[petalas]

Summary

Prove the exact joint bin-count projection in the historical fixed U8 T=29 whole-helper model. This answers the explicit Open next question in woklerx's #1844, using its affine family and Hanhong Zhao's #1160 tight-pattern theorem as cited premises.

 x = number of capacity-16 {9,7} bins
 y = number of capacity-8 {4,4} bins
- 4824 <= x <= 12992 attained with y=8286
+ every integer pair 4824 <= x <= 12992, 0 <= y <= 8286 attained

Claim type: scoped constructive proof with exact finite corroboration. The new step is the sharp y bound and a paired four/eight replacement proving that both coordinates vary independently. It preserves every bin quota, every helper demand, the entire remainder profile and the signed price. Only research/t29-joint-bin-spectrum/ is added. No selected frontier or kappa change.

This contribution was produced by Nick Petalas. Credit woklerx for #1844's affine construction and open question, Hanhong Zhao for #1160's model and tight-pattern theorem, Chafik Boukhalfa for #795/#890, Jonathan Capalbo for #942, woklerx for #992 and Hao Yan for #858. No predecessor census or executable was replayed as an audit.

Advance claim. Scientific base: canonical main 81e9bd69e4a5b00ba9c713a1620e36d576d5522e, priority revision 7. The selected-packing card is explicitly changed to this arithmetic side question. Main was refetched unchanged before committing. Live claims, open work, merged-main terms and merged/all-state T29 searches found no competing joint-projection result in the inspected coverage. Sparse local notes, arbitrary diffs and unpublished work remain outside complete coverage.

Evidence

Before, #1844 left the two-coordinate projection open. After, the README proves necessity and constructs every integer pair. The external conversion pool has exactly 20,640 eights for every x; at most 8,286 paired replacements are needed.

Local warning-strict normal and optimized runs each pass:

• 82,138 new joint constructions with complete capacity, helper, remainder, price and named-projection invoices;
• 8,169 uniform conversion-pool identities;
• 15 local exchange cases and 11 rejection controls.

The finite grid includes every y on four x slices and six selected y values for each remaining x. It is corroboration, not exhaustive pair coverage; the written exchange proves the complete rectangle. Generated stdout receipts bind all five package files by SHA256. Both registered commands pass, timed locally at 1.65 seconds each on CPython 3.12.15/macOS arm64. AST syntax/function-size inspection passes. No standalone linter or type checker is installed.

Exact reproduction from the repository root, standard library only:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/t29-joint-bin-spectrum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/t29-joint-bin-spectrum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/t29-joint-bin-spectrum
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/t29-joint-bin-spectrum --check-source-history

Expected markers: PASS_T29_JOINT_BIN_SPECTRUM, PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed, PASS_RESEARCH_COMMIT_BYTES. The source-history check passes at c0bc320c52a8f12a839574efd9fec4f7b9163c84. Official committed-package export and offline export integrity pass. Normal and optimized direct checks also pass from a read-only Git-free export with PATH=/nonexistent, preserving all five source hashes. The current selected source-binding orientation also passes; that is metadata orientation only.

Limitations and unrun obligations

This is a historical capacity-tagged arithmetic packing theorem. It does not supply owner geometry, chronology, charts, physical matching, restoration, precision, a complete caller or recursive savings. Those obligations remain unpriced. The theorem depends on the cited predecessor tight-pattern statement and affine starting family; their source bytes are pinned, and no new acceptance of their proofs is claimed. No selected cold operator replay, native/Lean/Docker build, formal proof-kernel or independent human review is claimed. No registered local obligation remains unrun.

Open next questions

1. Determine the full three-coordinate projection also tracking the capacity-48 {9,9,9,9,8,4} count. Start by retaining that coordinate in the four/eight ledger; the present exchange does not classify its possible values.
2. Impose one literal historical owner and chronological constraint. Start with its legal windows and charge its physical implementation; those data are absent here.

Related open work

No open PR studying this historical T29 bin projection was found in the final targeted live search. Merged #1844 supplies the starting x family; merged #1160 supplies the tight-pattern model. Neither is being re-audited.

CI notes

CI pending for head c0bc320c52a8f12a839574efd9fec4f7b9163c84. Local registered replay and committed-byte/source-history checks pass. Hosted CI has not yet certified this contribution.

Merge danger

Two-way door. Blast radius: research. The change adds one isolated opt-in package and does not edit shared code, frozen evidence or the selected record.