Before the first diagnostic: five predetermined cutoffs, 144,576,2304,9216,36864; two coefficient families; two capacities; exact rational algebra; two interval logarithm implementations at 35 and 70 requested digits. No optimizer. One numerical thread, estimated under 30 seconds. Abort and report rather than expand if a bounded run cannot finish within 60 seconds.
The diagnostic separates cost above psi(N) from cost above N. Neither a small finite ratio nor agreement of the two logarithm implementations is an asymptotic result. Exact coefficients and rational enclosures are retained; the pinned source reconstructs the exact prime-log vectors before evaluation. decimal summaries are display approximations, not interval endpoints.
Initial diagnostic completed in 1.31 seconds. At N=36864, the balanced-prefix raw excess over N was approximately 904.873880 and the perfect-power-cap excess was 877.252193. This motivated a second, distinct test: uniform averaging over cutoff choices from ceil(y/2) to y, keeping the same five N values. It tests sensitivity to concentration of the balancing correction, not a larger-N extrapolation. Averaging improved one of five cutoffs. A third diagnostic compared the finite cutoff choices individually, using the exact small-integer excess formula. Selection improved two and tied three. The durable run below includes four families, with the same five N values.
The final diagnostic took about 7.15 seconds. The durable run is capped at 60 seconds and 1 MiB captured output, with one requested numerical thread. No free-coefficient optimizer, external execution service, or Ostoyae is used.
Independent review of the actual manuscript confirmed the cap and coefficient bounds but found two scope defects: an undefined cutoff at support two and an overstated information cutoff outside the square-root regime. Both were fixed. The averaging range now starts at max(2,ceil(y/2)); the general arithmetic cutoff is max(y,floor(N/a)). Regression tests include support two.
The supplementary profile argument in PROFILE_NOTE.md is preserved but is not part of this run's evidence or the main construction's dependencies.
After check-001, a distinct arithmetic discriminator keeps the endpoint coefficients fixed and excludes composites witnessed by small prime factors. It uses prime sets {2} and {2,3,5,7}, tests the cap for every integer through 4096, and keeps the original five N values. Predicted runtime under 10 seconds, with the same 60-second wall and one-thread durable-run limits.
Durable scaling run
The scientific baseline is PR 231 (4efa72a). The recorded execution base is PR 232 (ede6551), which restores recovered test bytes and relocates their precision guard without changing the mathematical source.
Check-001 completed in 7.182455 seconds: 257 cap identities, eight averaging identities, and 191 candidate cutoffs with 764 interval comparisons across two implementations and two requested precisions. All assertions passed.
id: paid-scaling-2026-09-14-check-001
hunt: paid_shortfall_scaling
started: 2026-09-14T00:21:57.084779+00:00
finished: 2026-09-14T00:22:04.267234+00:00
ran:
- .venv/bin/python hunts/paid_shortfall_scaling/scaling.py --output hunts/paid_shortfall_scaling/computations/check-001/results.json
outcome: All finite identities and complete-cost checks passed; the uniform rate remains unproved
artifacts:
- hunts/paid_shortfall_scaling/computations/check-001/manifest.json
- hunts/paid_shortfall_scaling/computations/check-001/results.json
- hunts/paid_shortfall_scaling/computations/check-001/stdout.txt
- hunts/paid_shortfall_scaling/computations/check-001/stderr.txtSmall-factor runs and source preservation
Check-002 completed in 1.263508 seconds. All 8,192 nested-cap comparisons through integer 4096 and all five complete-cost evaluations passed. Independent review then found that a generator argument would be exhausted by validation, silently disabling exclusions. The recorded tuple inputs were unaffected. The current function materializes its input once; regression tests exercise generators and reject a composite divisor supplied through an iterator.
Check-002 is historical, not a fresh-source validation of the fixed script. Its exact original script bytes are preserved as check-002/small_factor.source.txt. The manifest and outputs were not edited. The preservation test checks that snapshot against the manifest's original input hash, the other two unchanged inputs against their hashes, and every recorded output against its hash. Validating check-002 against current source with --root must report the changed input. Check-003 reruns the fixed script and is the current evidence.
id: paid-scaling-2026-09-14-check-002
hunt: paid_shortfall_scaling
started: 2026-09-14T00:27:04.952655+00:00
finished: 2026-09-14T00:27:06.216163+00:00
ran:
- .venv/bin/python hunts/paid_shortfall_scaling/small_factor.py --output hunts/paid_shortfall_scaling/computations/check-002/results.json
outcome: All tuple-input cap and cost assertions passed; original source retained before an iterator-handling correction
artifacts:
- hunts/paid_shortfall_scaling/computations/check-002/manifest.json
- hunts/paid_shortfall_scaling/computations/check-002/results.json
- hunts/paid_shortfall_scaling/computations/check-002/small_factor.source.txtCheck-003 completed in 1.370157 seconds with all 8,192 capacity comparisons and the same five complete costs passing. It records the corrected script.
id: paid-scaling-2026-09-14-check-003
hunt: paid_shortfall_scaling
started: 2026-09-14T00:28:39.516149+00:00
finished: 2026-09-14T00:28:40.886306+00:00
ran:
- .venv/bin/python hunts/paid_shortfall_scaling/small_factor.py --output hunts/paid_shortfall_scaling/computations/check-003/results.json
outcome: Corrected iterable handling; all finite cap and complete-cost assertions passed without changing the reported costs
artifacts:
- hunts/paid_shortfall_scaling/computations/check-003/manifest.json
- hunts/paid_shortfall_scaling/computations/check-003/results.json
- hunts/paid_shortfall_scaling/computations/check-003/stdout.txt
- hunts/paid_shortfall_scaling/computations/check-003/stderr.txtVerification
The focused sequential suite passed 249 tests, with four slow cases deselected, in 16.86 seconds. It covered the two new test modules, the prior paid-shortfall module, all 52 recovered-source hashes, the recovery precision guard, the two restored checkers, knownness, numbering, hygiene, HuntSpec, probe discipline, and executable doors. The governance suite passed 335 tests in 3.73 seconds. The contribution contract passed, including its 21 tests with two deselected. scripts/make_context.py --check passed.
The provided manifest validator passed with current-root hash checks for check-001 and check-003. Check-002 passed structural validation; its historical source and output hash checks passed in the preservation regression, as explained above. Check-002 and check-003 result JSON objects agree exactly after removing only their runtime field. No prior research outputs were edited.
No local full numerical suite or Lean build was run. Remote full-suite status is reported separately from these scoped checks and is not inferred from them.