Octad pattern census, Borel core/bridge reduction, constructive Perkel matrix units, proof-carrying clique batching, and minimum marked-resolvent tomography.
Every multiplicity pattern is an edge-disjoint clique packing of K8. Canonical augmentation gives exactly 69 S8-orbits. Actual one-factorization realization remains a larger census.
P19 is the shared binary core. P57 adds eighteen nonneighbor bridge variables. This is an exact solver contract, not a verdict.
Nine Eij and nine field units satisfy the full matrix-unit algebra. The invariant positive form has 2×2 principal minors 15/19 and determinant 7/19.
Sixteen genuine star-complement compatibility graphs, sizes 48–159, have frozen witnesses and independently checked upper-bound proof DAGs.
A triangle is the smallest marked resolvent that separates W33 from triangle-free Gewirtz, and it uniquely reconstructs a W33 K4 line.