[RossRichardson]

Scale-dependent seed switching was proposed as a way to improve recursive cost using a log-scale average of seed exponents. This package gives an exact counterexample to that unconditional substitution in the stated positive recurrence: two complete profiles with roots 1/4 and 3/4 yield p_eff=log_256(19+3 sqrt(33)) in (5/8,2/3), rather than 1/2. The predecessor's explicit oscillation-error bound permits the discrepancy; a sublinear error estimate is needed before dropping it.

A separate positive-tree comparison proves that arbitrary per-node switching cannot beat the best fixed seed in a finite positive catalogue with zero glue, bounded child ratios and unit terminal cost. That limited extremal conclusion survives independently of root averaging. Both child ranks, all multiplicities and terminal overshoot are retained; a unit additive toll leaves the counterexample exponent unchanged. No physical seed, supplier admission, outer transfer or new kappa is claimed.

Validation: registered CPython 3.12.12 replay with -I -O -B; a separate mathematical worker's exact rank-history implementation (1,023 schedules and 112,311 total rank histories); 11 semantic controls across the two implementations; exact committed-byte preflight; Git-free read-only CPython 3.14.6 replay. These support written all-scale proofs, not formal-kernel or independent human review. No selected cold replay was needed or performed; selected binding check passes.

python3.12 -B scripts/research_package.py --package research/scale-dependent-seed-moment-audit

Scientific base c7df0bb is retained. Canonical main refreshed to a7fc79c at 06:48:47 UTC, priority revision 5; relevant guidance and selected record unchanged, and the package patch applies cleanly. Inspected predecessor: Philippe Van Velzen/philippeVV single-letter lane at ed21d62; full source/blob/hash pins and attribution are in SOURCES.json. No predecessor implementation is imported. Live overlap before submission covered 491 open PRs incl drafts, 2,088 closed/merged PRs and 1,526 Discussion records with complete pagination, plus local notes; metadata snapshots exclude arbitrary diffs and are not a novelty guarantee.

Canonical finding/claim: #2278 (comment). Prepared by Ross Richardson with substantial OpenAI Codex GPT-6.1 Sol Max assistance and one independent AI mathematical worker. Apache-2.0.

Review follow-up: current head 36622e0bd376cf56a0acf54045469bac094d0d94 corrects only the coverage field names in REVIEW.md: homogeneous_cases=240, homogeneous_max_index=120. Complete registered replay and eight-file committed-byte validation pass at this head; separate proof/source audit found no substantive defect. Scientific/code/metadata files and result remain unchanged. Current CI is separate in the follow-up comment.

[RossRichardson]

Review completed; pushed one evidence-wording correction at 36622e0bd376cf56a0acf54045469bac094d0d94. REVIEW.md now names the actual fields homogeneous_cases=240 and homogeneous_max_index=120, rather than implying 240 depths. The seven mathematical/code/metadata files are unchanged.

A separate asymptotic audit found no substantive mathematical defect. It confirms the exact periodic matrix [[36,8],[1,2]], dominant eigenvalue 19+3*sqrt(33), exponent bracket (5/8,2/3), real-scale/terminal-overshoot argument, unit-toll comparison, and the finite positive-catalogue arbitrary-node theorem under its stated zero-glue/bounded-ratio/unit-terminal hypotheses. The predecessor's error-qualified theorem remains distinct from the unqualified averaging implication; both predecessor file hashes/blob IDs match.

The complete registered CPython 3.12 optimized replay passes at the corrected head: 1,023 schedules / 112,311 complete rank histories, both implementations and 11 semantic controls. Independent extra probes cover 204 node-choice cases, 32 off-grid rational points and an overshoot negative control. Committed-byte validation covers all eight files; the current-main synthetic merge leaves this package byte-identical.

python3.12 -B scripts/research_package.py --package research/scale-dependent-seed-moment-audit

Full log SHA256: 5f846d2197a3318cf69326bb9a66a139558b0fd937e020281cc1213734f89d75.

All 61 hosted checks were queued at this exact head when checked; no hosted success is inferred. This is the stated positive-recurrence counterexample and scoped catalogue theorem, not a realizable physical seed or new kappa. Scientific pins, predecessor attribution and Apache-2.0 notices remain unchanged. Ross Richardson with substantial OpenAI Codex assistance, including a separate mathematical audit; no independent human or formal-kernel review claimed.