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Close the 24 coordinate-kernel span-six pools by a 3+1 restriction - #5552
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Mister-Meeseeks merged 1 commit intoOct 25, 2026
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Change and mathematical scope
PR #5231 leaves 24 coordinate-kernel pools unresolved for eight rank-one calls at address span six. This scoped written proof excludes that orbit: the literal projectors and every interleaved coordinate projector preserve a coordinate 3+1 decomposition. Relations of the triple subword inject into the full two-dimensional relation space. PR #720's disjoint-subset lemma bounds the triple's rank sum by its calls; the singleton has the complementary budget. Total rank is at most eight, contradicting three invertible 3-by-3 targets. Both singleton-data and singleton-auxiliary cases are covered.
The predecessor lemma is cited, not independently reproved. Combined span-six closure additionally depends on #4119, #5075 and #5231. Span five, additional auxiliaries, native costs and new kappa remain outside scope.
Task and overlap
User-selected #5231 handoff; refined bit-new-register-model scope, priority revision 7. Base: 13be6ac. Head: fb607e2. Bounded live PR text searches found no overlapping closure; full Discussion pagination and arbitrary diffs were not searched.
Validation
CPython 3.14.4 on macOS arm64: normal/optimized direct replay passes all 24 pools, 2,800 short formal words, seven inverse controls, model-boundary controls and a nine-call span-six positive control. The verifier authenticates three declared predecessor inputs. All five canonical source blob IDs were recomputed. Metadata validate(check_pins=False) passes in a local Git fixture. Seven published blobs match checked local bytes.
Full canonical-history metadata/source-history replay and hosted Python 3.12 replay were not run locally; shell GitHub DNS is unavailable. Hosted replay is registered. No independent human or proof-kernel review is claimed.
default reproduction from repository root:
a) python3 -I -B research/coordinate-kernel-span-six-closure/verify.py
b) python3 -I -O -B research/coordinate-kernel-span-six-closure/verify.py
c) python3.12 -B scripts/research_package.py --package research/coordinate-kernel-span-six-closure
Credit
Credit woklerx, unpaidmoon, Nick Petalas, David Govea, JoyboyBrian, Ali Romman, eumemic and huxint. Prepared for Nicholas Stone with substantial OpenAI Codex assistance; Apache-2.0. Only one seven-file research package is added.
Open next questions
TASK.json and PROOF.md retain span five as the next question: its relation rank is three, beyond the inherited lemma. No further search or selected-result edit is included.