Research: split-key kernel search on the T74 pivot-566+554 base word; 108 kernel-sized deletions proven infeasible (no new kappa)

[ianlibanio]

Research: split-key kernel search on the T74 pivot-566+554 base word (no new κ)

Stacked on #1957. This branch carries #1957's commit unchanged, with Rohan Arun's authorship, because the tools and words come from it. Review only the second commit: research/t74-split-key-kernel-search/.

Why this question

In the open κ claims built on PR1673's capped outer (#1928 and predecessors), κ sits within 1e-10010 of the T74 complex supplier's own saving. So only a cheaper T74 local word can raise it. The last two real gains were single helper-donor kernels: #1625 at pivot 566 and #1672 at pivot 554.

• Each kernel deletes a read whose removal saves a rank-3 split (−6,763 units).
• It restores exactness with inserts that cost one rank-2 split (+4,913).
• Net: −1,850 units per call, about +4.1e-8 in κ.

#1957 found no third kernel. Two things limited its coverage:

• its compensation family only reads registers holding a multiple of the deleted read's source value;
• it priced only 327 of the 3,451 A-section gates.

What this adds

1. The rest of the A-section deletion relaxation. It prices the other 3,124 gates; all 3,451 now solve exactly.
   • 32 cold-pivot pre-cancels at −6,763 (A1215–A1246); 27 are new, since Research: local-move exhaustion on the pivot-566+554 T74 supplier (0 gains in 4,461 exact moves, 107 compensation searches) and idle-optimal 249/417 dispatch (no new kappa) #1957 searched 5.
   • 48 accumulation reads at −8,857 (A3403–A3450); 40 are new.
2. A split-key compensation search. tools/skkernel.py generalises Research: local-move exhaustion on the pivot-566+554 T74 supplier (0 gains in 4,461 exact moves, 107 compensation searches) and idle-optimal 249/417 dispatch (no new kappa) #1957's abkernel.py.
   • Inserted reads may use any value held by, or added into, the deleted read's source or target register.
   • The identity Σ λ_j col_j ⊗ key_j = d ⊗ v splits into one column equation per basis vector of the keys' span.
   • Values and columns come from the deleted word. Interacting inserts are banned: an insert that writes x may not precede an insert that reads x.
   • Every solution is re-priced exactly with Research: local-move exhaustion on the pivot-566+554 T74 supplier (0 gains in 4,461 exact moves, 107 compensation searches) and idle-optimal 249/417 dispatch (no new kappa) #1957's abkernel.price.

Results (RESULTS.json)

• 108 strict searches cover every kernel-sized deletion (gain −3,050 to −8,857) in A and B: 32+48 in A; 2+4+4+2+16 in B.
  • 104 are proven infeasible by CP-SAT.
  • 4 are shape-infeasible (the −6,100 B reads).
  • 0 exact profitable kernels. The only exact solutions are delta-0 relocations of the deleted read.
• Positive control. On Conditional κ = 0.001676938111022438: pivot 554 helper-donor kernel (k554-signed-001) stacked on the previous record #1672's pivot-566-only word, deleting B342 rediscovers k554-signed-001 exactly: the same five inserts, delta −1,850. Flipping one coefficient makes the exact pricer reject it as inexact.
• Diagnostic. For B783 and B802, the per-insert cap was loosened to 3× budget with no summed budget. The only exact compensation within 12 inserts is re-inserting the deleted read. So for these two reads the obstruction is algebraic, not an oracle-pricing artifact.

Scope and limitations

• Base word only: gauge44-word-566, 249 of the 666 calls. The extra word was not searched.
• First-order model. Interacting inserts are not covered. Without the ban, the model proposed zero-oracle-cost B783 compensations that were inexact only because of interaction.
• Bounded family:
  • one level of keys;
  • at most 12 inserts;
  • |λ| ≤ 4 in half steps;
  • no multi-deletion kernels;
  • data butterflies and delivery reads are out of scope.
• Finite searches on fixed words, not a supplier lower bound.
• Task card. selected-packing, with the scope changed: the object is the pending T74 supplier rather than the selected k8 record. The reason is in TASK.json. No packing gain was found, so no ceiling.ceiling() fraction is reported.

Verification

Base 5e5672c1 (main) + #1957 head e6a4d348 (cherry-picked as ae8d2de0). Package head 0c5516bf.

python3.12 scripts/research_package.py --package research/t74-split-key-kernel-search
# PASS_REGISTERED_RESEARCH_PACKAGE research/t74-split-key-kernel-search outcome=negative-result

That registered replay is standard library only. It checks:

• every package file;
• the 5 repository inputs of the frame solver;
• all 62 Research: local-move exhaustion on the pivot-566+554 T74 supplier (0 gains in 4,461 exact moves, 107 compensation searches) and idle-optimal 249/417 dispatch (no new kappa) #1957 files against Research: local-move exhaustion on the pivot-566+554 T74 supplier (0 gains in 4,461 exact moves, 107 compensation searches) and idle-optimal 249/417 dispatch (no new kappa) #1957's manifest;
• RESULTS.json, recomputed from data/.

The solver stages need NumPy and OR-Tools, so they are documented rather than registered. They were run locally (Linux x86-64, Python 3.14, numpy 2.5.4, ortools 9.15) and all passed:

• verify.py --stage relax: 12 sampled A deletions re-solved exactly.
• verify.py --stage control: rediscovers k554 and rejects the tampered copy.
• verify.py --stage negative: A1215 and B783 re-proved infeasible from scratch.

python3 -B -m unittest discover -s tests -p 'test_research_*.py': 53 tests OK.

Overlap was refreshed before submission. The nearest prior work is #1337 and discussion #1314: kernel censuses on the older #1297/#1305 words in a restricted family.

Open next questions (structured in TASK.json)

1. Interacting inserts: add a residual-repair second round.
2. The extra word: about 20 minutes of relaxation on 11 cores, plus the searches.
3. A recursive key family for the four shape-infeasible −6,100 reads.

Credit

• Rohan Arun (Research: local-move exhaustion on the pivot-566+554 T74 supplier (0 gains in 4,461 exact moves, 107 compensation searches) and idle-optimal 249/417 dispatch (no new kappa) #1957): toolkit, B relaxation data and the A sample, all reused unchanged.
• Tooling: exact frame solver and delivery-rewrite search for gauge44/k8 local words (no new κ) #1186: exact frame toolkit.
• mdumitrean (Conditional κ = 0.001676896669021299: pivot-566 helper-donor kernel on PR1319's T74 exact fill (+4.14e-8) #1625): pivot-566 kernel.
• jeremykothe2 / LeverRank (Conditional κ = 0.001676938111022438: pivot 554 helper-donor kernel (k554-signed-001) stacked on the previous record #1672): pivot-554 kernel, used as the positive control.
• Conditional kappa 0.001676855229071321: verified exact fill on 74 replicas #1319/Conditional kappa 0.001676854978651125: four composed helper-donor cold kernels #1297 lineage: T74 words.
• Vlado Vojdanovski (Conditional kappa = 0.001677160151714381 via capped phase-cell fusion #1673): the outer.

Prepared by Ian Libânio with substantial AI assistance (Claude Opus 5.5 via Claude Code).

Workflow feedback (optional). The largest friction was that #1957's exact-frame toolkit lives only in an unmerged PR, so this package has to stack on it. A small merged home for that toolkit would make follow-up kernel work on T74 words independent of open PRs.

🤖 Generated with Claude Code

[ianlibanio]

CI note: two failures are expected and come from main, not from this PR.

I merged current main (0fdd7d82) into this branch as 5f2f6084, so the shallow-history, certificate-regeneration and ledger fixes from main are included. Two checks still fail on main itself:

• verify (*, clones): make split-skip-check stops with ValueError: Source pin: scripts/audit_fused_block.py. That script changed on main in b1072bb2, but research/split-skip still pins the old hash.
• r420-two-sided-basis: research/r420-two-sided-basis/verify.py prints REJECT: source pin: certificates/selected-result.json. The selected record changed in the batch1099 integration, but the package still pins the old bytes.

I reproduced both locally on the merged branch. This PR doesn't touch scripts/, certificates/, research/split-skip/ or research/r420-two-sided-basis/; git diff origin/main HEAD is empty for those paths. I left the stale pins alone because re-pinning other packages' evidence is a maintainer decision.

This PR's own package passes its registered replay: PASS_REGISTERED_RESEARCH_PACKAGE research/t74-split-key-kernel-search outcome=negative-result.