[petalas]

Summary

Close the unequal-shift part of Philipp / mightyOil's PR1322 Open next question 1. In both orientations, no independently chosen integer shifts of at most three of its 237 eligible helpers improve the final historical k8 gauge vector. Every improving constant-gauge move must change at least four helpers.

Claim type: scoped computer-assisted negative proof with an exact finite certificate. The new two-neighbor clipping certificate proves, for every such displacement d,

F(k+d) - F(k) >= 8 sum_h max(0, |d_h|-3) >= 0.

F includes both entry/exit scales and the unit-coefficient discount. The infinite domain reduces rigorously to [-3,3]. The predecessor's tariff and final vector are cited inputs; no predecessor census, invoice or endpoint suite is audited or replayed as the deliverable. The new contribution was produced by Nick Petalas.

Only research/k8-unequal-triple-closure/ is added. No new kappa or selected-frontier change.

Evidence

• Before: PR1322 established common-shift triple closure and left unequal triples and arbitrary pair stationarity at its final vector open. After: exact closure for arbitrary integer shifts on any three helpers, plus the quantitative bound above.
• Per orientation: 1,540 clipping inequalities have minimum lower bound 8; all 237 singleton minima, 27,966 pair minima and 49,898 connected-triple minima are zero. This includes 1,659 singleton, 1,370,334 pair and 17,115,014 connected-triple assignments. A separate graph enumeration classifies all 2,190,670 triple subsets; the proof handles disconnected shift assignments by their nonnegative components.
• Forty synthetic graphs give 13,720 literal-objective comparisons. Tariff and unjustified-clipping controls pass. Thirty-two actual-objective fixtures include shifts of magnitude 2^32. The verifier imports no predecessor executable, uses exact integers and generates hash-bound stdout receipts.
• Local CPython 3.12.15/macOS arm64: warning-strict normal and optimized direct checks pass, about 4.4 seconds each. Both registered commands pass, with identical complete receipts. Each has a 120-second timeout, well below the CI bound.
• Committed bytes, executable modes and declared source ancestry pass at 69ad888ce9dd15f7fe68ad5b4d2f3a43bda7533a. Official export and both declared-input checks pass. Normal/optimized replay in the read-only Git-free export with empty PATH passes in 5.086/5.160 seconds. All nine export files retain their bytes and modes; full receipts agree. Deliberately corrupted source bytes are rejected before scientific work and restored afterward.

From the repository root, using the exact local interpreter path; another CPython 3.12 executable can replace it:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/k8-unequal-triple-closure/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/k8-unequal-triple-closure/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/k8-unequal-triple-closure
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/k8-unequal-triple-closure --check-source-history --check-modes

Expected markers are PASS_K8_UNEQUAL_TRIPLE_CLOSURE, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Replays need only the standard library. SOURCES.json binds the original archive and final gauge vector to exact blobs at canonical base a5bff3bf133e482541c09eba4e773884bdb2b1b7.

Limitations and unrun obligations

The result fixes the historical PR1322 scalar graph, helper eligibility, constant integer gauges and scalar lowerer. It does not establish global optimality or cover four changes, temporal gauges, changed coefficients or another word. The current selected P104 supplier is different. No circuit, paid recursive profile or supplier budget is improved.

The predecessor tariff/conjugation contract remains a premise. Native/owner integration, complete precision, stock, tape, compiler and outer obligations remain outside scope. Selected scientific replay, predecessor scientific suites, Lean and Docker were not run. Independent human and formal proof-kernel review are unclaimed. The five selected framework assumptions remain.

Credit Philipp / mightyOil's PR1322/1310/1263, Hanson Sun / bbMisk's PR1214, woklerx's PR1161, Sagar Thakrar's copy-init lowerer, Jeremy Kothe, Rohan Arun, Hao Yan / icekylinx, Douglas Colkitt and the retained supplier/native/frame/compiler lineage. Their source files and attribution are unchanged.

Related open work

• #3917 maps native/caller obligations of a four-end historical k8 frame assignment; this result fixes the scalar graph and changes no frame or caller.
• #1204 composes joint entrances, X54 and T37 owners on a different historical k8 word; no owner or kappa value is recomputed here.
• #1364 applies T37 packing to the older three-demotion word; this constant-gauge exclusion supplies no packing or supplier comparison.
• #2409 changes a production native auditor on the retained historical lineage; this verifier reads immutable archive data and imports no auditor.
• #3729 bounds integer idle packing for the historical supplier; this result fixes that recursive profile and studies its scalar gauge bill. No packing value is recomputed or compared.

Related open work was refreshed at closeout. No new account PR created after the advance CLAIM remains open, and no competing value for this unequal-triple objective was found in the checked open-work coverage. No extra predecessor computations were run for these comparisons.

Open next questions

Can four or more independently shifted helpers improve this final fixed-graph vector? Start with one explicit displacement and its complete boundary-inclusive F difference. No larger-block witness or global certificate is supplied, and this result does not justify an automatic historical supplier expansion.

CI notes

Hosted CI passed at 69ad888ce9dd15f7fe68ad5b4d2f3a43bda7533a: plan and replay (research/k8-unequal-triple-closure) both succeeded in run 40964867457. Confirmed once with gh pr checks 6225 --repo CrocSwap/integer-mult-bounds; the run head, completed status and successful conclusion were also verified. The hosted package replay certifies execution of the registered normal/optimized finite checks.

The maintainer merged the PR on 2026-10-30. Local direct/optimized, registered, committed-byte/source-history and read-only export checks also pass. The merge and hosted replay do not discharge the inherited native, compiler, precision/tape or full-theorem obligations stated above.

Merge Danger

Door: two-way. Blast radius: isolated research package. Shared code, frozen sources and the selected frontier are unchanged.