FINDING / HANDOFF: no split-key helper-donor kernel on the T74 566+554 base word (108 deletions, no new kappa)

[ianlibanio]

Status: FINDING / HANDOFF, 2026-10-14 21:00 UTC, priority revision 3. No κ is claimed.

Public artifact: PR2142, package head 0c5516bf5bfb5c9c4d09446d3b21028032060fec, research/t74-split-key-kernel-search/. It is stacked on #1957 and carries #1957's commit unchanged.

Question. In the PR1673-outer claims (#1928 and predecessors), κ ≈ the T74 supplier's saving. So a third helper-donor kernel, like #1625 (pivot 566) and #1672 (pivot 554), would be a real κ gain of about +4.1e-8. #1957 found none, but with two limits:

• compensating reads had to hold a multiple of the deleted read's source value;
• only 327 of the 3,451 A-section gates were priced.

What I did on the base gauge44-word-566 word:

1. Finished the A-section deletion relaxation: 3,124 new gates, all solved. That gives 32 cold-pivot pre-cancels at −6,763 (A1215–A1246; 27 new) and 48 accumulation reads at −8,857 (A3403–A3450; 40 new).
2. Searched a wider "split-key" compensation family. Inserted reads may use any value held by or added into the deleted read's source or target register. The rank-one identity then splits into one column equation per basis vector of that span.
   • Values and columns come from the deleted word.
   • Interacting inserts are banned.
   • 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 frame-cost oracle prices candidates, and every solution is re-priced exactly with abkernel.price.

Result.

• All 108 kernel-sized deletions in A and B (gain −3,050 to −8,857): 104 are proven infeasible by CP-SAT and 4 are shape-infeasible. There are no 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, the same search rediscovers k554-signed-001 exactly (−1,850).
• B783 and B802 are blocked by algebra, not pricing. With the per-insert cap at 3× budget and no summed budget, the deleted read itself is the unique exact compensation within 12 inserts.

Scope. Finite searches on one fixed word, first order only: one level of keys, at most 12 inserts, |λ| ≤ 4 in half steps. This is not a supplier lower bound.

Handoffs (open, unclaimed):

1. Extra word (extra-word-566, 417 of 666 calls): rerun the relaxation and the B783/B802/A-pre-cancel searches. This needs only compute, about 20 minutes on 11 cores plus searches.
2. Interacting inserts: without the ban, the model proposed zero-oracle-cost B783 compensations that failed exactness only because of insert interaction. A residual-repair second round could test that family.
3. The four −6,100 B reads (B822/840/858/876) are shape-infeasible with one-level keys. A recursive key family might reach their missing output rows.

Thanks to Rohan Arun (#1957) for the toolkit and data, and to mdumitrean (#1625) and jeremykothe2 / LeverRank (#1672) for the kernels.

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

[mdumitrean]

FINDING (taking this handoff's items 1–3) · 2026-10-15 01:46Z · priority revision 4 (main 0fdd7d82). No κ is claimed. Author-checked local replays; not maintainer-reviewed.

Public artifact: fork branch t74-local-word-round3 @ 3d1daf37, folder research/t74-local-word-round3/. Start with its README; each search has a REPORT.md, evidence JSON and scripts. Not a PR.

Same #1672 words (76c01583 / e86142b3) and the same premise as your search: B, tail and prepend edits only; A, the cut and the compensation fixed; the #1186 frame model.

1. Extra word (item 2). It is the base word's program up to the frames of B397–B408 and the start ranks of 627/629/630/631. All 1,797 B and 3,451 A deletion relaxations match the base word, and all 108 of your models rebuild identically (A-extra/scripts/diffcheck.py, about 4 s per job). So your 104 infeasibility proofs and 4 shape results carry over. The one extra-only structure, readers 624/625/626/628 of those four registers, has no kernel: 92 relocations are neutral or infeasible, and 5 multi-deletion CP-SAT runs are infeasible.
2. Interacting inserts (item 1). A residual-repair second round on 25 deletion sets. The B783/B802 cores you saw cannot be repaired within 12 inserts. The −4,925 early reads (B1/B3/B8/B39/B44) do have 16 map-exact second-order words at oracle cost 0, but all of them fail exact frames on cone caps. With a cone-cap-aware filter, every repair is proven infeasible.
3. Recursive keys (item 3). Wider span-based holders make the four −6,100 reads (B822/B840/B858/B876) shape-feasible, with about 33.6k candidates each. Round 1 is then proven infeasible for all four. A depth-2 key run did not finish.

Also negative on the same premise: the #1186 toolkit's fusions, read-set exchanges and triangle hybrids (6,081 candidates), and compensated non-commuting re-timings, port re-timings of the 13 never-written copy-window pivots, merges and cleanup moves (9,705). For those 13 pivots, 10,821 units per call of each deletion screen is provably unreachable by any B, tail or prepend edit.

Next. Every negative so far shares the premise that A, the cut and the compensation stay fixed. The copy-window bound leaves A-section re-routing as the remaining lever. Handoffs are under Open next questions in the README.

Prepared by Marius-Paul Dumitrean with substantial AI assistance (Anthropic Claude, via Claude Code).

[mdumitrean]

FINDING, rounds 4–7 (follow-up to my round-3 reply above) · 2026-10-15 12:18Z · priority revision 4 (main 0fdd7d82). No κ is claimed. Author-checked local replays; not maintainer-reviewed.

Public artifact: fork branch t74-local-word-rounds4-7 @ 23b9ce54, folder research/t74-local-word-rounds4-7/. Start with its README, which has a per-search REPORT.md, evidence and scripts. Not a PR.

Correction relevant to this thread (oracle artifact). #1957's per-insert cost oracle returns 2^60 for some cone-valid inserts, because it measures against the deleted word's re-solved frames. That is what made B822/B840/B858/B876 "shape-infeasible" in #2142. With those sentinels re-priced exactly (14–15 legal inserts per target), all four are proven infeasible: CP-SAT on growing row restrictions, with HiGHS confirming. So #2142's conclusion holds, on a correct basis. Models that prune by this oracle's cost should treat 2^60 as unpriced.

Rounds 4–7 in brief (same #1672 words; details in the README):

• Beyond round 3's premise: A-section re-routing, neutral walks, port dirt, frame slack, cut and precondition, compensation-aware and depth-2 kernels, 3–4 repair rounds, A-section inserts, large-deletion and block censuses, peephole rewrites of A and B, width swaps and all 45 extra-word interfaces. No profitable edit. The superadditive port-delivery pairs (e.g. B1407–1408 jointly −49,210) are settled infeasible apart from 24 neutral compensations.
• Width-9 narrowing: a valid narrowed owner map exists (PR1319 checks pass, about +8.4e-8 κ per helper), but no construction zeroes the helper's entrance response once narrowed. The generator chain is reproduced byte-for-byte; the factory admits no such helper.
• Hosting and outer: the T74 fill is optimal, and idle is already maximally concentrated. κ ≤ q is family-specific, but the complex leaf cannot be done by the bit primitive, so κ < C holds for this algorithm family.
• Auditor gap found on the way: audit_local.py does not frame-check non-centre cut gates (the native chain catches it). An additive guard is in PR Verifier: audit_local.py accepts non-centre cut gates; additive guard and regression fixture (no new κ) #2478.

What remains needs new constructions: a new factory packet (most downstream searches are not vendored, about 1–2 weeks) or a new parent algorithm (at most +6.6e-7 κ).

Prepared by Marius-Paul Dumitrean with substantial AI assistance (Anthropic Claude, via Claude Code).