Research: complete Discussion #2278's sampled four-port, ten-term P111 plane-upgrade case

[commdata2338]

Change and mathematical scope

The common-r P111 plane-upgrade screen in discussion #2278 leaves its four-port, ten-two-sided-term case checked on only 4,965 of 49,648 effective punctures. This package completes that domain with an independent necessary-subproblem proof: none of the 49,648 compatibility graphs contains eight cheap pair terms. Ten two-sided label terms therefore save at most 17 against cost 80 and cost at least 63 under the rational pairing tariff.

With the predecessor's explicitly retained four-unit residual-charge lemma, total decoder cost is at least 67. A four-port improvement needs at most 66, so this closes that computational branch of the conditional screen. The residual-charge lemma itself is not independently proved here, and this is not a universal arbitrary-support decoder barrier.

The package also supplies short certificates for the line-reader and target premises after any four live-line deletions: five disjoint rational bases for each of the 511 nonzero label functionals, and five disjoint rank-81 masked-equation blocks. It reconstructs the selected E8 geometry and checks all certificates without importing predecessor code. Only research/bit-plane-fourset-bound/ is added. No selected result, supplier, frozen source or κ changes.

Task and overlap

Contribution commit: this PR's head. Base main: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4. Main, selected result and guidance were unchanged at the final refresh on 2026-10-15 at 16:30 UTC. The claim is in discussion #2278.

Exact predecessor: Philippe Van Velzen's public fork, 0a60443cf4fad2ff65bff317996c1dce02a4a09f, research/complex-budget-law/bit/plane-step1/. Its current branch head 2f71bc21dea63204403aefcb83323b6560e70fe4 has no change to that subtree. The selected word is PR #589 head a5afc8c6169ea61ba7c51dfd3cade02cc7b13b46 on #555, bound by Git blob 85a4499bf9e4911a705d50567a2c2ef6cf23739a and SHA256 e3e301e62bc2b41dbf1ceac0acf3d35824ebeab1848c5a2142d4577b109da52f. Six predecessor definition/coverage files are retained byte for byte with exact pins; they are not executed.

Task card:

• Task / scope key: bit-helper-contract / 3475f46c8358c6a9211b; concrete scope class-U/P111/four-plane-ports/ten-two-sided-terms of Discussion FINDING / HANDOFF: the complex budget is a* ≈ ν/Λ — a lever map for step changes #2278.
• Base and dependency heads: those above; no pending physical supplier is admitted.
• Object and question: all four-subsets of the 111 selected lines that are pairwise nonorthogonal and perpendicular to an omitted common r; can the source's ten-term relaxation achieve savings 18 on P union S?
• Scope status: changed. The explicit open bit-decoder handoff replaces the generic helper-star interface example; the register-model alternative was passed over because the concrete handoff directly tests a possible escape from the bit bound.
• Changed restriction: complete the predecessor's 10% sample, retaining the support class and rational pairing tariff.
• Method/domain: exact integer reflection closure, rational rank certificates, F2 rank-one-complement reduction, and complete compatibility-graph search on all 49,648 effective punctures (65,150 common-r incidences). No search cutoff.
• Overlap: the named predecessor is the direct partial result. This account's PR #1421 reconstructed the selected rank ledger; this result instead concerns a changed decoder. PR #2423 and PR #2653 on it concern physical R438 rehosting. PR #2652 concerns PR Audit: PR442 hosting windows make the partial width-seven bit rows unreachable for replica-local hosting on the frozen recipe (no new kappa) #1736's mixed-width pending projectors. They do not share the decoder question.
• Optimistic completed budget: ten-term cost at least 63 plus the retained residual charge 4 gives 67; ΔC >= 4 > 32/9, so the partial-upgrade deficit cannot improve. No added physical charge is discarded to claim a construction.
• Success/falsification/cap: complete exact exclusion or an eight-pair witness; one bounded session, one CPU for replay, no GPU or third-party packages. A real nine-pair E8 dual system outside the puncture domain is the positive control.
• Artifact and credit: research/bit-plane-fourset-bound/; Philippe Van Velzen and the selected E8 lineage credited below. Reproduction is the registered package command below.

Final overlap queries: four-port, four-set, class-U, 49,648, plane-step1, preceded by --task bit-helper-contract and exact source-script/size queries. The read-only snapshot covers 812 open PRs including drafts (2026-10-15T16:22:35.669380+00:00) and 1,276 Discussion records (2026-10-15T16:27:10.714268+00:00), with complete pagination. Expected offline warnings remain: later changes, arbitrary code diffs and unindexed content can be missed. The relevant board thread and concurrent claims were compared; no duplicate scope was found.

Validation

Environment: CPython 3.12.3, Linux x86_64, standard library only. All commands below passed; git diff --check and a staged-file completeness check for all six retained predecessor files also passed.

python3 -B scripts/research_baseline.py
python3 -I -B scripts/research_package.py --package research/bit-plane-fourset-bound --check-only
python3 -I -B scripts/research_package.py --package research/bit-plane-fourset-bound

The baseline is source-binding/arithmetic orientation only. The registered replay reconstructs the complete domain and graph sequence, checks all 2,555 rational reader bases and five target blocks, verifies the full stored receipt, and runs optimized-mode controls. Expected full output:

PASS_BIT_PLANE_FOURSET_BOUND sets=49648 eight_pair_witnesses=0 pair_cap=7 savings_cap=17 conditional_cost_floor=67 no_new_kappa

The graph search visits 939,575 nodes, using only proper-coloring and cardinality upper bounds. Its result is checked against an independent brute-force subset oracle on all 1,024 labeled five-vertex graphs. A literal E8 nine-pair system is accepted and its exact outer products reconstruct the binary target. Six rejection controls cover corrupted source bytes, a singular form, wrong cross-equation types, duplicate readers, forbidden type-11 pairs and nonorthogonal self-pairs. The incumbent rational COPY tariff reproduces as nine sevens. Acceptance uses explicit exceptions, not removable assertions.

The case digest is 8a2a2786b8cfa4552403b927f4ab1bbb2c5305251a088028be5ceaf6a5b4e3e1; the graph digest is bf8537f519979894e4798155bb46564405f8a6e8e88a28a65ed2bce0150680bc. These identify regenerated finite evidence; they do not replace its replay.

Not checked: the predecessor's residual-charge lemma, other term-count branches, the fifteen exceptional three-sets, arbitrary line supports, physical frame/hosting/row-stock or compiler admission, precision/analytic transfer, full supplier cold replay, and formal proof-assistant checking. The result is author-checked with independent code, not independently human-reviewed.

Credit

Philippe Van Velzen (@philippeVV) supplied the common-r upgrade question, classes P/U, ten-term relaxation, prior 4,965-case sample and real dual-system positive control. The source files retain their original text and attribution. DreamingOfClouds supplied PR #589 and associated format/decoder work, on eumemic's PR #555, with DaysSky and the retained E8/hosting lineage. This account's prior selected ledger work is credited without claiming it as new. New material is Apache-2.0; source licensing is retained.

Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the derivation, the implementation, the checking and the writing; it was packaged and submitted with Claude Code. No independent human review is claimed.

Open next questions

The smallest next proof task is to derive the punctured-r class-U residual charge independently, including arbitrary S-supported terms; the total-cost corollary still depends on it. The fifteen exceptional three-sets are a distinct remaining branch. A construction with non-label line supports could evade this result, but must satisfy the actual port-space mask and rational cost. The package README and TASK.json give concrete first steps and blockers.

Workflow feedback (optional)

The explicit sampled-case handoff made a bounded central contribution possible. A proof-obligation table alongside the source's broad summary would make it easier to distinguish completed branches, sampled checks and inherited lemmas. The full four-set check becomes small after testing the necessary eight-pair subproblem first.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG