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ROOTED BICUBIC PLANAR MAPS VIA DYCK PATHS 17 Figure 27.Roote","work_id":"04fdf330-f95f-49ef-82d6-1e0b067c3617","year":2012}],"snapshot_sha256":"1d1c52ecebd693cb9836a6a6fbbb5136eb2f76a2c0023eae978d21701e72a10a"},"source":{"id":"2605.17515","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T22:31:26.474105Z","id":"31bcd3f8-ed4e-489d-9a65-f50f18163151","model_set":{"reader":"grok-4.3"},"one_line_summary":"An explicit bijection maps rooted bicubic planar maps on 2n vertices to colored Dyck paths of semilength 3n, proving Tutte's decomposition into 3-connected components via Bell transformations.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Rooted bicubic planar maps on 2n vertices correspond bijectively to Dyck paths of semilength 3n with colored ascents of length divisible by 3.","strongest_claim":"We establish an explicit bijection between rooted bicubic planar maps on 2n vertices and Dyck paths of semilength 3n with ascents of length divisible by 3, where each 3j-ascent is colored using one of g_j colors corresponding to the rooted 3-connected bicubic maps on 2j vertices.","weakest_assumption":"The construction assumes that the numbers g_j of rooted 3-connected bicubic maps on 2j vertices are already known or recursively available independently of the full map enumeration, so that the coloring step does not presuppose the decomposition being proved."}},"verdict_id":"31bcd3f8-ed4e-489d-9a65-f50f18163151"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:44a17ceca64ded1c034c28c30c27c721a52304a0489feb3059693b2198b6d219","target":"record","created_at":"2026-05-20T00:04:43Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f5bf455d7247aa89007f591adf8779724d2f0cb8472c24a8abeb55324b8ad032","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-17T15:59:46Z","title_canon_sha256":"ec18d003ed15f1c9b7c807a7880858ac2df5bab07e7472dd46795d61bc2a0f23"},"schema_version":"1.0","source":{"id":"2605.17515","kind":"arxiv","version":1}},"canonical_sha256":"c66bb91f78c8e6bd2f634412c7120eac4f849938301f11e5a2123d0f8cfe9a29","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c66bb91f78c8e6bd2f634412c7120eac4f849938301f11e5a2123d0f8cfe9a29","first_computed_at":"2026-05-20T00:04:43.230488Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:04:43.230488Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AZSni/GRzQae9iOjRdqJy8gH/om39BtMGliI5R6TBX3Qg4nrKr05Xr4K9FoKuk2JrJKK1HzLBlIa20LOU8cCCA==","signature_status":"signed_v1","signed_at":"2026-05-20T00:04:43.231370Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.17515","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:44a17ceca64ded1c034c28c30c27c721a52304a0489feb3059693b2198b6d219","sha256:66110ec8593e46cb79645e5a14fbaf789f1cd297fb0890e2d6dc7117b53488d6"],"state_sha256":"5c0d495b2631df6cd3d97ed010668eac7c25c5717711744080270d04e81aaf7e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"W4EV+suf7anWIVOm21edlZYYnmjM7r/Yuq+FqZUpNY4K5BiQF22V/6F4AcJ1JmHMMMpClyRYKhQCIpOAqhbvCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T18:33:16.321370Z","bundle_sha256":"befa68db11ae5efa5a6f4198c119d0bbefdb850520ee997538c3f9547a98d07d"}}