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 |
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