Exhaustive small-representation relative-degree certificate

Exact statement

For all 531,522 one- and two-dimensional integer linear representations over the binary alphabet with every entry in {−1,0,1}, the Krylov algorithm agrees with a direct definition-level oracle for every candidate r<d; no mismatch or nonunique candidate set occurs.

StatusReproduced from public code over the complete stated finite domain

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

This replaces dependence on an unavailable random checker with a complete, explicitly bounded census. It does not recreate the old run's sampling distribution.

Hypotheses and scope

Dimensions are one and two.
Every entry of λ, γ, M_a, and M_b lies in {−1,0,1}.
The direct oracle tests every candidate r<d and every suffix word of length below d; these columns span the reachable space.

Proof or evidence

The checker enumerates 81 one-dimensional and 531,441 two-dimensional representations. It finds 24 degree-zero cases in dimension one, and 290,304 degree-zero plus 5,568 degree-one cases in dimension two, with zero mismatches and zero nonunique candidate sets.

Exhaustive certificate and checker →

Verification notes

The algorithmic test and definition-level oracle are implemented separately. The saved certificate includes the enumeration domain, exact counts, and a SHA-256 transcript digest.

Limitations

Finite exhaustive evidence does not replace the all-dimension proof.
The certificate says nothing about entries outside {−1,0,1} or dimensions above two.
It is a new audit and must not be described as a reproduction of the unavailable 100,000-instance run.

Open work

Extend only if a new adversarial domain tests a distinct proof boundary; larger random samples alone would not mature the theorem.