[petalas]

Summary

Scoped constructive proof with finite corroboration. The new two-phase countdown shares one physical work tape between each coordinate's genuine and padding phases in ywz65's PR #2636 aggregate scan.

- 17 tapes; 2D mutable counter cells
- sufficient action bound 1024N+128D+128
+ 12 tapes; D mutable counter cells
+ sufficient complete action bound 192N+16D+80

The same five canonical q-1 descriptors remain immutable. Their fixed-width complements provide R-q without another descriptor. Singleton and binary-complete coordinates carry correctly. The uniform proof includes initialization, payload traffic, translated head returns and zero-counter cleanup. The new construction was produced by Nick Petalas. No new kappa, selected-frontier update, minimum-tape theorem or wall-clock speed claim.

Package: research/single-counter-aggregate-padding/. Only seven new isolated files. Scientific base: 721c03822c421b7f7fcbb2598993b947c4d88419, refreshed unchanged before publication. Advance CLAIM. Delivered scope matches the claim; research stopped at the first verified improvement.

Evidence

• Before: PR Implement linear aggregate padding on fixed bit tapes #2636's written 17-tape, two-counter contract is a cited predecessor premise. Its scientific verifier and counts were not replayed or audited.
• After: the new literal tape machine passes an independent mixed-radix oracle on 516 full scans, 1300 additional exhaustive payload scans and 257 complete coordinate cycles. Thirteen rejecting controls include a wrong complement and omitted cleanup. Every successful full scan checks all twelve heads, immutable input/descriptors, protected cells, D zero work cells and the complete action bound.
• Standard-library CPython 3.12.15, macOS arm64. Warning-strict normal and optimized direct commands pass with identical generated hash-bound receipts, locally timed at 0.72s and 0.84s respectively. These are check durations, not performance comparisons. Both registered checks and metadata/provenance validation pass.
• Committed-byte/source-history check passes at 43d8381379669700abc5e2cfa0b734d4a8c27cdf, covering all seven files and scientific-base ancestry. Official export integrity/input coverage passes. Normal and optimized read-only Git-free replays with empty PATH give identical receipts. Export SHA256: c3f26ea29a72381615aa84f14252181bc88525a535013441814ebe4500284376.
• Syntax, function-size and whitespace checks pass. The longest function is 44 lines. Ruff is unavailable; no lint pass is claimed.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/single-counter-aggregate-padding/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/single-counter-aggregate-padding/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/single-counter-aggregate-padding
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/single-counter-aggregate-padding --check-source-history

Any CPython 3.12 path may replace the local interpreter. Expected markers: PASS_SINGLE_COUNTER_AGGREGATE_PADDING, PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES. Each registered command has a 60-second limit.

Limitations and credit

The uniform claim rests on the written proof; finite checks corroborate it. Canonical aggregate descriptors, owned zero counters and fresh output are inputs. Descriptor production, allocation, ancestor parking, the completed fast callee, selected caller integration and a complete precision/stock/tape invoice remain external. Malformed calls need not restore state. A compiled finite-state table, physical allocator, independent human review and formal proof-kernel review are unclaimed. Selected scientific cold replay, historical predecessor suites, Lean and Docker were not run.

Credit ywz65's PR #2636 scan and PR #2457 aggregate reduction, commdata2338 / #2399, Jose Manuel / #2234, Adam Gall / #2087 and the retained descriptor/parking/supplier lineage. Binary countdown amortization and complement arithmetic are classical. The new contribution is their shared physical counter, successful ownership contract and complete uniform bound. All predecessor files and credit lines retain their bytes.

Open next questions

• Can an actual completed exchange integrate the twelve-tape scan? Start with its simultaneous tape inventory, head returns, owned intervals and ancestor parking. The completed callee/ownership interface remains the blocker.
• Can several counters share one physical work tape while retaining linear movement? Start with two unequal widths and charge travel between delimiters. No serialization construction or bound is supplied here.

Related open work

The refreshed 407-record open-PR snapshot and latest 100 open PR bodies contain no other PR studying the PR2636 five-coordinate scan's shared counters or tape count. Merged #5090 treats a different ROW padding controller, #2786 a completed logical row caller, and #2740 a restricted arbitrary-dirty padding alphabet. None is a new result verified by this package; no extra computations were run to compare predecessors' numbers. Sparse cached local coverage and unpublished work remain outside this overlap claim.

Completion refresh checked all 7 other open account PRs created since the CLAIM at 2026-10-28T16:09:32Z: #6149, #6150, #6152, #6153, #6154, #6155, #6156. Their titles and stated scopes concern different objects or quantities. None studies this five-coordinate aggregate scan, its counter cells or tape count, and none supplies a conflicting same-instance number. No extra scientific computations were run for comparison.

CI notes

CI is uncertified at contribution head 43d8381379669700abc5e2cfa0b734d4a8c27cdf. In run 40724923418, plan failed and replay (research/single-counter-aggregate-padding) was skipped. The package replay did not run in CI. The checks were read once on completion, and the failed-job log confirms exactly two onboarding helper test failures:

• test_research_current_status.CurrentStatusTests.test_current_section_matches_selected_record expects the literal **0.003125804** in docs/research/current-status.md; the file instead puts 0.003125804 = 781451/250000000 inside one bold span.
• test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record expects make frontier6030-verify in docs/ci.md; that document still describes PR5599 and make frontier5599-verify.

Main commit 721c03822c421b7f7fcbb2598993b947c4d88419, titled Admit conditional PR6030 kappa after independent application audit [skip ci], updated the selected record without satisfying these documentation tests. These inherited failures are unrelated to this isolated seven-file package. PR #6120 run 40691776027 was directly checked and has the same two test failures and exact FAILED (failures=2, skipped=23) summary. Open PR #6050 targets the documentation synchronization.

All previously reported local checks remain passes. No package or main-document fix, new commit, or push was made for this CI blocker. A maintainer re-run is needed after main is fixed; no hosted package certification or scientific acceptance is claimed.

Merge Danger

Door: two-way. Blast radius: isolated research package. Seven additive files and opt-in replay; no shared implementation or selected record changes.