O-01d0 is a precise algebraic gate inside a programme connected to the Polydegree Conjecture. It asks whether four weighted homogeneous polynomials have no common projective zero for every index from six onward. This release does not answer that question. It records what can now be proved, what has been falsified, and exactly what remains. A fixed-index theorem shows why one geometrically empty polar fibre would force a complete intersection that is finite free over its deformation parameter. Producer computations supply such fixed-index consequences at twenty-one and twenty-five. A third case, twenty-nine, is preserved by a sealed source-and-transcript record but is not rerun. The main structural step rewrites the lower arithmetic obstruction as triple coprimality for three consecutive members of an explicit one-variable Hermite family. The paper proves that any structural triple gcd is squarefree, controls two exceptional boundary fibres, and derives a fifth-derivative connection whose reciprocal form has only five boundary columns. Several attractive shortcuts fail exactly. Low-degree scalar transfer, a canonical polynomial-remainder tail, bounded-degree projective transfer, and the claim that every prime in the structural window works are all rejected by reproducible counterevidence. Large finite scans point instead to a pair-content and four-factor theorem, but finite survival is not a proof. The remaining route has four independent gates: that lower theorem, one simultaneous good prime for the upper determinant, graded multiplier lifts in both odd residue classes, and a complete projective chart cover. Ordinary and optimized producer replay agree, public continuous integration passes, and GitHub and Zenodo expose matching assets. Those facts establish availability and internal replay, not O-01d0, AF5 transversality, any linked conjecture, independent reproduction, formal verification, specialist review, peer review, novelty, priority, or impact. This synthetic-voice briefing is a communication aid, not mathematical evidence.