[petalas]

Summary

Scoped proof with a finite exact certificate: an explicit eight-bit Boolean membership table has exact minimum maximum vertex degree eight over every finite tree tensor realization. The contract allows arbitrary internal coefficients and any number or width of Boolean wire bundles. Every wire counts toward degree, and each external bit attaches once.

all 127 table cuts have full rank
  -> cut capacities at a weighted tree centroid sum to eight
  -> some vertex has degree at least eight
one tensor attains eight

This supplies an original explicit obstruction for the acyclic table-splitting subquestion in Jacob Sussman's PR #5817. The general cut-rank and centroid methods are classical. The new Boolean witness and its application were produced by Nick Petalas. No predecessor result is audited or replayed as the deliverable.

Evidence

Before: the source leaves vector-table splitting as an open proposed route. After: TABLE.json specifies all 256 Boolean values, with 121 ones, and every exact tree realization retains a degree-eight vertex.

Local CPython 3.12.15 checks pass:

• Every unordered nontrivial cut, with side-size counts 8/28/56/35 and full ranks 2/4/8/16.
• 127 square minors, selected modulo 101 and rebuilt by a separate input-word scan, have exact integer determinants nonzero modulo 101.
• 2,785 weighted-tree cases, twelve rejection controls and an exact four-site cyclic boundary control.
• Warning-strict normal and optimized direct checks, about 0.061 seconds each; both registered commands pass within their 60-second limits.
• Metadata and source pins, committed bytes and source-history availability, official export integrity, and both read-only Git-free export modes pass. The eight exported files, including the export manifest, retain their bytes and modes. The verifier prints generated package-hash-bound JSON receipts.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/boolean-table-tree-degree/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/boolean-table-tree-degree/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/boolean-table-tree-degree
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/boolean-table-tree-degree --check-source-history

Another CPython 3.12 path is equivalent. Expected marker: PASS_BOOLEAN_TABLE_TREE_DEGREE, with 127 cuts, degree minimum 8, 2785 small weighted trees and 12 controls. All scientific inputs are package-local; the pending question-source commit is motivation only, not an executable or proof dependency. SOURCES.json records its exact README blob and SHA256.

Limitations

The tree implication is a written all-tree proof; the explicit table ranks are a finite certificate over characteristic zero and F_101. The example has eight axes and does not prove a family of Boolean examples at every larger dimension or an asymptotic degree-cap obstruction. A literal integer-valued cyclic control demonstrates that the tree hypothesis cannot simply be dropped. No general Orthogonal Vectors lower bound, cyclic-network exclusion, new kappa, physical supplier, recursive invoice change or selected-frontier update is claimed.

Independent human and formal proof-kernel review are unclaimed. Lean, Docker and the selected supplier's scientific cold replay were not run; no obligation to those checks is discharged by this package. The original formal-interface-lemma card is explicitly changed to this mathematical side question in TASK.json. Base is de5ec2e, priority revision 8.

Related open work

• Research: if the reported ETH disproof holds, no κ record moves; Alman–Li's depth-2 WHT barrier survives on OVC; what falls, what survives and what might open up (no new κ) #5817, Jacob Sussman, exact source be98f61: poses vector-table splitting for bounded-degree tensor contraction. This contribution settles a concrete acyclic replacement obstruction; it neither rechecks the source's conditional complexity map nor settles the complete proposed route. No common numerical invariant is compared.
• Prove the sharp fifth-root threshold for tensor phase recovery #5916 studies operator-norm phase recovery from tensor-square and tensor-cube comparisons. Its tensor-power quantity differs from the Boolean-wire tree-degree minimum here; no common-instance numerical comparison is claimed.

The completion refresh inspected this account's eight other open PRs created since the advance CLAIM: #5910 through #5916 and #5918. Their subjects are sector workspace, colored batching, rounded-shear history, serialized carry state, cyclic quotient size, carry frequency, tensor phase recovery and clockwise XOR synthesis. None studies this table's exact tree realization degree or reports a conflicting same-instance value. No additional computation was run to compare them.

Open next questions

• Can this particular table have an exact cyclic realization at maximum degree seven with a useful complete wire count? Begin with one fixed graph and check its entire 256-entry endpoint.
• Which explicitly structured membership families have low-rank tree cuts and small complete factorizations? Begin with one family and its exact consumer contract. No structural promise is supplied here.

CI notes

At exact head 6bce04acf6ffe10ba2b649c060c37701fb913db6, run 40506876386 finished with plan failed; replay skipped.

The only failed step was Onboarding helper tests. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22). The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight assertion failures concern frozen-card/current-route expectations, missing-reference handling and sparse omission lists. This matches the reported shared onboarding failure on main de5ec2e7a6074068452f45b1a191b5c72832df4a; the step, file, ten failure/error entries and final summary were directly inspected in this run. No package-specific failure was reported.

The package replay was skipped. Hosted replay has not certified this package; only the local scientific and publication checks listed above have run. This is not a green hosted workflow or mathematical acceptance. Maintainers will fix the shared failure. No onboarding test, shared source, workflow or package file was changed for this closeout.

Advance CLAIM. Credit Jacob Sussman for the question, I. V. Oseledets for classical tensor unfolding-rank background, and classical weighted tree centroids. The proof states the cut argument directly and depends on no unreviewed complexity-theoretic claim.

Merge danger

Door: reversible. Blast radius: research-only. Adds one isolated opt-in package; existing code, pinned evidence and the selected frontier retain their bytes.