Datasets:

cjc0013
/

laplacian-matching-theorem-frontier-v15

claim_id stringlengths 15 47	evidence stringlengths 55 183	limitation stringlengths 40 110	next_step stringlengths 32 96	stage stringclasses 4 values	statement stringlengths 43 154	status stringclasses 4 values
predecessor-v14	cjc0013/laplacian-matching-theorem-frontier-v14@453f9cdbfb552e385d4d2b980b7c0142a6cde33a	The predecessor does not contain the complete v15 Hall-pair partition.	Control the complete Schur image.	prior_dataset	The original v14 release is the immutable predecessor of this expanded frontier.	proved
m-matrix-harmonic-comparison	data/certificates/block_schur_dirichlet_to_neumann_certificate.json	This comparison does not alone prove the graph-local lower envelope is nonnegative.	Prove the remaining core/interface inequality.	v15	For lambda=max_j b_j/(D1)_j, the residual harmonic extension satisfies D^(-1)b <= lambda 1.	proved
bounded-monotone-envelope	199728 applicable sign partitions; zero failures; minimum 17/9.	Finite exact verification is not a dimension-free proof.	Extract a graph-local symbolic proof.	v15	The monotone lower envelope is nonnegative on every applicable frontier coordinate through order eleven.	verified
order-degree-envelope-split	data/certificates/bounded_replay.json; ten groups; zero baseline, identity, coupling-sign, or envelope failures.	The grouping is finite and does not establish a dimension-free inequality.	Generalize the cell constants to symbolic functions of order and maximum degree.	v15	Each of the ten observed order/degree cells admits an exact scalar decoupling of the graph-local envelope.	verified
sharp-unattached-hall-pair-family	data/certificates/block_schur_dirichlet_to_neumann_certificate.json; exact formulas, three positive cubic cases, and 29991 zero-failure symbolic checks.	The proof does not cover attached Hall neighbours, nonuniform residual signs, or larger critical Hall cores.	Extend the symbolic decomposition to the remaining attachment and sign classes.	v15	The graph-local envelope is positive for every unattached two-vertex critical Hall core with uniform core signs and uniform opposite residual signs.	proved
mixed-residual-sign-hall-pair-family	data/certificates/block_schur_dirichlet_to_neumann_certificate.json; lambda<2, core ratio at least 2, 119964 zero-failure symbolic cases, and 49932 exact blocks.	The proof does not cover the remaining uniform-residual-sign attachment classes or larger canonical cores.	Close the attached uniform-opposite-sign classes and then treat mixed core signs dimension-free.	v15	The graph-local envelope is positive for every uniform-sign critical Hall pair with mixed residual signs and any Hall-neighbour residual attachment count.	proved
attached-uniform-opposite-sign-hall-pair-family	data/certificates/block_schur_dirichlet_to_neumann_certificate.json; generic lambda<2 comparison, two exact boundary bounds, 59982 zero-failure symbolic cases, and 16662 exact blocks.	This is one component of the two-core/two-residual Hall-pair topology, not a proof for larger canonical cores.	Combine it with the baseline sign classes into an exhaustive Hall-pair partition.	v15	The graph-local envelope is positive for every attached uniform-sign critical Hall pair with uniform opposite residual signs.	proved
complete-two-by-two-hall-pair-sign-partition	199728 exact blocks assigned once; all 25704 lambda>1 blocks absorbed; 329901 symbolic cases and zero failures.	The partition does not cover canonical Hall cores larger than the critical pair topology.	Lift the sign-local arguments to larger canonical Hall cores.	v15	Four disjoint dimension-free families prove the graph-local envelope for every two-core/two-residual critical Hall-pair sign and attachment pattern.	proved
harmonic-unit-cap	3096 exact failures through order eleven; maximum 14/9.	This refutes a shortcut, not the surviving comparison.	Use lambda rather than a unit cap.	v13	The residual harmonic extension is always coordinatewise at most one.	refuted
rank-one-interface	data/certificates/core_residual_rank_one_compression_obstruction.json	The arbitrary-rank Schur route survives.	Preserve full coupling topology.	v13	Every core-residual interface has rank one.	refuted
trace-cut-shortcut-coverage	4 exact-trace failures, 4 weighted-cut failures, and 32 triangle-corrected failures.	The full M-matrix and monotone-envelope certificates still pass; these are not theorem counterexamples.	Use the order/degree-conditioned envelope rather than the weaker shortcuts.	v15	The exact-trace, weighted-cut, and triangle-corrected sufficient bounds cover every order-eleven signed matrix.	refuted
graph-local-envelope	Exact order-eleven frontier support plus a complete dimension-free certificate for the two-core/two-residual Hall-pair topology; no certificate for larger canonical cores yet.	This remains the branch obligation only beyond the completed Hall-pair topology.	Prove or falsify the attachment-local inequality for larger canonical Hall cores.	next	Every admissible core coordinate satisfies A1_i >= lambda(-B1)_i.	open
vc3-theorem-alternate-route	cjc0013/vc3-laplacian-matching-theorem@4494842b0bd0bdb7d935b68a153f8581a6c84a03	The later proof does not establish the v14 graph-local envelope inequality.	Retain the v14 inequality as an independent branch-level open problem.	v15	For every connected simple G with tau(G)<=3, mu_1(G)<=Delta(G)+nu(G).	proved

Laplacian Matching Theorem Frontier v15

Produced by the Ouroboros AI Research System, under human direction.

This private-review dataset advances cjc0013/laplacian-matching-theorem-frontier-v14, pinned at revision 453f9cdbfb552e385d4d2b980b7c0142a6cde33a. It does not claim the unrestricted Laplacian matching theorem and does not claim that the current unmatched-vertex branch is closed.

What changed in v15

Exact replay rows now expose the attachment-local Schur decomposition instead of only scalar summaries.
For the Hall residual block, with b=-B^T 1, r=D 1, h=D^(-1)b, and lambda=max_j b_j/r_j, the dimension-free M-matrix comparison h <= lambda 1 is proved.
This reduces effective-core positivity to the graph-local inequality A 1_i >= lambda (-B 1)_i.
The exact frontier now reaches order eleven and evaluates the lower envelope on 199,728 applicable sign partitions: zero comparison failures, zero nonnegative-envelope failures, and unchanged sharp minimum 17/9.
A new finite certificate conditions on the ten observed (order, Delta) cells. In each cell, the lambda <= 1 branch uses A1 >= -B1; the lambda > 1 branch verifies max lambda <= min A1/(-B1) with exact rational constants.
The sharp finite witness is now embedded in a dimension-free proved family: a two-vertex critical Hall core, one unattached Hall neighbour, uniform core signs, and uniform opposite residual signs. A 29,991-case symbolic replay has zero failures and minimum cleared gap 17/90.
A second dimension-free proof closes every Hall-neighbour attachment count for uniform core signs and mixed residual signs. It covers 49,932 exact blocks, including 21,864 lambda > 1 cases, and its 119,964-case symbolic replay has zero failures. The proof combines the strict bound lambda < 2 with A1/(-B1) >= 2.
A third dimension-free proof closes the attached uniform-opposite-sign cases. The generic comparison is again lambda < 2 <= A1/(-B1); the sole boundary coordinates at Delta=4,d=3 satisfy the sharper exact bounds 10/9 < 10/7 for one attachment and 20/23 < 7/4 for two. It covers 16,662 exact blocks, including 1,890 with lambda > 1, and 59,982 symbolic cases with zero failures.
Together with a baseline lambda < 1 proof for mixed core signs or uniform cover signs, the four disjoint Hall-pair families partition all 199,728 exact blocks and absorb all 25,704 lambda > 1 blocks. The combined 329,901 symbolic cases have zero failures. This completes the two-core/two-residual Hall-pair topology, while larger canonical Hall cores remain open.
The earlier false shortcut h <= 1 remains retired: 3,096 exact cases exceed one, with maximum 14/9. Order eleven also exposes 4 exact-trace, 4 weighted-cut, and 32 triangle-corrected sufficient-bound failures without producing a graph-local-envelope failure.

v15 theorem-route context

A later, independently replayed packet proves the VC3 theorem through a different Hall-classification and Schur route. The authority is pinned to cjc0013/vc3-laplacian-matching-theorem at revision 4494842b0bd0bdb7d935b68a153f8581a6c84a03. This does not prove the v14 graph-local inequality; it makes that inequality an optional branch-level problem rather than a remaining requirement for the VC3 theorem.

Current claim boundary

Proved dimension-free: canonical Hall-core reduction, rank-agnostic block Schur algebra, Hall-residual M-matrix positivity, and the new harmonic comparison h <= lambda 1; also a complete four-family sign partition for every two-core/two-residual critical Hall-pair topology.
Verified exactly: the graph-local monotone lower envelope is nonnegative over the complete order-eleven frontier, including a ten-cell order/degree split certificate.
Open: extend A 1_i >= lambda (-B 1)_i beyond the completed Hall-pair topology to every admissible larger canonical Hall core, or find a genuine counterexample.
Proved by a later independent route: for connected simple graphs with tau(G) <= 3, mu_1(G) <= Delta(G) + nu(G).
Not proved: the unrestricted all-graphs theorem. The v14-specific graph-local branch also remains open and is not represented as closed.

Verification surfaces

Exact rational replay is the primary mathematical evidence. CrossHair supplied a bounded symbolic falsifier for the two-residual-coordinate attachment inequality. Temporal-model checking guarded proof-state transitions; fixed-point provenance analysis checked claim propagation; answer-set checking validated the research obligation ordering. Probabilistic model checking was intentionally not applied because no calibrated theorem-workflow probabilities exist.

See FRONTIER.md for the derivation, CHANGELOG.md for the v14-to-v15 delta, data/claim_ledger.jsonl for machine-readable claim status, and REPRODUCE.md for exact regeneration instructions.

Review status

The bundle is prepared for private human review. Public visibility, mathematical endorsement, authorship, licensing, and any stronger claim remain human decisions. No priority or novelty claim is made.

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