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Tags:
graph-theory
spectral-graph-theory
mathematics
mathematical-certificates
research-progress
symbolic-computation
claim_id stringclasses 8
values | evidence stringclasses 7
values | limitation stringclasses 8
values | next_step stringclasses 8
values | stage stringclasses 5
values | statement stringclasses 8
values | status stringclasses 4
values |
|---|---|---|---|---|---|---|
base-vc3 | cjc0013/vc3-laplacian-matching-theorem@4494842b0bd0bdb7d935b68a153f8581a6c84a03 | This is a bounded vertex-cover theorem, not the unrestricted theorem. | Extend the structural method beyond the fixed-cover slice. | prior_dataset | The target inequality holds for connected simple graphs with vertex-cover number at most three. | proved |
canonical-hall-core | data/certificates/canonical_hall_deficient_core_decomposition_certificate.json | The decomposition does not by itself prove effective-core positivity. | Analyze the induced Schur complement. | v11-v12 | The current deficiency-one branch admits a canonical Hall-deficient-core decomposition. | proved |
rank-agnostic-schur | data/certificates/block_schur_dirichlet_to_neumann_certificate.json | The effective core still needs a positive supersolution. | Prove a dimension-free lower bound for the effective core. | v12 | The block Schur criterion is valid without a rank-one coupling assumption. | proved |
hall-residual-m-matrix | data/certificates/block_schur_dirichlet_to_neumann_certificate.json | Residual positivity alone does not settle the Schur complement. | Control the correction B D^(-1) B'. | v12 | The Hall residual block is a positive-definite nonsingular M-matrix in the dimension-free range. | proved |
rank-one-shortcut | data/certificates/core_residual_rank_one_compression_obstruction.json | The obstruction refutes only the shortcut, not the block criterion. | Keep the full coupling topology. | v12 | The core-residual interface can always be compressed to rank one. | refuted |
bounded-effective-core | 1854 source graphs; 14832 sign partitions; 5460 rank-at-least-two couplings; zero failures; minimum 17/9. | A finite exact frontier is not a dimension-free proof. | Extract an attachment-local universal inequality. | v13 | Every effective core in the exact order-eight frontier has positive unit image. | verified |
harmonic-unit-cap | 84 exact failures; maximum 20/17; an explicit small witness reaches 10/9. | This refutes a sufficient shortcut, not effective-core positivity. | Bound the complete Schur image rather than the harmonic vector alone. | v13 | The residual harmonic extension is always coordinatewise at most one. | refuted |
dimension-free-effective-core | Exact frontier support only; no dimension-free certificate yet. | This is the remaining obligation for the current branch. | Derive and falsify-test an attachment-local lower envelope for S 1. | next | Every admissible effective core satisfies S 1 >= 0. | open |
Laplacian Matching Theorem Frontier v13
Produced by the Ouroboros AI Research System, under human direction.
This is a progress-and-reproduction dataset for the next stage after cjc0013/vc3-laplacian-matching-theorem, pinned at revision 4494842b0bd0bdb7d935b68a153f8581a6c84a03. The earlier dataset proves:
For every connected simple graph G with tau(G) <= 3,
mu_1(G) <= Delta(G) + nu(G).
This new dataset does not claim the unrestricted theorem. It records how the research moved beyond the vertex-cover-three slice, what has been proved, what has only been verified on an exact finite frontier, which tempting shortcuts failed, and the one current dimension-free proof obligation.
What changed since the VC3 dataset
- The search moved from a fixed small vertex cover to the deficiency-one unmatched-vertex branch.
- The branch was reduced to a canonical Hall-deficient core and expressed as a block Schur-complement problem.
- A rank-agnostic Dirichlet-to-Neumann criterion was proved, replacing a false rank-one compression idea.
- Dimension-free positivity was proved for the Hall residual block, including isolated, cover-attached, single-neighbourhood, and arbitrary-neighbourhood attachment families.
- The exact order-eight frontier now verifies the stronger effective-core unit inequality on all 14,832 sign partitions, including 5,460 rank-at-least-two couplings, with no failures.
- The harmonic-extension shortcut z <= 1 was falsified in 84 exact cases. The largest exact value is 20/17, so that shortcut has been retired.
Current frontier
- Exact source graphs: 1,854.
- Exact sign partitions: 14,832.
- Coupling rank at least two: 5,460.
- Block-Schur failures: 0.
- Minimum exact scaled effective-core unit image: 17/9.
- Open dimension-free obligation: prove S 1 >= 0 for every admissible effective core (or find a genuine counterexample).
- Unrestricted/unbounded branch status: open.
FRONTIER.md gives the full path from the earlier theorem to this frontier and the next research program. data/claim_ledger.jsonl gives the same ladder in a machine-readable form.
Files
- FRONTIER.md: complete mathematical approach, evidence boundary, and next steps.
- CHANGELOG.md: stage-by-stage changes from the VC3 release through v13.
- data/claim_ledger.jsonl: machine-readable proved/verified/refuted/open claims.
- data/verification_summary.json: compact exact counts and status.
- data/certificates/: hash-verified compact certificate artifacts.
- scripts/: portable copies of the exact reproduction dependency closure.
- REPRODUCE.md: reproduction instructions.
- AI_DISCLOSURE.md: AI-generation and human-review disclosure.
- SOURCE_PROVENANCE.json, manifest.json, and SHA256SUMS: provenance and integrity.
Evidence boundary
The dimension-free results labeled proved are structural or symbolic claims backed by their certificate artifacts. Results labeled verified are exact finite enumerations and are not promoted to unrestricted proofs. Results labeled refuted have explicit exact counterexamples. The remaining claim is labeled open. No priority or novelty claim is made.
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