[Gonzalo-Mier]

Change and mathematical scope

Screens the rank-4 child calls in both orientations of the pinned #1094 gauge44 local program. Fourteen target frames contain either four or seven calls; the verifier retimes only intervening scalar additions whose register references do not touch a later child input. It replaces four independent rank-4 products by a block-diagonal rank-16 product, and three by rank 12. The work is a zero-charge local screen, not a physical supplier.

Result

The exact source census groups 80 calls into 22 blocks (14 rank-16 and 8 rank-12), preserving all 320 units of rank mass. The standalone ideal ceiling gain is 2.0187%, below the task card's 5% screen. Composing this profile arithmetically with the pinned rank-2/rank-3 zero-charge profile yields a 6.6289% ideal ceiling gain, but still waives the physical input layout, routing, frames, hosting, precision and dirty restoration. No supplier or new kappa is claimed.

Validation

• python3 scripts/research_package.py --package research/complex-rank4-retimed-blocks --check-only
• python3.12 scripts/research_package.py --package research/complex-rank4-retimed-blocks
• python3 scripts/check_research_commit.py --package research/complex-rank4-retimed-blocks --check-source-history

The registered replay passed both the pinned predecessor's rank-2/rank-3 screen and this package's archive/member pins, both orientations, exact scalar retiming, symbolic block-diagonal identities, rank-mass and rational ceiling checks.

Task and overlap

Scientific base: canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision 6. A refreshed text scan at 2026-10-19 10:45 UTC covered 733 open PRs and 2,596 Discussion records. The closest work is selected-word rank-1 batching (#1779) and strict chronological rank-2/rank-3 blocks (#1960/#2014); this screen is limited to same-target-frame rank-4 groups with exact scalar-reference retiming. Text search is not a novelty certificate and does not inspect every code diff or unpublished work.

Credit

The selected gauge44 word and its source lineage retain their original authorship. The predecessor rank-2/rank-3 result retains its attribution in its package. This scoped rank-4 census and verifier were prepared by Gonzalo Mier with substantial OpenAI Codex assistance; no independent human review or formal proof assistant is claimed.

Open next questions

Can one concrete group (for example target frame 431) be emitted as an actual rank-16 child with its left/right operands, output owners, scratch lifetime and arbitrary-dirty restoration fully specified? Trace its child recipes through producer and consumer, write the block inputs/output map, and price every copy, frame conversion, hosted slot, precision term and restoration operation. The frozen event trace does not currently supply that physical ownership/lifetime contract.

[Clear1nsight]

FINDING / HANDOFF — source-bound joint-child rank interface

PR4045 tests one transfer of this screen to the pinned PR3945 staged544 providers: helpers 600, 601, 602 and 604 share four rank-4 climbs from span(45) to span(409,80,40,5,2).

The inherited native rank counts Fourier axes. Exact evaluation of the common residual on all 512 basis columns gives row/column support 16; one rank-16 child has support 65,536. Monomial pre/post maps preserve row-support multisets, so arbitrary monomial routing/wrappers and auxiliary coordinates returned identically on all dirty inputs cannot implement that one-call substitution. Selector batching retains active rank 4. This is a scoped transfer obstruction; the matrix identity and explicitly zero-charge scope of this PR remain credited.

The hypothetical allowance is only 4H4-H16 ≈ 0.03824944758 before additional costs. A useful next step needs an explicit nonmonomial or multi-query encoder, its actual consumer/dirty return, and a complete invoice or separately justified saving. Proof and obligation ledger. No new κ or full-provider replay is claimed.

Hosted CI passed at exact head91afbb643f7f4ccf6f34b685ecd959eb790a0f94: https://github.com/CrocSwap/integer-mult-bounds/actions/runs/39243988237.