[petalas]

Summary

Scoped constructive proof with finite corroboration. Resolve PR #6151's second open question by placing all five coordinate counters on one physical tape, with one head and every move charged.

- 12 tapes; D counter bits; sufficient bound 192N+16D+80
+  8 tapes; D counter bits; sufficient bound 256N+32D+256

Here L_i=bit_length(q_i-1), D=sum L_i and N=product 2^L_i for the five positive coordinate cardinalities. Both padding and restriction preserve immutable descriptors/input, return every head and clear all counter bits. The sufficient action bound is larger; the gain is four fewer tapes with linear complete movement. A neighboring counter-block round trip costs exactly 4L+8, paid by a complete inner cycle.

Only research/shared-tape-aggregate-padding/ is added. The uniform-transfer-audit card is explicitly changed to this component construction, relevant to precision_tape and owned_erasure. No selected assumption, supplier or new kappa is claimed. Research stopped at this first verified improvement. Advance claim.

Evidence

• Before: PR Reduce aggregate padding from seventeen tapes to twelve (no new κ) #6151's twelve-tape contract and local two-phase countdown are cited predecessors. No predecessor verifier or scientific census was replayed.
• After: 700 full scan cases, 348 exhaustive payload scans, 1032 adjacent-block round trips and fifteen rejecting controls pass. Checks include singleton counters, unequal widths in every position, translated origins, immediate reuse, arbitrary discarded padding, one actual shared head, all eight returned heads, byte preservation and complete action bounds.
• Registered normal/optimized commands and direct warning-strict checks pass locally on CPython 3.12. Each direct check took about 1.1 seconds. Normal and optimized generated, hash-bound receipts agree exactly. Committed-byte and source-history checks pass on e0c7c4a9c5bb7ac19b92170f04bd2c8de59263df.
• Read-only Git-free exports pass normal and optimized checks with empty PATH from an external working directory. Every exported byte stays unchanged, and both receipts equal the working-package receipt.

Exact reproduction from the repository root:

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

Another standard-library CPython 3.12 path may replace this local path. Expected standalone marker: PASS_SHARED_TAPE_AGGREGATE_PADDING. The all-width theorem rests on PROOF.md; finite execution corroborates it.

Limitations

Canonical descriptors, owned delimited zero work and fresh output are entry premises. Allocation, descriptor production, completed fast callee execution, ancestor parking, physical caller integration and the complete supplier invoice remain external. Malformed calls need not restore state. No minimum-tape theorem, optimal-action claim, compiled transition table, formal kernel or complete multiplication theorem is supplied. No selected cold replay, native/Lean/Docker suite or independent human review was run.

Related open work

No overlapping open PR studying the same five-coordinate shared-counter scan was found in the fresh focused searches and open-PR/claim checks. PR #6151 is the merged predecessor; PR #5090's merged fifteen-tape ROW adapter has a different descriptor/callee boundary. The completion check also read the other three open PRs from this account created since the advance claim, #6796, #6797 and #6798; none studies this shared-counter scan or its tape/action quantities. Search absence is not a novelty certificate.

Open next questions

• Integrate this scan into an actual completed exchange, with its simultaneous tape inventory, ancestor parking and cleanup charged.
• Determine whether the five immutable descriptors can also share one tape with linear complete movement. Start by pricing simultaneous descriptor/work navigation through every phase change and carry.

CI notes

Both hosted checks passed at e0c7c4a9c5bb7ac19b92170f04bd2c8de59263df: plan and replay (research/shared-tape-aggregate-padding), in run 42228143986. The completed PR checks were read once and confirmed. Local registered normal/optimized checks, matching direct receipts, read-only Git-free exports and committed-byte/source-history checks also pass. Hosted replay certifies the registered finite checks; the all-width claim remains the scoped written proof. No check failed or remains pending.

Credit

This contribution is produced by Nick Petalas. PR6151 supplies the two-phase countdown and shared-tape question. ywz65's PR2636 supplies the earlier aggregate scan and PR2457 the aggregate-before-padding reduction; commdata2338/PR2399, Jose Manuel/PR2234 and Adam Gall/PR2087 retain descriptor, parking and supplier credit. Binary countdown amortization is classical. Predecessor files and credit lines remain unchanged. Apache-2.0.

Merge Danger

Door: two-way. Blast radius: isolated. Seven additive package files and an opt-in replay hook; no shared implementation or selected-frontier edits.