For the five fixed selected-prefix cutoffs N=144, 576, 2304, 9216, 36864, every ordinary composite in the local repair interval is exposed by primes through floor(sqrt(X)) while every prime power remains, and the saturated local cap equals the exact Mangoldt weight, with zero remaining local cap surplus.
hunts/paid_shortfall_saturation (Record 96 of 98 in chronological sequence)
Guiding Question
Does full trial division through floor(sqrt(floor(N/y))) make the local repair cap equal the Mangoldt weight for the five fixed selected-prefix cases?
Method & Verification
Unchanged selected prefixes were saturated by trial division through floor(sqrt(X)) at supports 12, 18, 48, 95, 192, with exact prime-log vectors and both python-flint and mpmath.iv interval routes overlapping at 35 and 70 digits.
Lineage & Relationships
Primary Sources (at pin 8fa46e134)
Editorial Notes
No RESULTS.md at the pin. Headline filled from the case log and RUNS.md. Ordinary-composite retention failures, prime-power retention failures, and cap-equality failures were all zero. Global positive surplus S remains nonzero; only the local repair cap saturates. No general rate claim. Enclosure-supported applies to the two-backend log pricing of the exact vectors, not to a new analytic bound.
Date Provenance
commit 7b026791c3cca71c1c31f61175b919b79f40b649, hunts/paid_shortfall_saturation/, author 2026-09-13T23:10:55-05:00