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Research: prove a sparse P112 completion rank law through 52-port blocks - #1447
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100-PR batch review at Merged to Reviewed scope: P112 root-code minimum support 53 and correction rank theorem, with pinned hyperplane classification and independent exact premises. The unrestricted rank-seven sharp example changes constrained diagonal entries and is explicitly not a masked decoder improvement. 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. |
On the canonical P112 root lines, an odd-only rank-two correction on exactly 53 rows and two columns gives completion rank nine instead of ten, and no correction with at most two changed columns and at most 52 rows fails rank additivity. Orthogonal column pairs force rank ten; odd pairs admit nine or ten. Every nonzero odd-only rank-one correction gives rank nine within 52 rows and columns, so PR #1447's suggested 53-by-53 rank-one follow-up cannot occur. Exact certificate over all 112 columns and 6,216 pairs with an independent checker. No COPY saving or kappa. Adds research/p112-two-column-completion/ only. Submitted by @commdata2338 with substantial AI assistance (OpenAI Codex research and implementation; packaged with Claude Code).
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For an arbitrary complex correction Q supported on at most 52 P112 rows and 52 columns, rank(2G + E_AQE_B^t) = 8 + rank(Q). The proof derives root-code minimum support 53 under the retained E8 flat-classification premise. Exact matching/rectangle controls and a correctly labeled unmasked 53-support rank-drop example delimit the result. This bounds total rank-one factors, including free terms, rather than paid copy cost.
Validation: Python 3.12 registered replay with source pins and scoped controls; selected-result integrity or baseline bindings and diff checks pass. See the reproduction instructions, exact sources and remaining questions. No selected source, certificate or kappa changes; no full selected circuit replay is claimed. Prepared by woklerx with substantial OpenAI Codex assistance, with predecessor attribution retained.