Audit: recursive parking boundaries and exact fixed-tape overhead (no new kappa)

[commdata2338]

Change and mathematical scope

The retained tape package proves individual push/pop round trips but leaves recursive handoffs open. This package supplies a scoped written proof and an explicit park/call/restore controller. Arbitrary dirty roles are restored around a length-preserving child, and the wrapper costs exactly 5V - n_active + 9W - 2 transitions. Equal positive roles give child volume V/W and a wrapper cost of at most 14V, independent of the length of the parked ancestors.

A separate all-length counterexample shows why restoring every byte and head is not enough for a uniform time bound. The same finite child can scan an arbitrarily long parked ancestor and return unchanged, so the child contract needs the distinction. This is not a defect claim against the manuscript, which explicitly forbids ancestor scans.

Status: a scoped written proof with exact finite checks, plus the counterexample, for an interface (transfer) result. No eliminated recursive calls, exponent gain or new κ are claimed.

Precise delta: only research/recursive-parking-boundary/ is added (eight files). It holds:

• the written quantified proof;
• a source-table extractor and a literal tape simulator;
• an explicit active-stream move and a finite phase controller;
• a deterministic replay receipt, six focused semantic test groups, and standard package metadata.

No existing source, selected record, frozen artifact or supplier profile changes. The first-done and trace-locality arguments are written proofs, and no new Lean theorem is claimed.

Cost accounting: the exact cost includes one stationary dispatch per phase, including the child's return. Child execution, row splitting, descriptors and marker lifecycle costs are separate. Empty streams have charge 9W-2; they are not covered by the positive-volume 14V inequality. Unequal role lengths keep the boundary semantics but do not imply V/W child volume.

Remaining obligations.

• The nested fixture uses abstract row split/merge shims and does not price or implement them.
• Still open:
  • physical complete-row routing, descriptors and recursive call-stack control;
  • the concrete selected-call binding and a uniform child-time proof;
  • new Lean composition and trace proofs;
  • a full native or selected cold replay, and the precision and analytic transfer.
• Source extraction is a restricted parser, not a Lean elaborator.
• The selected κ is unchanged, and no full multiplication theorem or runtime speedup is claimed.

Task and overlap

• Task / scope: uniform-transfer-audit / ce7411fb7cf6f3460bfc, refined to fixed-tape-parking/dirty-role-boundary/all-length-locality. Chooser nonce e101f9f408701b26cf61d9bd1f07e040, track adversarial-audit, priority revision 2. It passed over supplier-boundary-audit because this account's PR Audit: selected PR589 bit-hosting pending projectors across all hosted helpers (no new kappa) #1307 and the active selected-geometry claims cover nearby ground.

• Base and exact dependencies:

  • base bce54fb3f895cbe5cd4c0eb477c3afeed100190f; contribution 3a45bd089ad56c38b55da8d80d7b056f2d90e5fa;
  • the retained machine-transfer stack sources introduced at 2d970ab3ca0271cfb2b609dc9aae1b7073b7774d (rfu08), with exact current Git blobs and SHA256 in SOURCES.json;
  • selected PR Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094, head 0fc329d0bec93ca7cd288eaea1295172e8789810, context only.

  There is no pending PR dependency.

• Concrete object and question: the fixed-W inactive parks, the active move, the child boundary, the active return and the reverse-order pops. The aim is to separate semantic restoration, intermediate locality and uniform cost.

• Exact implication audited: do literal primitives plus child endpoint restoration suffice for the normalized recurrence's equal-volume and linear-overhead premises? The counterexample disproves the weaker endpoint-only time implication.

• Scope status: refined. It takes the card's tape/restoration branch and the composition boundary the retained README explicitly names as missing. The finite tests support the written all-length proof; they do not replace its quantifiers.

• Method and domain: four-symbol, head-local, unit-move multitape semantics, with a written theorem for every fixed W >= 2, arbitrary lengths and finite ancestor contents. Source extraction covers all 12 push and 384 pop state-symbol entries. The exact finite domains and mutation controls are in RESULT.json.

• Expected cost: (W-1)(5n+7) + 4n+4 + (2W+1) for equal lengths. Exact restoration, the original stream order and ancestor independence are all required.

• Nearest work:

  • The retained package already proves primitive round trips; this adds intervening-child composition, active movement, finite dispatch and trace framing.
  • PR Research audit: sharp physical counter costs with direction reversals (no new kappa) #861 (7113da8d458f5c73fe70b1bd5371ea1e44bc75d4) audits counters and excludes recursive scheduling; no part of it is imported.
  • Scalar-word, dirty-fanout and verifier-concurrency schedulers are different objects.

• Claim and final check: the claim is in discussion #1356. A live overlap check just before submission (850 open PRs and 98 discussion records at 23:47 UTC, complete pagination) found no competing work.

Validation

From the repository root, Python 3.12.3, standard library only:

python3 -B scripts/research_package.py --package research/recursive-parking-boundary --check-only
python3 -B scripts/research_package.py --package research/recursive-parking-boundary
python3 -I -B -O research/recursive-parking-boundary/verify.py
python3 -B scripts/research_baseline.py

The registered replay runs the verifier and all six focused groups. The finite evidence is:

• all 396 literal source-table entries and 8,001 primitive round trips;
• 22,308 equal-role and 300 unequal-role boundaries;
• 90 nested fixtures, with 4,044 inner boundaries;
• 13 controls.

The deterministic receipt matches under normal and optimized Python. An isolated read-only relocation passes, and altering one recorded count or one live source byte is rejected. The registered replay and the metadata/pin check were run again on the final commit before submission, and both passed. Hosted CI and independent human review are not reported as passed.

PASS_RECURSIVE_PARKING_BOUNDARY: 396 source-table entries; 22308 equal-role boundaries; 4044 nested boundaries; 13 controls rejected
PASS_REGISTERED_RESEARCH_PACKAGE research/recursive-parking-boundary outcome=completed

Credit

• rfu08's retained PR Verify finite profiles and formalize multiplication transfer interfaces #64 transfer lineage, for the literal stack machines and their Lean proofs.
• The upstream fixed-tape schedule and the ranked-pair transfer review.

New files are Apache-2.0; predecessor sources and notices are unchanged.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the proof, the implementation, the verification and the writing; it was packaged and submitted with Claude Code. No independent human review is claimed.

Open next questions

From the package README:

1. Formal boundary composition and trace frame rule. This would connect the existing kernel-checked stack primitives to recursive handoffs. First step: formalize the marked-stream move and the quantified intermediate head bounds, then induct over boundary_program. Blocker: the new composition and trace statements have not been implemented or compiled in Lean.
2. One physical complete-row split/merge caller. The boundary earns V/W only under equal complete rows. First step: a fixed W=2 caller with an explicit row payload width, keeping every spectator and counting the counter and reset movement. Blocker: the nested test uses abstract routing shims, and descriptor and dispatch-stack integration need their own uniform cost proof.

Workflow feedback (optional)

• What worked: the package generator, the exact source pins and the offline overlap mode.
• Largest detour: narrowing the broad all-size and precision overlap search before finding the retained tape README's explicitly open scheduler-composition boundary.
• Suggestion: add that README and its named open boundary to the uniform-transfer card's starting points. Alternatively, keep a small source-linked index that separates proved primitives, open composition and physical-caller work.

[Mister-Meeseeks]

100-PR batch review at 3a45bd089ad56c38b55da8d80d7b056f2d90e5fa.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Conditional park/call/restore tape-boundary proof with exact 5V−n_a+9W−2 wrapper cost, source-table replay and nested tests. Ancestor-preserving endpoints alone do not imply a uniform child-time bound. Physical row routing and Lean composition remain separate obligations.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.