Strong spectral formulation of the FZF formalism] {Strong spectral formulation of integer factorization via discrete self-adjoint operators

We develop a rigorous spectral formulation of the Function Zero Factorization (FZF) formalism, in which the factorization of an integer N is reinterpreted as a discrete spectral problem. Moving beyond heuristic approaches, we adopt a literal functional analysis perspective: based on the indefinite quadratic form L (b, i) = b² - i² - 4N, we construct discrete self-adjoint operators defined on finite Hilbert spaces whose eigenvalues exactly encode the admissible integer zeros associated with the factorization. Our results demonstrate that the presence and multiplicity of the null eigenvalue (0) directly correspond to the factorizations of N contained within admissible windows. We introduce the notion of a "spectral gap" around zero as a measure of structural informativeness. A key finding is the existence of a universal arithmetic threshold: under the admissible parity condition, the spectrum of the diagonal operator is contained in 4ℤ, establishing an absolute minimum gap |λ| ≥ 4 for any non-zero eigenvalue. Finally, to analyze the stability of the spectral kernel, we introduce a discrete Schrödinger-type regularization, incorporating local coupling via graph Laplacians. This approach allows us to formulate geometric criteria for the isolation and degeneracy of the solution space, connecting number theory with the spectral theory of quantum graphs. Key Highlights Strong Spectral Theorem: Establishes a bijection between the factorization of N and the kernel of a discrete self-adjoint operator TN. Universal Arithmetic Gap: Proves that, for the diagonal operator, no "quasi-zeros" exist below the energy level |E| = 4, providing a rigid structural invariant. Discrete Hilbert Space: Formalizes the problem in l² (DN), applying rigorous operator theory to Diophantine equations. Schrödinger Regularization: Proposes a Hamiltonian model H = εΔ + V to study the localization of admissible states against the spectral background. Mathematical Subject Classification (2020): 47B15 (Hermitian and self-adjoint operators), 11A41 (Primes), 81Q10 (Self-adjointness of Schrödinger operators), 05C50 (Graphs and linear algebra).