Completion and Marked Factors for Fourth-Root Hadamard Matrices

These two working papers study matrices with entries ±1 and ±i whose rows are mutually orthogonal. They separate the completion of a prescribed partial matrix from the unresolved construction of such matrices at every even order. The main paper gives direct algebraic completions through defect three, parity criteria at defect four, and a two-ray completion criterion at defect five. Gaussian relation ideals describe second marks for the three critical integral residual root types. For E₈, root moments and a geometric flow argument make the prime-three condition automatic; only the seven and eleven residue tests remain conditional. Explicit complementary factors at orders ten and eighteen realize the theory. Affine-plane inflation preserves the entire extension graph and produces an actual 301-by-306 factor, attaining the least odd multiplier for its prescribed complementary space. A spectral argument proves mark preservation in the stated conference families. The companion studies modular sign selection, row and column repair, standard Gaussian enlargements, and exact orthogonal initializers. It classifies homogeneous congruence and index-nine switch lattices, gives an analytical five-column repair bound, and proves a global obstruction to one-sided dyadic flattening of sparse factors. Explicit two-sided identities escape that obstruction for specified seeds at orders 6·2ᵏ. Periodic complementary products locate every paired zero and norm-two alphabet defect. Residual Gram matrices, norm saturation, Gaussian lattice duality, root systems, binary geometry, and correlation identities organize the proofs. The main paper contains its essential internal dependencies; the companion cites its completion criteria. Exact computational certificates remain explicit where finite classifications require them, and reproducible sources accompany both papers. Uniform complementary-factor construction, general standard enlargement, and global alphabet repair remain open. No previously unknown Hadamard order or all-even existence theorem is claimed.