Negative result: P112 fixed-reader two-total COPY at source union 48 — two-column loss bound exactly 5 (attained), full-rank lower bound 65, COPY >= 64 extends (PR2382 OQ1)

[thakrarsagar]

Summary

Status: Negative result (no new κ, supplier or decoder; the selected record is unchanged).

This closes band 48 of PR2382's open next question 1, using an independent stdlib re-implementation that imports no PR2382 code and uses exact Fraction and F2 arithmetic. The setting is the pinned P112 fixed-reader exact two-total decoder XX^t = XY^t + (XA)Z^t, with Y = X - ZA^t, A 8x2, Z 112x2 and deletion union |U| = 48.

• Geometric loss at 48 is 8, up from 5 at 44..47. One example is a four-dimensional 48-port support of reader 0 containing a cut, which gives loss vector (4,0,1,1,1,1,0,0). There are also seven 8-reader single-word unions.

• The two-column retained-loss bound is exactly 5. Every configuration with loss 6 or more contains a fully specified column subset whose exact rank is 3. A rank-two ZA^t cannot match such a subset. The completeness argument is a case split on the largest closed support:

  • size above 36: cut words only, by the 12-port margin;
  • size exactly 36: explicit enumeration;
  • size 32 or less: tuples of single words.

  The bound uses only rank(ZA^t) <= 2.

• An explicit decoder attains loss 5, at a bill of 74. It is a two-column decoder with rank Z = rank A = 2 and rank[X Z] = 10, in shape 4+1. The identity holds on all 12,544 pairs. Retained ranks are (4,7,8,8,8,8,8,8) and added ranks are (8,7).

  • PR2382's "at most 4" bound, which PR2382 states only through 47, does not extend to 48.
  • This representative (A = [e_0, e_1]) lies in PR1197's disjoint-coefficient class. The same W in an overlapping basis keeps loss 5, but that representative's added bill was not computed.

• Lower bound at 48 is 65. This is a provable lower bound in the full-rank on-column (PR1197 residual) case, 64 - 5 + 6, using the inherited price minimum of 6. Whether 65 is attained is open; the only witness costs 74.

  • Shape 4+1 costs at least 66. This is a written argument; its 12/12 cut sign-balance premise is machine-checked.
  • The shapes 3+1+1, 2+1+1+1 and 1^5 are not analysed this way.

• The "COPY at least 64" statement extends to union 48. The dependency chain is:

  • PR1787 (merged) and PR1827 for unions of at most 43;
  • this package's independent cap-47 replay for 44..47;
  • the new band-48 result;
  • PR1197's inherited reductions.

  It does not depend on the open PR2382.

• Positive control at cap 47. This reproduces PR2382 independently: geometric maximum 5, bound 4, floor 66 for M = 44..47, and single-word census 2032/280252/9072/42/0.

Base and pins

• Base: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2 (origin/main is unchanged at submission).
• Input: research/p112-single-lift/reference-geometry.json, sha256 4f44af368e046002553d632ee90c6470990827cea3fae41f0f54018522d7a080.
• Provenance pins: selected-result, the task catalogue and the PROOF.md files of PR845, PR1197, PR1827 and PR1787. All are listed in SOURCES.json with git blob and sha256.
• Package: research/p112-two-total-union48-seed30/. Only this directory is added; upstream/ and the frozen packages are untouched.

Reproduce

python3 -I -B research/p112-two-total-union48-seed30/verify.py

Expected final line (about 165 s on one core, peak RSS about 175 MB, CPython 3.14):

PASS_P112_TWO_TOTAL_UNION48 geometric_max_47=5 geometric_max_48=8 two_column_loss_bound_47=4 two_column_loss_bound_48=5 loss5_witness_bill=74 full_rank_copy_floor_48=65 no_new_kappa

RESULT.json is compared exactly. A metadata check is also available: python3 scripts/research_package.py --package research/p112-two-total-union48-seed30 --check-only reports PASS_PACKAGE_METADATA_AND_PINS.

Negative controls (must be rejected)

• Corrupted geometry. A one-byte change to the geometry is rejected by the sha256 pin.
• Perturbed minor. On rows {0,1,2,3,12,13,14,15} and columns {0,2,3}, replacing the third column by the sum of the first two drops the rank from 3 to 2.
• Three-column fake profile. A 4+1+1 correction reaches loss 6 but has rank W = 3, so it is not a two-column correction.
• Corrupted receipt. The same receipt_matches function used for the real comparison must reject a copy of the receipt with the floor field incremented.

Independent review checks:

• an end-to-end mutation of RESULT.json gives FAIL RESULT.json mismatch;
• an X sign flip, a label swap and a change to one witness Y entry are each rejected;
• separate GF(2) code reproduces the witness bill and the price minima (6, and 7 when both sources have at least 12 active labels).

Search record

• Domain: every retained-loss configuration of closed supports for the 8 P112 reader codes, with union of at most 47 and of at most 48. Readers, subsets and orderings are unrestricted, and no symmetry is assumed.
• Coverage: exhaustive under the lead-support case split above. At cap 48 this includes 699,132 single-word pairs and 7 eight-tuples.
• Stopping rule: stop when band 48 is closed. That happened: the bound of 5 is proven and attained.
• Not run: bands 49 and above.
• Seed: 30 selected the task only; the search itself is deterministic.

Scope, inherited assumptions, unresolved obligations

Inherited, not re-proved here:

• the separate-total COPY model (DreamingOfClouds lineage);
• the PR845/PR1432 price lemma r >= p(m);
• PR1197's rank-one, off-column and disjoint-coefficient reductions.

Open:

• whether 65 is attained at 48;
• whether 3+1+1, 2+1+1+1 and 1^5 are realizable (the rank-two condition is necessary, not sufficient);
• bands 49 and above. The census has no supports of size 37-39, so the s > 36 branch stays cut-only; the s = 36 branch widens to a budget of 13.

Other limits:

• Helper, access, frame, restoration and recursive costs are free in this screen.
• The all-size/compiler, hosting, precision, analytic-transfer and bit H1-H5 obligations are unchanged and out of scope.
• The finite enumeration plus the written argument is not a formal proof.

Overlap and credit

Overlap snapshot, about 15:13Z on 2026-10-15:

• PR2382 (Sricharan Vagvala @theon199, head 2d7807f6, open) stops at 47 and names band 48 as its next step. It is cited, and its ≤47 numbers are reproduced independently; its code is not imported.
• No other PR or discussion examines band 48.
• Adjacent open PRs Decoder floors are tight in the frontier format: one F2^9 label table, factors in col M of a rank-8 realisation (P112 64, E8 72, P104 (#968's unit) 56; two-call batches 128) — no κ claim #2338, Prove the sharp COPY cost of the fixed P112 two-column completion #2321, Prove sharp P112 COPY power for the right-space free-correction extension #2515, Research: a stronger P112 two-call COPY floor for arbitrary cross forms #2395 and Research: extra factors do not lower the fixed P112 COPY-power optimum #2398 study other P112 COPY questions (col-M factors, PR2021's fixed completion, two-call cross forms). They do not cover this band.

Predecessors:

• Philipp/mightyOil: PR1827 and PR1197;
• Josie Elliston: PR1787;
• Mihir Srivastava: PR845, including the geometry snapshot of DreamingOfClouds' PR717;
• woklerx: PR998;
• Hanson Sun/bbMisk: PR1432;
• commdata2338: PR1375;
• Joel Pulikkan/GamingPuzzled: PR620 (COPY-64 lineage).

Sprint: track 20-39 (complex supplier), seed 30. Claim: #1163 (comment)

Contributor: @thakrarsagar, with substantial AI assistance from Claude Opus 5.5 via Claude Code, run in an automated sprint loop. The mathematics, implementation, checks and writing were AI-assisted, and the reviews were AI-run. Apache-2.0.

🤖 Generated with Claude Code