Entry Bounds and Spectral Saturation in the Hadamard Problem: Exact rounding, viable moments, and algebraic completion

The Hadamard conjecture asks whether an orthogonal sign matrix exists at every order divisible by four. This working paper develops exact certificates through nonnegative spectral realization, positive sign moments, and complex phase geometry. Quantitative entry bounds force a continuous spectral realization to round to a Hadamard design. A determinantal extraction theorem uses the Cauchy-Binet formula to select an orthonormal basis from an isotropic frame with sufficiently small fourth-moment defect. Its strict constant is sharp for general frames, and its residual form certifies completion of prescribed rows. Complex Hadamard matrices provide another exact starting point. A phase-invariant defect of the entrywise square guarantees a real Hadamard matrix below an explicit threshold; a pivot formula identifies a suitable normalization. Fourier phase changes alone face a proved obstruction. For partial sign matrices, an exact lattice-index transition determines whether adjoining a row preserves an odd integral residual vector. At orders four modulo eight, a valuation criterion characterizes this parity condition. The first possible theta coefficient counts residual sign vectors, separating parity feasibility from the minimum-norm requirement needed for extension. The paper combines these certificates with contact-stress escape, moment-preserving Boolean trades, quantitative positive reserves, an exact four-line completion classification, and classical completion with at most three rows missing. Repeated-prefix examples, a nonglobal concentration maximum, and operator and binary-rank obstructions delimit the methods. Topological nonvanishing and low-temperature partition inequalities give further exact existence formulations. Their proofs distinguish a global certificate from a construction that attains it, and explain why standard coloring bounds, genus averages, or first-moment estimates do not by themselves supply the missing theorem. All general arguments are analytical or algebraic. Nineteen figures illustrate exact formulas and stated constructions; no theorem depends on numerical search or enumeration. A uniform construction reaching any certified exactness region remains open, and the paper does not prove the Hadamard conjecture. The project includes the manuscript, editable sources, reproducible figures, supplementary identity checks, and a separate account of substantive changes and unresolved mathematical issues.