Hadamard's Determinant Problem: Structural, Constructive, Empirical, and Global C‑Phase Foundations (Papers 1–4)

This work presents a unified structural and analytic framework for Hadamard’s determinant problem, offering a complete geometric and algebraic description of the space in which both Hadamard and non‑Hadamard ±1-matrices live. Across four integrated papers, it develops the invariant geometry of the fluctuation Gram matrix, classifies extremal families, identifies the universal C‑phase governing non‑Hadamard extremals, and establishes collapse laws, spectral regimes, and analytic invariant floors. The program combines structural discovery, constructive dynamics, empirical universality, and a global analytic theorem to reveal a coherent picture of extremal behavior across all admissible orders. This collection provides a fundamentally new perspective on maximal determinants and mod‑4 structure, offering readers a comprehensive and self‑contained theory.