Research: sharp nine-coordinate threshold for five-subset shared nodes (seed 48)

[ToikY]

Change and mathematical scope

A doubly degenerate shared node with constant-weight-five labels first admits the historical orthonormal-residual interface at nine coordinates. The proof excludes h<8 using PR920's floor and excludes h=8 by a complement isometry to the triple class in PR1139. An explicit h9 witness provides all four orthonormal residual bases, with dimensions 3,3,4,4.

Status: structural research note; no new kappa or paid supplier. Only research/five-subset-shared-node-seed48/ is added. Global label coverage, simultaneous frames, source access, helpers, copies/restoration, recursive pricing, hosting and analytic/precision obligations remain open.

Task and overlap

Seed48 (fresh OS draw), structural decoder track. Exact base: ce7ad1b. Scope key: five-subset / doubly-degenerate / signed-rank-one / disjoint-spans / h8-h9 / orthonormal-residuals. CLAIM.json records three candidate tasks, the local claim, complete domain and stopping rule.

Nearest predecessor: @woklerx's #1139, pinned at 82da1e8, proves the triple-only ten-coordinate classification. This changes the label alphabet and adds a sharp weight-five threshold and exact census. It does not lower PR920's unrestricted norm-one h8 floor. PR920 (@ArnavSwiggy), historical head 9138174, supplies the residual criterion and four-plus-four bound. No pending implementation is imported.

Open PR metadata refreshed through #1171, with no matching five-subset package. CONTRIBUTING/autoresearch guidance changed during the sprint; the new discussion board was checked before submission (only its three starter threads, no overlapping claims). The explicit seed48 assignment and fixed base are retained.

Validation

Python3.9.6, standard library, offline:

python3 -I -B research/five-subset-shared-node-seed48/verify.py
python3 -I -B research/five-subset-shared-node-seed48/independent.py

Both pass normally and with -O. The independent labeled set census agrees on 0 rectangles at h8 and 453,600 at h9. The canonical frame enumeration tests all 59,520 dimension-four candidates; every canonical rectangle has exactly 48 legal frames. Both implementations independently check the witness residual spans and Gram entries.

Six mathematical controls, all 65,536 h8 isometry vector pairs, all 3,136 complement-kernel pairs, optimized relocation, six malformed-certificate rejection exits, source pins, selected-record guard and staged whitespace check pass. Final read-only review caught and verified the fix for a missing-basis cardinality check. The written threshold proof and finite census are distinct evidence.

No repository-wide or selected cold supplier replay was run for this isolated historical geometry package. No hosted CI success or formal multiplication theorem proof is claimed. Selected frontier and upstream/ remain unchanged.

Credit

Credit Douglas Colkitt's historical parity kernel/obstruction, @ArnavSwiggy's #920, @woklerx's #1139 and the seed47 frame-completion note. Exact sources are pinned in SOURCES.json. Prepared for @ToikY with substantial OpenAI Codex assistance in mathematics, implementation, independent algorithms and review. Apache-2.0. No independent human peer review claimed.

[Mister-Meeseeks]

100-PR batch review at 72bfdf66d9823244ee8dab4c8898af834e34853b.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Constant-weight-five shared nodes attain the sharp nine-coordinate threshold; an alphabet change from triples, not a lowering of the unrestricted eight-coordinate floor.

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.