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"verified_results": [ { "id": "all-orders-auxiliary-rainbow-extension", "declaration": "TotalColoring.MinimalExtraction.hasValidRainbowColoring_of_inAuxiliaryClass", "supporting_declaration": "TotalColoring.IsOutsideEdgeMinimalNoncolorable.false_of_critical_allD", "module": "TotalColoring.CriticalAllDClosure", "source_path": "TotalColoring/CriticalAllDClosure.lean", "introduced_in_commit": "310b82c174ab2281581900897d4646875575e89b", "signature_summary": "{V : Type u} -> [Fintype V] -> [DecidableEq V] -> (D : Nat) -> (H : SimpleGraph V) -> [DecidableRel H.Adj] -> (J : Finset (Sym2 V)) -> InAuxiliaryClass D H J -> HasValidRainbowColoring D H J", "scope": "arbitrary finite vertex type and every formal member of InAuxiliaryClass", "palette": "Fin (D + 2)", "output_kind": "propositional existence", "conclusion_expansion": "There exists a proper edge assignment on H with colors Fin (D + 2) whose colors are pairwise distinct on distinguishedEdgeSet H J." }, { "id": 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member H and explicit selector membership in J", "palette": "ExtensionPalette D = Fin (D + 2)", "output_kind": "propositional existence of a semantically valid total-coloring assignment", "conclusion_expansion": "There exists assignment : Assignment G (Fin (D + 2)) with assignment.Valid; the theorem does not construct the extension or identify D with the maximum degree of G." }, { "id": "conditional-independent-seed-endpoint", "declaration": "TotalColoring.exists_valid_assignment_of_independentSeedPeel", "maximum_degree_wrapper_declaration": "TotalColoring.exists_valid_assignment_of_maxDegreeIndependentSeedPeel", "module": "TotalColoring.IndependentSeed", "source_path": "TotalColoring/IndependentSeed.lean", "availability": "present on current main commit da0595da1fa93c51a00be78193071bdea74aeb02, tree 34824cd54afb9a244a3fb58608a8d4da9bd6cfa6; the named proof branch commit and tree are historical source provenance", "proof_branch": "agent/independent-seed-endpoint", "proof_branch_commit": "cc4dd7ae1d858ea0583549f88707952e2414bf60", "proof_branch_tree": "9af6a84e1305aed9a0156dcd59c279de792dea4a", "default_branch_contains_declarations": true, "signature_summary": "{V : Type u} -> [Fintype V] -> [DecidableEq V] -> (G : SimpleGraph V) -> [DecidableRel G.Adj] -> {q : Nat} -> 0 < q -> (phi : EdgeAssignment G (Fin q)) -> phi.Valid -> (A : Finset V) -> G.IsIndepSet (A : Set V) -> IndependentSeedPeelCertificate G A q -> exists assignment : Assignment G (Fin (q + 1)), assignment.Valid", "scope": "a finite simple graph with decidable vertex equality and adjacency, a positive old-palette size q, a supplied proper Fin q edge assignment, a supplied independent seed A, and a supplied peel certificate covering exactly the vertices outside A", "hypotheses": [ "Fintype V", "DecidableEq V", "DecidableRel G.Adj", "0 < q", "phi : EdgeAssignment G (Fin q)", "phi.Valid", "A : Finset V", "G.IsIndepSet (A : Set V)", "cert : IndependentSeedPeelCertificate G A q" ], "palette": "Fin (q + 1)", "output_kind": "conditional propositional existence of a semantically valid total-coloring assignment", "conclusion_expansion": "There exists assignment : Assignment G (Fin (q + 1)) with assignment.Valid. The peel certificate supplies a duplicate-free order covering exactly V minus A and the strict later-neighbor bound at every step. The proof reverse-colors that order using old-palette colors missing from incident edges, while A receives the one fresh color.", "maximum_degree_specialization": "Setting q = G.maxDegree + 1 gives palette Fin (G.maxDegree + 2), hence a conditional Delta + 2 conclusion.", "maximum_degree_wrapper_signature_summary": "{V : Type u} -> [Fintype V] -> [DecidableEq V] -> (G : SimpleGraph V) -> [DecidableRel G.Adj] -> (phi : EdgeAssignment G (Fin (G.maxDegree + 1))) -> phi.Valid -> (A : Finset V) -> G.IsIndepSet (A : Set V) -> IndependentSeedPeelCertificate G A (G.maxDegree + 1) -> exists assignment : Assignment G (Fin (G.maxDegree + 2)), assignment.Valid", "maximum_degree_wrapper_boundary": "The wrapper only specializes q in the generic endpoint. The proper edge assignment, its validity, the independent seed, its independence proof, and the peel certificate remain explicit inputs.", "declaration_local_missing_inputs": [ "existence of a proper EdgeAssignment G (Fin (G.maxDegree + 1)); the library does not prove Vizing's theorem", "existence of an independent seed A with an IndependentSeedPeelCertificate G A (G.maxDegree + 1)" ], "claim_boundary": "The declaration is not an unrestricted Total Coloring Conjecture theorem and does not strengthen the unconditional scope of TotalColoring.exists_valid_assignment_of_highDegree." }, { "id": "conditional-total-independent-selector-decoder", "declaration": "TotalColoring.exists_valid_assignment_of_totalIndependentSelectorPeel", "maximum_degree_wrapper_declaration": "TotalColoring.exists_valid_assignment_of_maxDegreeTotalIndependentSelectorPeel", "path_certificate": "TotalColoring.AlternatingRainbowPathSelectorCertificate", "path_wrapper_declaration": "TotalColoring.exists_valid_assignment_of_alternatingRainbowPathSelector", "maximum_degree_path_wrapper_declaration": "TotalColoring.exists_valid_assignment_of_maxDegreeAlternatingRainbowPathSelector", "module": "TotalColoring.TotalIndependentSelector", "source_path": "TotalColoring/TotalIndependentSelector.lean", "availability": "present on current main commit da0595da1fa93c51a00be78193071bdea74aeb02, tree 34824cd54afb9a244a3fb58608a8d4da9bd6cfa6; the named proof branch commit and tree are historical source provenance", "proof_branch": "agent/total-independent-selector-decoder", "proof_branch_commit": "d008514c7a1cf834007bf0bd8de0d10a93926711", "proof_branch_tree": "1847934c78da03fe80bb67236868700c79016129", "default_branch_contains_declarations": true, "signature_summary": "{V : Type u} -> [Fintype V] -> [DecidableEq V] -> (G : SimpleGraph V) -> [DecidableRel G.Adj] -> {q : Nat} -> (phi : EdgeAssignment G (Fin q)) -> phi.Valid -> (S : Finset V) -> G.IsIndepSet (S : Set V) -> (F : Finset G.edgeSet) -> EdgeFinsetIsMatching F -> EdgeFinsetAvoids F S -> (K : Finset V) -> (g0 : V -> Fin q) -> SelectorOldColoringGoodOn G phi S F K g0 -> SelectorCorePeelCertificate G S K q -> exists assignment : Assignment G (Fin (q + 1)), assignment.Valid", "scope": "a finite simple graph with decidable vertex equality and adjacency, a supplied proper Fin q edge assignment, an independent fresh-color vertex selector S, a matching fresh-color edge selector F avoiding S, a supplied actual-list old coloring on the core K minus S, and a supplied core-relative peel certificate", "hypotheses": [ "Fintype V", "DecidableEq V", "DecidableRel G.Adj", "phi : EdgeAssignment G (Fin q)", "phi.Valid", "S : Finset V", "G.IsIndepSet (S : Set V)", "F : Finset G.edgeSet", "EdgeFinsetIsMatching F", "EdgeFinsetAvoids F S", "K : Finset V", "g0 : V -> Fin q", "SelectorOldColoringGoodOn G phi S F K g0", "cert : SelectorCorePeelCertificate G S K q" ], "peel_inequality": "(G.neighborFinset v intersect ((K minus S) union tail.toFinset)).card < q - G.degree v", "palette": "Fin (q + 1)", "output_kind": "conditional propositional existence of a semantically valid total-coloring assignment", "conclusion_expansion": "Vertices in S and edges in F receive the fresh color. The supplied core coloring and reverse greedy peel give allowed proper old colors to every other vertex, yielding a valid Assignment G (Fin (q + 1)).", "maximum_degree_specialization": "Setting q = G.maxDegree + 1 gives palette Fin (G.maxDegree + 2), hence a conditional Delta + 2 conclusion.", "maximum_degree_wrapper_boundary": "The proper edge assignment, independent set, matching selector, avoidance proof, actual-list core coloring, and core-relative peel certificate remain explicit inputs.", "path_certificate_fields": [ "k + 1 indexed injective vertices and k indexed edges", "exact consecutive path-edge endpoints", "the path vertices span the supplied core", "pairwise-distinct original path-edge colors", "a globally unused old spare color", "alternating old-lift and fresh-edge membership", "matching proofs for both alternating edge families", "fresh edges avoid the path start", "successor core colors equal predecessor edge colors", "a SelectorCorePeelCertificate rooted at the path start" ], "path_wrapper_boundary": "The wrapper proves the donor exchange and decoding from the explicit certificate; it does not prove existence of the proper edge coloring, selector/core, peel certificate, or alternating rainbow path.", "declaration_local_missing_inputs": [ "existence of a proper EdgeAssignment G (Fin (G.maxDegree + 1)); the library does not prove Vizing's theorem", "existence of compatible selector vertex and edge sets", "existence of a proper actual-list core coloring and core-relative peel certificate", "existence of the indexed alternating rainbow-path certificate" ], "claim_boundary": "The declarations are conditional decoders, not an unrestricted Total Coloring Conjecture theorem, and they do not strengthen the unconditional scope of TotalColoring.exists_valid_assignment_of_highDegree." }, { "id": "conditional-partial-edge-selector-and-exact-normalization", "forward_declaration": "TotalColoring.partialEdgeSelectorAssignment_valid", "last_color_forward_corollary": "TotalColoring.totalIndependentSelectorAssignment_valid_of_validOutside", "normalization_structure": "TotalColoring.PartialEdgeSelectorNormalization", "reverse_constructor": "TotalColoring.partialEdgeSelectorNormalization_of_valid", "maximum_degree_reverse_declaration": "TotalColoring.maxDegreePartialEdgeSelectorNormalization_of_valid", "module": "TotalColoring.PartialEdgeSelector", "source_path": "TotalColoring/PartialEdgeSelector.lean", "availability": "present on current main commit da0595da1fa93c51a00be78193071bdea74aeb02, tree 34824cd54afb9a244a3fb58608a8d4da9bd6cfa6; the named stacked proof branch commit and tree are historical source provenance", "proof_branch": "agent/partial-edge-selector-normalization", "proof_branch_commit": "c3dbe69c15f96e3c71d8481ae4e517ee2f4fdbf2", "proof_branch_tree": "11007a4aa381984a8d66aa1db297312cebe8d8b5", "default_branch_contains_declarations": true, "forward_signature_summary": "{V : Type u} -> [DecidableEq V] -> {G : SimpleGraph V} -> {q : Nat} -> (fresh : Fin (q + 1)) -> (phi : EdgeAssignment G (Fin q)) -> (S : Finset V) -> (F : Finset G.edgeSet) -> (g : V -> Fin q) -> phi.ValidOutside F -> G.IsIndepSet (S : Set V) -> EdgeFinsetIsMatching F -> EdgeFinsetAvoids F S -> (forall v notin S, SelectorOldColorAllowed phi F v (g v)) -> (forall adjacent v w outside S, g v != g w) -> (partialEdgeSelectorAssignment fresh phi S F g).Valid", "forward_scope": "an arbitrary graph with decidable vertex equality, an arbitrary chosen fresh color, an old edge assignment proper only outside F, an independent vertex selector S, a matching edge selector F avoiding S, and supplied allowed/proper old vertex colors", "ignored_data": "The values phi.color e for e in F are unrestricted and overwritten by the fresh color. Vertex allowedness likewise examines only incident edges outside F.", "palette": "Fin (q + 1)", "forward_output_kind": "conditional semantic validity of an explicitly decoded assignment", "reverse_signature_summary": "For [Fintype V], [DecidableEq V], [Fintype G.edgeSet], a supplied a : Assignment G (Fin (q + 1)) with a.Valid, any fresh : Fin (q + 1), and any fallback : Fin q, construct PartialEdgeSelectorNormalization a fresh.", "reverse_exactness": "The fresh vertex and edge color classes become S and F. Every other color is pulled back through fresh.succAbove; the fallback is used only on overwritten entries. The structure proves every forward premise and stores partialEdgeSelectorAssignment fresh oldEdge S F oldVertex = a literally.", "maximum_degree_reverse_boundary": "The maximum-degree declaration starts from an explicit valid Assignment G (Fin (G.maxDegree + 2)); it is notation for reverse decomposition and does not construct the assignment.", "declaration_local_missing_inputs": [ "for the forward theorem, every selector, properness, avoidance, and old vertex-color witness remains explicit", "for the reverse theorem, the valid total assignment being normalized remains an explicit input", "neither direction supplies an arbitrary-graph Delta + 2 coloring" ], "claim_boundary": "The forward declaration is conditional. The reverse declaration is exact but logically starts from the desired coloring. Neither is an unrestricted Total Coloring Conjecture theorem or a new coloring-existence result." }, { "id": "conditional-adapted-spare-vertex-endpoint", "declaration": "TotalColoring.adaptedSpareVertexColor_proper_iff", "supporting_definitions": [ "TotalColoring.adaptedSpareVertexColor", "TotalColoring.EqualMissingConflictsCovered", "TotalColoring.HeadLabelsClean", "TotalColoring.HeadLabelsProper" ], "module": "TotalColoring.AdaptedSpareEndpoint", "source_path": "TotalColoring/AdaptedSpareEndpoint.lean", "availability": "present on current main commit da0595da1fa93c51a00be78193071bdea74aeb02, tree 34824cd54afb9a244a3fb58608a8d4da9bd6cfa6, merged by PR #13 as dc6e3a4b7510b3d2df4fb5074afaee2027e9b442; the named proof branch commit and tree are historical proof provenance", "proof_branch": "agent/donor-global-formalization", "base_main_commit": "61e79beac7d4759568187bd43a5a40f23bf83af1 (historical base of the proof branch)", "base_main_tree": "cb2d7d06998c213e68a7372f743f67f9cff815f7 (historical)", "default_branch_contains_declaration": true, "signature_summary": "{V : Type u} -> [DecidableEq V] -> {C : Type v} -> (K : SimpleGraph V) -> (A B : Finset V) -> (missing head : V -> C) -> Disjoint A B -> ((forall adjacent v w, adaptedSpareVertexColor A B missing head v != adaptedSpareVertexColor A B missing head w) iff (K.IsIndepSet (A : Set V) and EqualMissingConflictsCovered K A B missing and HeadLabelsClean K A B missing head and HeadLabelsProper K B head))", "endpoint_assignment": "Vertices in A receive none, vertices in B receive some (head v), and every other vertex receives some (missing v); membership in A has definition-level priority and the theorem assumes Disjoint A B.", "equivalent_conditions": [ "A is independent in K", "every adjacent pair with equal supplied missing colors has at least one endpoint in A union B", "the supplied label at every b in B differs from the supplied missing color of every adjacent vertex outside A union B", "the supplied head labels differ on adjacent vertices of B" ], "output_kind": "exact equivalence for properness of the supplied vertex endpoint assignment", "declaration_local_missing_inputs": [ "construction or existence of the sets A and B", "construction or existence of a physical donor matching or donor-label transport", "construction or existence of compatible seed, matching, missing-color, or head-label data", "construction or existence of a proper edge coloring", "construction or existence of a valid total coloring" ], "claim_boundary": "This theorem is vertex-side only. It proves no donor-matching construction, seed or matching existence theorem, total-coloring existence theorem, unrestricted Delta + 2 result, or Total Coloring Conjecture." }, { "id": "supplied-pair-singleton-extension-seam", "declaration": "TotalColoring.Auxiliary.PairSingletonWitness.classEdge_mem_distinguishedEdgeSet", "supporting_declaration": "TotalColoring.Auxiliary.PairSingletonWitness.extension", "module": "TotalColoring.PairSingletonExtension", "source_path": "TotalColoring/PairSingletonExtension.lean", "introduced_in_commit": "dc2a318be1dd1475b90c492ad460c4180a3fbdec", "structural_layer_commit": "7aa102b0211c36c6d69f03bc051a5c2706f62c9d", "structural_layer_tree": "4b6440a0df108f47f5c120e7e0187c058a462138", "numeric_source_commit": "acb08de85f99c7db578e042e67d3f172c5599fcd", "numeric_source_tree": "7207e2a282ff829fba9737e93154f46f385ef879", "numeric_public_integration_commit": "343b7b87b82532a9d8eade6d4cd43679eae1f7c9", "numeric_public_integration_tree": "9863f500b2671048f4dc386f497eb6523b065099", "numeric_lean_content_identical_across_source_and_integration": true, "signature_summary": "{V : Type u} -> {G : SimpleGraph V} -> [Fintype V] -> [DecidableEq V] -> (P : Auxiliary.PairSingletonWitness G) -> (v : V) -> P.extension.classEdge v in distinguishedEdgeSet P.auxiliaryGraph P.distinguished", "scope": "a supplied PairSingletonWitness on a finite original vertex type", "palette": "not applicable", "output_kind": "concrete auxiliary graph and conflict-preserving Extension data, selector membership, qualitative auxiliary-class structure, exact degree and class-count identities, and conditional high-degree structural-class membership", "structural_declarations": [ "TotalColoring.Auxiliary.PairSingletonWitness.distinguished_exact_coverage", "TotalColoring.Auxiliary.PairSingletonWitness.matchingPart_isEdgeMatching", "TotalColoring.Auxiliary.PairSingletonWitness.matchingPart_off_center", "TotalColoring.Auxiliary.PairSingletonWitness.matchingPart_avoids_center_neighbors", "TotalColoring.Auxiliary.PairSingletonWitness.distinguished_decomposition", "TotalColoring.Auxiliary.PairSingletonWitness.degree_none_eq_singletonVertices_card", "TotalColoring.Auxiliary.PairSingletonWitness.degree_some_eq", "TotalColoring.Auxiliary.PairSingletonWitness.card_add_degree_none_eq_two_mul_distinguished_card", "TotalColoring.Auxiliary.PairSingletonWitness.degree_none_le_card_distinguished", "TotalColoring.Auxiliary.PairSingletonWitness.maxDegree_le_of_bounds", "TotalColoring.Auxiliary.PairSingletonWitness.center_degree_bounds" ], "conditional_class_declaration": "TotalColoring.Auxiliary.PairSingletonWitness.isAuxiliaryClassMember_of_numeric", "conditional_class_conclusion": "IsAuxiliaryClassMember D P.auxiliaryGraph none P.distinguished P.matchingPart", "conditional_class_hypotheses": [ "P.distinguished.card = D", "P.auxiliaryGraph.maxDegree <= D", "2 <= P.auxiliaryGraph.degree none and P.auxiliaryGraph.degree none <= D" ], "class_count_and_bounds_declaration": "TotalColoring.Auxiliary.PairSingletonWitness.isAuxiliaryClassMember_of_class_count_and_bounds", "class_count_and_bounds_hypotheses": [ "P.distinguished.card = D", "G.maxDegree + 1 <= D", "Fintype.card V + 2 <= 2 * D" ], "high_degree_declaration": "TotalColoring.Auxiliary.PairSingletonWitness.isAuxiliaryClassMember_of_highDegree", "high_degree_signature_summary": "{V : Type u} -> {G : SimpleGraph V} -> [Fintype V] -> [DecidableEq V] -> (P : Auxiliary.PairSingletonWitness G) -> [DecidableRel G.Adj] -> [DecidableRel P.auxiliaryGraph.Adj] -> P.distinguished.card = G.maxDegree + 1 -> Fintype.card V <= 2 * G.maxDegree -> IsAuxiliaryClassMember (G.maxDegree + 1) P.auxiliaryGraph none P.distinguished P.matchingPart", "high_degree_scope": "a supplied P : PairSingletonWitness G on a finite original vertex type, with decidable vertex equality and decidable adjacency for G and P.auxiliaryGraph; beyond these ambient typeclass requirements, the class-count equality and density inequality are the only additional proposition-valued hypotheses; no partition or nonempty hypothesis is part of this wrapper", "high_degree_hypotheses": [ "P.distinguished.card = G.maxDegree + 1", "Fintype.card V <= 2 * G.maxDegree" ], "high_degree_conclusion": "IsAuxiliaryClassMember (G.maxDegree + 1) P.auxiliaryGraph none P.distinguished P.matchingPart", "declaration_local_reduction_seams": [ "this supplied-witness theorem does not construct its PairSingletonWitness input", "this supplied-witness theorem does not itself select the empty-vertex branch" ], "conclusion_expansion": "P.auxiliaryGraph is defined on Option V; P.extension is an Auxiliary.Extension G P.auxiliaryGraph; every selector lies in the corresponding distinguishedEdgeSet; each copied original vertex has exact distinguished-edge coverage; matchingPart is an off-center matching whose endpoints avoid center neighbors; and distinguished is matchingPart union the full center incidence star. Exact degree and center/class-count identities derive the former maximum-degree and center-degree obligations. The high-degree wrapper supplies IsAuxiliaryClassMember with parameter G.maxDegree + 1 from the listed class-count and density hypotheses. The separate checked EquitablePairSingleton adapter constructs P and proves J.card = D from a supplied equitable independent partition in the range D <= Fintype.card V < 2 * D. The project-level terminal theorem discharges witness construction and the empty branch by a different route." }, { "id": "supplied-equitable-partition-high-degree-transfer", "declaration": "TotalColoring.Auxiliary.EquitableIndependentPartition.exists_valid_assignment_of_highDegreePartition", "supporting_declarations": [ "TotalColoring.Auxiliary.EquitableIndependentPartition.class_card_eq_one_or_two", "TotalColoring.Auxiliary.EquitableIndependentPartition.toPairSingletonWitness", "TotalColoring.Auxiliary.EquitableIndependentPartition.distinguished_card", "TotalColoring.Auxiliary.EquitableIndependentPartition.highDegreeWitness_distinguished_card", "TotalColoring.Auxiliary.EquitableIndependentPartition.highDegreeWitness_isAuxiliaryClassMember", "TotalColoring.Auxiliary.EquitableIndependentPartition.highDegreeWitness_inAuxiliaryClass" ], "module": "TotalColoring.EquitablePairSingleton", "source_path": "TotalColoring/EquitablePairSingleton.lean", "introduced_in_commit": "a441fbfa1e404dc7610e0c32c80dd692cd938c20", "exact_tree": "0e9b04a2acabae0cb0612e5e3cbf0344cc2f94f7", "signature_summary": "{V : Type u} -> {G : SimpleGraph V} -> [Fintype V] -> [DecidableEq V] -> [Nonempty V] -> [DecidableRel G.Adj] -> (Q : EquitableIndependentPartition G (G.maxDegree + 1)) -> Fintype.card V <= 2 * G.maxDegree -> exists assignment : Assignment G (ExtensionPalette (G.maxDegree + 1)), assignment.Valid", "scope": "a nonempty finite graph G with decidable vertex equality and adjacency, an explicitly supplied equitable independent partition Q into exactly G.maxDegree + 1 nonempty classes, and the half-order density inequality", "hypotheses": [ "Fintype V", "DecidableEq V", "Nonempty V", "DecidableRel G.Adj", "Q : EquitableIndependentPartition G (G.maxDegree + 1)", "Fintype.card V <= 2 * G.maxDegree" ], "palette": "ExtensionPalette (G.maxDegree + 1) = Fin ((G.maxDegree + 1) + 2), hence G.maxDegree + 3 colors", "output_kind": "conditional propositional existence of a semantically valid total-coloring assignment", "conclusion_expansion": "There exists assignment : Assignment G (ExtensionPalette (G.maxDegree + 1)) with assignment.Valid. This reusable conditional theorem constructs the pair/singleton witness, proves the exact distinguished count, establishes InAuxiliaryClass, and invokes the checked auxiliary-to-total transfer. By itself it does not construct Q or handle the empty graph; those project-level seams are discharged by TotalColoring.exists_valid_assignment_of_highDegree.", "declaration_local_missing_inputs": [ "existence of Q for every nonempty graph satisfying the intended high-degree hypotheses, potentially via a complement-matching construction", "the separate empty-graph base case" ], "project_level_status": "discharged by the terminal high-degree theorem" }, { "id": "end-to-end-high-degree-total-coloring", "declaration": "TotalColoring.exists_valid_assignment_of_highDegree", "supporting_declarations": [ "TotalColoring.MatchingLowerBound.exists_complement_matchingGraph_edgeFinset_card_eq", "TotalColoring.Auxiliary.exists_pairSingletonWitness_of_highDegree", "TotalColoring.Auxiliary.exists_valid_assignment_of_highDegree_nonempty", "TotalColoring.exists_valid_assignment_of_isEmpty" ], "module": "TotalColoring.HighDegreeTotalColoring", "source_path": "TotalColoring/HighDegreeTotalColoring.lean", "introduced_in_commit": "43a49aa6f7e7d8cf5c95a22a71a3a92a02db7c45", "verified_proof_tree": "4624044788ab42c0dc116cfbf7f38c696065263c", "verified_source_archive_sha256": "302bc3f00bf5d8c1ce563d2bc84d1370e627c81d219e8d8085b286a21d530077", "verified_publishable_commit": "06d43af7f4ea8fefea9e07e2bc29bdc960548171", "verified_publishable_tree": "89a32c7a78e294a8b1484092ec79afaa3b4ace5a", "verified_publishable_source_archive_sha256": "c9950b9e8af364a0d2ef3c08d80f06786ea36d3e6a5bf728054f7824612b4331", "signature_summary": "{V : Type u} -> [Fintype V] -> [DecidableEq V] -> (G : SimpleGraph V) -> [DecidableRel G.Adj] -> Fintype.card V <= 2 * G.maxDegree -> exists assignment : Assignment G (ExtensionPalette (G.maxDegree + 1)), assignment.Valid", "scope": "every finite simple graph with decidable vertex equality and adjacency satisfying the half-order density inequality, including the empty vertex type and with no parity hypothesis", "hypotheses": [ "Fintype V", "DecidableEq V", "DecidableRel G.Adj", "Fintype.card V <= 2 * G.maxDegree" ], "palette": "ExtensionPalette (G.maxDegree + 1) = Fin (G.maxDegree + 3)", "output_kind": "propositional existence of a semantically valid total-coloring assignment", "conclusion_expansion": "The nonempty branch obtains an exact complement matching, builds the pair/singleton witness, enters the checked auxiliary class, and decodes a valid assignment. The empty branch uses a separate vacuous assignment theorem. This is a Lean-package theorem; it is not the author-locked paper theorem, a Delta + 2 theorem, the unrestricted Total Coloring Conjecture, or a novelty claim.", "proof_tree_trust_jobs": [ "5388311", "5388312", "5388313", "5388314", "5388315" ], "publishable_tree_verification_status": "passed a separate Wave 11 exact-tree gate, PR-head CI, and post-merge main CI; the external PR receipt names the certified tree without changing it" } ], "not_proved": [ "the Total Coloring Conjecture", "an unrestricted total-coloring theorem for all finite graph orders", "an unconditional high-degree or all-graphs total-coloring theorem with Delta + 2 colors", "Vizing's theorem or universal existence of the proper edge-coloring witness required by the conditional independent-seed endpoint", "universal existence of an independent seed and peel certificate satisfying the endpoint hypotheses", "universal existence of a compatible total-independent selector, actual-list core coloring, core-relative peel certificate, or alternating rainbow-path certificate for the selector wrappers", "existence of the valid total coloring assumed by the exact partial-edge normalization; a reverse decomposition cannot be cited as a coloring-existence theorem", "construction or universal existence of the adapted-spare sets, a physical donor matching, compatible seed or matching data, missing/head labels, or a proper edge coloring required before the adapted-spare vertex endpoint can be used", "an end-to-end Delta + 3 theorem without the half-order density hypothesis", "an author-approved or manuscript-integrated paper theorem; the checked package declaration is not identified with the manuscript theorem", "an executable total-coloring construction or a complexity bound", "the equality chi''(G) = Delta(G) + 3 or use of every palette color; the theorem gives existence into a palette of that cardinality", "the stronger D + 1 auxiliary palette", "novelty of any formal result" ], "trust_boundary": [ "Lean 4.32.0 kernel checking is authoritative for the formal declarations.", "Production Lean modules prohibit sorry, admit, custom axiom declarations, and native_decide.", "Executable certificate checkers have semantic soundness theorems.", "The independent-seed endpoint is a semantic existence theorem conditional on all of its supplied witnesses; it is not a witness generator for Vizing or the peel certificate.", "The total-independent selector and alternating-path wrappers are semantic existence theorems conditional on every displayed edge-coloring, selector, core, peel, and path witness; they do not generate those witnesses.", "The partial-edge decoder ignores old values on the selected matching but retains every other displayed forward hypothesis. Its exact normalization begins with a supplied valid total assignment and proves no existence statement for that input.", "The adapted-spare endpoint is an exact vertex-properness equivalence for supplied data only; it constructs no donor matching, edge coloring, total coloring, or unrestricted Delta + 2 witness.", "External JSON parsing and claims of census completeness remain outside the trusted Lean interface.", "A checked positive assignment validates that assignment; it does not prove an external enumeration complete." ], "verification": { "current_main_commit": "da0595da1fa93c51a00be78193071bdea74aeb02", "current_main_tree": "34824cd54afb9a244a3fb58608a8d4da9bd6cfa6", "independent_seed_merge_pull_request": 10, "independent_seed_merge_commit": "f4c56956ed535864daa0116b66fd895c81d9d719", "selector_merge_pull_request": 11, "selector_merge_commit": "35e127c177201c0ab1e35689741132ea2bb559c7", "partial_edge_merge_pull_request": 12, "partial_edge_merge_commit": 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