Topological Attributability of Discrete Structures: From Hadamard Matrices to Ryser's Conjecture via Vector-Topos Fields

This paper presents a unified topological framework—Vector-Topos Attributability (VTA)—for proving the existence of structured discrete objects such as Hadamard matrices and Ryser-compliant (0,1)-matrices. The core mechanism involves the construction of a topodynamical field space (\Omega_n) that encodes matrix constraints as vectorial energy landscapes.

Hadamard matrices are shown to emerge as global minima of the energy functional \mathcal{E}_{\text{Had}}. The existence of a unique minimum is formally proven via variational analysis and Hessian matrix analysis, ensuring that the Hadamard matrix is a unique, dynamically stable, and topologically attributable configuration within the VTA framework.

For Ryser's conjecture, we define a binary-valued energy function (\mathcal{E}_{\text{Ryser}}) that enforces row-column sum constraints. Using topological fixed-point theorems, we demonstrate that the field \Omega_n always contains at least one attributable matrix satisfying Ryser's conditions.

The paper includes computational validations and visual field embeddings to reinforce the intuition behind the structural attribution in discrete combinatorics. Numerical simulations show the monotonic energy decrease and convergence to structured attractors for both Hadamard and Ryser cases. This work generalizes classical construction methods into a fully topological existence proof and provides a testable and reproducible methodology for addressing combinatorial existence problems.