the introduction of a Hamiltonian gradient flow. We demonstrate that the population of internal semiprimes exhibits topological instability under this flow, forcing a transcritical bifurcation towards the boundary of the domain. This parity breaking mathematically guarantees the existence of a pure prime (P₁) within the interval, thereby validating Legendre's conjecture.