[petalas]

Summary

A scoped proof strengthens the positive-Q8 lower bound for imbalanced inverse-generator targets, with any finite number of arbitrary restored dirty roles and free binary scalar GL mixing. It builds on PR3578's homogeneous quadratic method, PR3589's mixed cubic ranks and PR3736's count region.

new: 2n_i+n_j >= 4t_i-t_j,  n_i+2n_j >= 4t_j-t_i
     2n >= 5t_i+2t_j,      2n >= 2t_i+5t_j

shape (7,1): previous necessary-count minimum 18; new lower bound 20

The total bound improves the inherited 11t/5 bound when one target multiplicity exceeds four times the other. The package also proves the exact integer minimum of this strengthened necessary-count relaxation:

N(t_i,t_j) = t_i+t_j + 2 max(ceil(3(t_i+t_j)/5), ceil(3t_i/4), ceil(3t_j/4)).

This is not an attainable-word classification. The actual (7,1) minimum remains in {20,22,24}. The infinite family (t_i,t_j;n_i,n_j)=(7k,k;7k,11k) passes the previous mixed cubic/floor/parity conditions and fails the new quadratic condition for every positive k.

Package: research/q8-negative-shape-bound. Base: 12fa8c5c844be21bfefb38a19503fbd867a11244, priority revision 7. Claim: #1166 (comment).

Evidence

• Before: the old region admits (n_i,n_j)=(7,11) for shape (7,1). After: the new inequality requires 25>=27 and rejects it; the strengthened relaxation attains 20 at (7,13).
• Exhaustive integer corroboration: 2,347,344 candidate surplus splits across 2,144 ordered shapes with total target count at most 64; exact formula agreement for every shape. The strengthened minimum improves on the old mixed region in 640 ordered mixed shapes in that domain.
• New rank-step controls: 4,369 rectangular binary pairs and 33,297 chronological invertible-middle cases. Positive controls: 44 complete formal Q8 direct words, 2,636 matrix entries and 2,624 explicitly checked regular basis columns, with two arbitrary dirty spectators. Six failure-control groups include 256 scaled strict exclusions.
• Normal and optimized registered commands pass locally in 0.713 s and 0.788 s. Ruff 0.14.0 E/F/B/UP passes. The registered package runner passes; all nine publication/input files pass committed-byte and source-history preflight.
• The official committed export and integrity check pass. Both scientific modes pass without Git on the exported payload, with all ten files and 59,676 bytes unchanged. A corrupted premise copy is rejected under optimized Python. Local and exported receipts agree exactly: 31ef4cccccc02fe5e65ddb0cc0bc1d03a2ca5c20f8a95f43de676c9a4f8f2430.

From the repository root, use the exact interpreter path:

~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/q8-negative-shape-bound/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/q8-negative-shape-bound/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/q8-negative-shape-bound
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/q8-negative-shape-bound --check-source-history

The verifier emits a generated receipt bound to package and declared input hashes and writes no files. The written proof inherits the predecessor chronological identities as premises; their independent re-verification is not this contribution.

Limitations and unrun obligations

No 20-call word, full endpoint census, physical caller or complete scalar/routing/restoration/hosting/precision invoice is supplied. Necessary-count witnesses need not be realizable. The four-target ten-versus-twelve questions and balanced sharpness remain open. No new kappa, supplier admission, selected-frontier change or complete multiplication theorem is claimed. The selected full cold replay, Lean and Docker builds were not run. Independent human and formal proof-kernel review are unclaimed.

Related open work

• Audit Q8 dirty spectator restoration boundary #3906 checks the one/two-target arbitrary-dirty restoration boundary. It does not study the new mixed shape inequalities. The overlapping counts are 3 and 6, numerically reproduced here and equal to that package and Attain the mixed-negative Q8 bound for t <= 2 #3697.
• Exact-search tooling for noncommuting words, and a complete six-distinct-value screen of the KB2 thirteen-call candidate #2444 provides generic noncommutative exact-search tooling and a separate GL2/S3 census. It does not study this Q8 count projection; no numerical quantity is shared.

The overlap refresh covered 340 open PRs and 2,972 Discussion records with complete pagination, plus current handoffs and targeted historical Q8 titles and bodies. This is a scoped overlap check, not a proof of novelty.

Open next questions

1. Does a full 20-call (7,1) word exist? Start from the legal n=20 splits and all eight formal endpoint coefficient equations.
2. Can quadratic and cubic rank equality occur simultaneously at t_i=4t_j, using the same chronological L in both orientations?
3. Can an improved formal word save a matched complete physical caller bill? First identify that caller and every charged boundary operation.

CI notes

All checks pass on 12269a5311577c5aabbba432558e89fb12de869c: plan in 2m36s and the registered package replay in 1m21s. Hosted run 39488342994 completed successfully. Its normal and optimized scientific receipts both exactly match the local and Git-free export digest 31ef4cccccc02fe5e65ddb0cc0bc1d03a2ca5c20f8a95f43de676c9a4f8f2430. The hosted registered-package pass is present in the log. No failures or manual reruns occurred. Local execution used CPython 3.12.15, Clang 23.1.3, on macOS 27.0.1 arm64; hosted replay used the registered Python 3.12 environment.

Merge danger

Door: two-way. Blast radius: package. This adds one standalone research package; shared code and selected-frontier files are unchanged.

Produced by Nick Petalas. Predecessor credit, exact source pins and the retained Q8/register lineage are recorded in the package. Apache-2.0.

[rcfc-jreed]

Independent scoped audit of #4234 Q8 negative shape bound at 12269a53: 12/12

Code-disjoint checks; does not import package verify.py (local fork under nested research/q8-negative-shape-bound/ with patched PACKAGE/ROOT). Package bytes from head 12269a5311577c5aabbba432558e89fb12de869c via GitHub contents API. CLAIM.

No packaging defect in scope. Stdout JSON receipt (PASS_Q8_NEGATIVE_SHAPE_BOUND) with receipt_sha256.

What passes: pins at base 12fa8c5c…; five SOURCES digests; fork regenerates status + receipt digest.

Not covered: Lean formalization; new κ; merge/close. Finite replay ≠ formal proof.

James Reed (@rcfc-jreed), with substantial AI assistance (Claude, in Cursor).