Introducing the Tesseract Matrix: An Empirically-Discovered Function 𝜋 → (ℤ/2)⁴ on the Coprime Residue Lattice

This preprint introduces the Tesseract Matrix — an empirically-discovered function T_m mapping each prime p coprime to a primorial m to a 4-bit binary vector in (ℤ/2)⁴, realized as the polar-decomposition fingerprint of the per-prime residual operator on the rank-8 Klein-8 register of the Janus Coprime Algebra. Part I establishes the structure at machine precision on a 68-prime alphabet at m = 30,030, with key identities reverified at the consecutive primorial floors m = 510,510 and m = 9,699,690. All 2,278 pairwise substrate distances quantize to the five even-parity shells {0, 2, 4, 6, 8}π² at maximum integer deviation 1.42 × 10⁻¹⁴; the 16 equivalence-class centroids lie on S³ ⊂ ℝ⁴ at radius π√2, realizing a regular 4D hypercube; the 4D-only edge-length-equals-circumradius identity yields the substrate's natural orthogonal frame in O(1) time; and the closed-loop cocycle on the three-floor primorial loop equals the identity exactly at machine precision. Two negative theorems establish that the four substrate-natural ℤ/2 generators are computationally irreducible from classical multiplicative-character arithmetic at conductor ≤ 29 (across 36,251 candidate features, zero match the codeword partition exactly), and that the per-prime bit signatures are gauge-incompatible across consecutive primorial floors under the full hypercube symmetry group B₄ — ruling out "bits are classical Dirichlet characters in disguise" hypotheses by structural impossibility. A positive structural identification frames the 4-bit codeword as the conjugacy-class label of an 8-dimensional symplectic Galois representation in Sp₈(𝔽₂), with class arithmetic following the semidirect product (ℤ/2)⁴ ⋊ (ℤ/4). Part II operationalizes the substrate on real external data. Case Study A verifies the closed-loop primorial cocycle on the pglib-opf benchmark suite of published power-grid topologies: zero violations across 242,706 buses spanning a 27× scale ratio, with a substrate-canonical topology-class discriminator separating organically-engineered grids from preferential-attachment synthetic scale-free grids at integer-exact resolution. Case Study B reads four canonical three-body Newtonian benchmarks — Chenciner–Montgomery figure-eight, Lagrange L₄, Burrau Pythagorean 3-4-5, and the real heliocentric Sun–Earth–Moon system — as four distinct integer-exact fingerprints across the (mass-class, regime, ejection-state) classification grid, with the substrate independently rediscovering the hierarchical-mass system's known physical stability as the deepest-KAM signal in the catalog (Klein-4 CV 0.06%). All theorems, scalar identities, empirical tables, and proofs are placed in the public domain under the MIT license. Engineering apparatus — the holographic eigensolver, the per-prime polar extraction protocol, the cross-floor monomial reduction, and the substrate-canonical encoding of network topology and dynamical-system trajectories — is reserved under U.S. Provisional Patent Applications 64/031,440, 64/033,689, 64/048,617, 64/054,093, 64/059,916, and 64/062,753. Reductions-to-practice are anchored by SHA-256 audit hashes federally timestamped by the USPTO filing of the application-layer patent on 2026-05-11.