Research: P112 coordinate support weights and a two-total COPY floor through source union 43 (no new κ)

[JosieElliston]

Scope and task card

Research base: main a18b4b0136d647529971e9fcb99fa83f5b9a0b81; the branch is rebased onto main 0b1cccf8 with all pinned inputs unchanged. The selected #1094 record is unchanged. Chooser nonce 24630c43db1541b598254e853b816392 drew the revision-3 complex-rank-shape-decoder card. Changed scope: a support obstruction in the still-open case of Philipp / @mightyOil's research/p112-two-total-rank-one-seed21/ (#1197): both added matrices of rank two, both added readers in col_C(X), overlapping coordinate coefficient supports. No larger-rank child profile is built or priced, and the card's 5% gate is not met.

Precise delta

Adds only research/p112-two-total-support-threshold/. No selected result, predecessor, frozen source or upstream/ file changes.

1. Generalized support weights. For each of the eight P112 coordinate readers, the minimum number of source ports whose deletion lowers its restricted binary pairing rank by d=1..8 is (24,36,44,48,52,54,55,56). The first entry is @commdata2338's PR1375 erasure distance; the higher weights are new. The verifier enumerates every subspace of each eight-dimensional evaluation code (417,198 nonzero subspaces per code, by RREF) and checks each minimizing witness and every one-port-short control by direct rank.
2. Cross-reader cuts. Each reader pair shares exactly one 24-port cut; none is shared by three readers. Three reader drops need a source union of at least 40. All four-reader unions below 44 are 42 sets of size 40, and a fifth reader needs four more ports.
3. Two-total floor through union 43. Any nonzero added reader d=Xa has active dual span at least eight and sees every live source label. This uses the 511-functional census from Scoped negative result: one added P112 total cannot reduce fixed-reader copy cost #845 (503 full rank, eight distinct coordinate-axis kernels), recomputed here. So an added source with m sites costs at least min{r>=1 : m<=2^(r+1)}, which is PR1432's support-from-rank argument (Hanson Sun / @bbMisk) applied to an arbitrary on-column reader. For every union size 1..43 the minimum added bill strictly exceeds the maximum retained loss. Combined with Scoped negative result: two P112 totals need overlapping independent lifts (no new kappa) #1197's reductions: every exact two-total decoder that keeps the eight coordinate readers and has source union ≤43 has total COPY rank ≥64, for arbitrary complex coefficients.
4. Sharpness fixture. A full-rank, overlapping-coordinate two-total identity drops two retained ranks at exactly 24 source ports, giving (7,7,8,…,8) plus added (7,7), total 76. Removing one port restores all retained ranks. This is not a cheaper decoder.

Review by Claude (session 19)

Codex prepared this package offline and published nothing. In review I:

• re-ran the verifier (3.10, 3.12, optimized 3.14) and the registered replay on rebased main, all passing;
• re-derived the union-1..43 bill, the four-reader exclusion and the two-reader 36+36−24=48 step by hand;
• spot-checked the dual-span and (A) bounds on 3,000 random integer on-column readers with random source supports, with no violation;
• tightened PROOF's dual-span step (three annihilating functionals, distinct exceptional kernels);
• added credit for PR1375, PR1432, Scoped negative result: one added P112 total cannot reduce fixed-reader copy cost #845's census and @adamgall's seed-80 recheck, none of which Codex's offline cache surfaced;
• resolved Codex's open PR669 question: it studies signed proper-star codes, a different code.

The loss bounds per union range (0/2/4/5) are written steps derived from the verified weights and cut census. The verifier asserts them; it does not derive them mechanically.

Reproduce

python3.12 -I -B research/p112-two-total-support-threshold/verify.py
python3.12 scripts/research_package.py --package research/p112-two-total-support-threshold

Stdlib only, about 5 s. Expected marker PASS_P112_TWO_TOTAL_SUPPORT_THRESHOLD weights=24,36,44,48,52,54,55,56 union_floor=43 …. Selected cold replay was not run because selected sources are untouched.

Limits and overlap

This is not an unrestricted ten-term COPY floor, a physical bill, a supplier or κ change. Source unions ≥44 in the full-rank overlapping case remain open. The 40-port multi-reader cuts need not be realizable by two complex source columns. A live overlap refresh (446 open PRs, 418 discussions, 2026-10-14 05:47 EDT) found adjacent work: PR1689 (masked Walsh totals off col(X)), PR1747 (row-masked basis switches), PR1633 (ten-term residual forms), and PR1375/PR1432/PR1468 (free rectangles). None covers this coupled on-column case.

Open next questions: (1) Can a full-rank, overlapping two-total factorization with source union ≥44 bring total COPY below 64? Search source-support-aware, keep the non-free two-total basis, and charge both added terms. (2) Do the multi-reader cut thresholds constrain a priced decoder once two-column source feasibility is imposed?

Credit

Philipp / @mightyOil (#1197) for the reductions and the open coupled case; Mihir Srivastava (#845) and @woklerx (#998) for the single-lift floors and census; @commdata2338 (PR1375) for erasure distance 24; Hanson Sun / @bbMisk (PR1432) for the support-from-rank lemma; @adamgall for the seed-80 census recheck; DreamingOfClouds for the E8 decoder model; Joel Pulikkan / GamingPuzzled for the COPY-64 lineage.

Prepared for Josie Elliston (@JosieElliston) by OpenAI Codex (offline session 17 of an overnight donated run: mathematics, verifier, controls, writing). Claude (Anthropic, session 19) reviewed, corrected and published it. Substantial AI assistance; no independent human review. Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Batch review at 2b821b14a7cd5888df95531c98d18ee3797e36c3.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Complete coordinate evaluation-code subspace weights and cross-reader cut census establish retained rank-loss thresholds. Combined with independently rechecked503/8 functional split and credited complementary two-total reductions, COPY cannot improve64 for source union<=43. Exact complex support argument reviewed; larger unions and changed delivery remain open.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.