[kumagi]

Change and mathematical scope

PR3486 proves the least admitted borrowing at the actual body depth but leaves its encoded implementation and setup bill open. This adds an opt-in Count-tape selector for that exact covering contract: fixed w, strict 2k<e, body n=floor((e-k)/w)>=m, and q^k>=S_D(n).

All input-dependent arithmetic stays on tapes. Encoded bisection recomputes every trial's body/depth/stock and materializes a fresh final plan. The setup proof includes full rejected-trial radix powers, final recomputation and cleanup: O(h^3+h^2(e+wh)^2) tape operations and a conservative 24-role allowance, including the borrowed input, within the inherited 64-role setup inventory.

The reduced e=13 case fits with k=6 where the earlier D(e) selector requires 8 and fails. At e=24, substitution cycles 8→11→8; the exact tape selector returns k=9. The threshold theorem remains PR3486's result.

Only research/encoded-body-depth-borrowing/ changes. No default backend, selected payload, supplier, cutoff, κ or selected-frontier change is claimed. Completed-callee, pointwise restoration and physical/ordinary C0 premises remain conditional.

Task and overlap

• Canonical base: 6f100d4bb5083f8c35d96fd95a539811c62f245c, priority revision 6.
• selected-stock-envelope, scope changed to PR3486's explicit handoff, exact head 991861d40a1ebdbbf35be9adc1726e6af5b71e49.
• Canonical claim. Entry coverage: 410 open PRs and 2,332 Discussion records, complete pagination; focused publication refresh found unchanged main/predecessor.
• The concurrent factor-two padding claim permits S/2<R<S and explicitly excludes a full encoded borrowing selector. This package retains R>=S; no claim transfers across those predicates.

Validation

At 232b3916e6331222278f6edd5a4b9d56731667d9, Python 3.12 standard-library checks pass:

python3.12 -B scripts/research_package.py --package research/encoded-body-depth-borrowing
python3.12 -I -O -B research/encoded-body-depth-borrowing/verify.py
python3.12 -B scripts/check_research_commit.py --package research/encoded-body-depth-borrowing --check-source-history

171 literal Count cases (84 fitted), 5,760 independent forward/slab cases, 12 rejection controls and an observed peak of 19 setup roles. Five actual implementation mutants reach the semantic oracle. Borrowed input, successful/fallback cleanup and an injected completed-trial failure are checked. A scan-origin bug caught by literal execution was corrected before publication and recorded in TASK.json.

Committed-byte/history checks, the 30-file export integrity check and optimized Git-free export replay pass. Runtime needs only unchanged main inputs; the pending predecessor proof is retained byte-for-byte without an external Git-object dependency. No selected GL64 payload, full selected cold replay or formal proof-kernel check was run. Arbitrary interruption/allocation-failure recovery inside inherited arithmetic is unclaimed.

Credit

Hiroki Kumazaki, with substantial OpenAI Codex analysis, implementation, execution, writing and separate same-session AI-agent checks; no independent human review claimed. Apache-2.0. Credit Nick Petalas (#3486/#3246), Tekalign (#3264), Jose Manuel (#2899/#2809), David Arias (#2753), Ross Richardson (#2420), and their retained BIT/geometry/compiler lineage. Exact attribution and source pins are in the package.

Open next questions

Bind this plan factory to one complete reduced ordinary caller, checking both padded returns, spectators and fallback. Separately, a saturated comparison could avoid full rejected-trial powers, but needs its own equivalence proof and final exact materialization. Both next steps are scoped in README.md; neither is claimed here.