[petalas]

Summary

Scoped proof with a finite machine certificate: one owned blank beside the random segment removes the growing stopping counter from the fixed (1,1) paired DDG reversal in JoyboyBrian's #6102.

mark the owned blank
walk forward, XOR the leaf output into the dirty output cell
reverse each edge using its phase and resident coin
at the delimiter, clear it and return both heads

For every terminating path of depth T, the explicit two-tape machine uses exactly 3T+4 finite-alphabet transitions, 2T+4 input-head moves, one owned delimiter cell and a fixed 1,563-state, 10,000-rule controller. The random tape and delimiter are restored; the dirty output becomes A XOR i. Using zxv's cited mean, expected transitions are 48479692514343769953106/4649714690825496825793 < 43/4.

The new contribution is the changed local tape boundary, counterless machine and exact tariff. zxv's #6072 supplies the graph, sampling law and mean; JoyboyBrian's #6102 supplies the edge-inverse lemma. Their exact bytes and credit are retained unchanged in an ancestor commit. The classical idea of reversing a recorded computation is not claimed as new. Produced by Nick Petalas.

Evidence

• Before: Reverse the paired DDG with a resident transcript and clean stopping counter #6102's reversal retains a stopping counter with O(log(T+1)) occupied bits and leaves its physical counter/tape service open.
• After: a local marker distinguishes the initial root from phase-wrap revisits. The new machine has no coordinate, elapsed-time or stopping-depth input in any rule key. Preparation and delimiter erasure are included in its bill.
• Every generated rule participates in injectivity/write-permission checks. All 4,088 short dirty-output runs and all 938 terminating prefixes through depth 470 pass exact output, tape/head restoration and tariff checks, including 942 reverse root revisits. Controls reject bad outputs/inputs, exhausted fixtures, a missing marker return and premature root halting.
• CPython 3.12.15: direct isolated normal/optimized and both registered modes pass. Direct checks took 0.19/0.22 seconds, below each registered 60-second limit. These timings establish timeout headroom, not a speedup.
• Committed-byte/source-history/file-mode preflight and official 14-file export integrity pass. Both normal/optimized read-only Git-free export replays pass, with all 15 export files including its manifest unchanged. RESULT.json is generated and binds proof, machine, verifier and retained source bytes by SHA256.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/marked-tape-ddg-reversal/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/marked-tape-ddg-reversal/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/marked-tape-ddg-reversal
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/marked-tape-ddg-reversal --check-source-history --check-modes

Expected PASS_MARKED_TAPE_DDG_REVERSAL, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Elsewhere, substitute a local CPython 3.12 executable. PROOF.md supplies the arbitrary-finite-path induction; finite fixtures corroborate the implementation.

Limitations

This is a partial two-tape finite-alphabet machine with a stationary four-symbol output cell and an owned blank adjacent to the input segment. Its fixed rule selection is finite control, not a RAM oracle. Individual binary-gate costs and binary-alphabet lowering are not counted.

The transcript and head excursion have no finite worst-case bound. Restoring consumed bits does not make them conditionally fresh after retaining the output. Actual caller placement, parameter dispatch, binary lowering and native/recursive pricing remain open. No supplier admission, new kappa, complete multiplication theorem, independent human acceptance or proof-kernel verification is claimed. The five selected framework assumptions remain. The predecessor probability/optimality suite, full selected/native replay, Lean and Docker were not run.

Open next questions

• Bind one actual nonadaptive residue pair to a caller providing the owned adjacent blank, stationary output head and conditionally fresh random input. Begin with its tape ownership layout.
• Lower the fixed four-symbol machine to an explicit binary microprogram and prove its constant, returned heads and scratch cleanup before comparing binary physical tariffs.

Related open work

• Reverse the paired DDG with a resident transcript and clean stopping counter #6102, JoyboyBrian: supplies phase-and-coin inversion and a counter-based logical lifecycle. This adds a changed physical tape boundary that removes its growing stopping counter; it does not audit the predecessor result.
• Joint denominator-13 decisions need fewer expected fair bits than separate calls #6072, zxv: supplies the paired DDG and sampling law. This uses only its literal fixed (1,1) graph and cites its mean.
• Share the paired residue sampler bank in 84,578 bytes (no new κ) #6124, Nick Petalas, 9ae1d82f4e248dd830e32c4d2a0330b62cddc1dd: shares forward graphs across parameter pairs. This keeps the original fixed (1,1) graph; reversal of that quotient and transfer of this tariff are not covered. The controller contracts differ, so no same-quantity numerical comparison is made.
• Classify actual /52 inverse residues and denominator13 applicability #6096, JoyboyBrian: classifies actual inverse residues and a conditional selector adapter. This does not establish that caller applicability or price the adapter.
• The denominator-13 inverse needs bias, aborts or unbounded fair-bit sampling #6068, zxv: proves finite-cap sampling obstructions. This retains unbounded stopping and does not alter that tradeoff.

CI notes

CI is uncertified at commit 116dbe505704a4aca043364b8cff97580990d6dd. In run 40692823700, plan failed in Onboarding helper tests and the package replay was skipped. The package replay did not run in CI.

The failure is inherited from main commit 721c03822c421b7f7fcbb2598993b947c4d88419, which admitted PR6030 with [skip ci] and updated the selected record without matching the onboarding documentation:

• test_research_current_status.CurrentStatusTests.test_current_section_matches_selected_record expects the exact marker **0.003125804** in docs/research/current-status.md. The document instead bolds 0.003125804 = 781451/250000000 together, so that marker is absent.
• test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record expects make frontier6030-verify in docs/ci.md, which still describes Raise conditional kappa to 0.00310045 with independent helper splits #5599 and make frontier5599-verify.

The failed-job log reports 678 tests, two failures and 23 skips. PR #6120's run 40691776027 fails the same two tests, confirmed from its failed-job log. This package does not modify either document. Check statuses were read once; no scientific CI success is inferred.

All package checks listed above passed locally, but this hosted run certifies none of them. The PR needs a maintainer re-run after main's documentation is fixed. No new commit or push was made for this inherited failure.

Merge danger

Isolated research-package addition. Removing the package reverts the change. Shared implementation, frozen predecessor files outside the new package and the selected result are unchanged.

Advance claim. The claim predicted a persistent delimiter; the delivered service also pays to prepare and erase it in one owned blank cell.