Factorization via discrete hyperbolic lattice]{Integer factorization, quadratic operators, and spectral boundaries in a discrete hyperbolic lattice

We present a structural reformulation of the integer factorization problem based on the hyperbolic Diophantine equation b² - i² = 4N. We introduce a discrete quadratic operator whose admissible kernel bijectively encodes the factorizations of N and characterizes primality as spectral uniqueness. Analysis of the associated lattice reveals that the method's stability is governed by the golden ratio φ, which acts as a natural threshold for the asymmetry between factors. We demonstrate that the violation of this threshold induces a metric signature transition to a circular regime, which imposes a rigid modular obstruction associated with the Sum of Two Squares Theorem. Finally, we identify the class of "aligned semiprimes", exceptional cases where the hyperbolic and circular geometries coincide, preserving arithmetic information across spectral boundaries. Key Highlights Spectral Formalism: Redefines integer factorization not as an algorithmic search, but as the determination of the kernel of a discrete operator LN defined on a causal lattice. The Golden Threshold: Identifies the golden ratio (φ ≈ 1. 618) as the critical boundary for the structural stability of the hyperbolic parametrization. Metric Signature Transition: Demonstrates that high asymmetry between factors forces a topological transition from a hyperbolic open regime to a circular compact regime. Aligned Semiprimes: Characterizes a rare class of integers where the spectral signature remains invariant under metric transition. Mathematical Subject Classification (2020): 11A41 (Primes), 11D09 (Quadratic and bilinear equations), 11E25 (Sums of squares).