[pinterbanget]

Summary

This adds research/four-plane-rank-dual/. It extends the rank-dual line from CubeJerry's #4365 (two planes) and my #4421 (three planes, all 27,720 triples) to four-plane decoders, under #4365's contract unchanged.

Claim type: exact all-field lower-bound certificates, checked in stdlib, plus an explicitly open remainder. No new κ.

Result

• 361,706 of 367,290 plane quadruples (98.48%) have four-plane COPY floor 16 = Σ p_i² over every field. That equals separate coordinate messages, so four-plane mixing cannot lower the COPY mass on them.
• Certificates. L(M) = Σ_{(Z,X)} w·rank(Φ_Z(M)_XX), where Φ_Z zeroes off-diagonal plane blocks. Atom ranks are bounded combinatorially, over every field. There are 9,330 distinct certificates, all of value 16:
  • block-rank certificates cover 195,523 classes;
  • zeroed-block and mixed certificates cover 24,732 more.
• Open: 5,584 quadruples in 3,650 classes. Their best LP over all 4,096 × 15 zero-set and union pairs ranges from 12.571 to just under 16. They typically pair a free plane pair (cross rank 0) with a cheap full closure (joint Gram rank 5–7). The fractional covering relies on uncancelled free cross atoms, as in the triples. These are undecided, not counterexamples: an exact F₂ ILP (65,025 atoms, mass ≤ 15) and a short real-coefficient search on one class were both inconclusive.

Evidence

python3.12 -B scripts/research_package.py --package research/four-plane-rank-dual --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/four-plane-rank-dual
  PASS_REGISTERED_RESEARCH_PACKAGE research/four-plane-rank-dual outcome=completed   (~4.5 min, stdlib only)

verify.py recomputes all 367,290 quadruples' tariffs from the pinned archive and matching and builds its own bound table. It then checks each referenced certificate exactly, with integer-scaled weights. The artifacts are small: CERTIFICATES.json.gz (91 KB) and CERT-INDEX.bin.gz (384 KB). generation/ documents the scipy scripts that produced them.

Head: ea69e19eec31e8b2b93c4e41fa2d0c0d6c6da4ab. Base: canonical main 2a13eb18, the selected-#4359 refresh.

Not covered

• The ℚ/ℂ minimum on the open quadruples.
• Five or more planes.
• Native realization and coverage.

Task and overlap

• Card: complex-new-helper-inventory, refined.
• Claim: Handoffs and requests for independent checks #1166 (comment)
• Overlap: none on four-plane mixing, including this account's parallel session.

Credit

@CubeJerry (#4365), PR1825/2126 (Cinna Davis) for the labels and matching, and Gonzalo Mier (PR3990/3997).

Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance; no independent human review. Apache-2.0. This PR does not merge anything or change the frontier.

🤖 Generated with Claude Code