Requiring the factorial ceiling only at floor(N/d) reduces the earlier all-cell excess by 34.4 percent, 29.9 percent, and 41.9 percent at N=1000, 10000, and 100000, and an exact N=27, y=9 example with eight prime-looking cells has minimum excess log(2), refuting the earlier dimension-count argument for zero excess.
hunts/quotient_certificate (Record 93 of 98 in chronological sequence)
Guiding Question
How much excess does positivity on unattainable cells force in the finite factorial-certificate LP?
Method & Verification
The attainable-quotient LP at square-root support retained 889 constraints checked in exact integer arithmetic, and an exact rank-seven row relation at N=27 produced a feasible vector attaining excess log(2).
Lineage & Relationships
Primary Sources (at pin 8fa46e134)
Editorial Notes
Status is finite improvement and correction, no asymptotic result; mapped to completed for the finite work. The N=27 counterexample is an ordinary exact argument; the percentage reductions are measured LP ceilings, so the composite grade is measured. Input is the factorial LP at main d393e1a under prime_pair_error/frontier.
Date Provenance
commit d49e699a1aebcd0e51e3241cf061ae48ce4d1095, hunts/quotient_certificate/RESULTS.md, author 2026-09-07T11:22:02-05:00