{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:NPIGUIGPHUDLEKHX73Y4WFYULS","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"4cb981462f3abdffd4857adc50ac897ea08c06eecab7281f8c033ace3513f589","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T00:14:56Z","title_canon_sha256":"859203e54228cffbd31679dcbce8fea0291e9b17e26885afb42f63123d1bb18c"},"schema_version":"1.0","source":{"id":"2605.14213","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.14213","created_at":"2026-05-17T23:39:10Z"},{"alias_kind":"arxiv_version","alias_value":"2605.14213v1","created_at":"2026-05-17T23:39:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.14213","created_at":"2026-05-17T23:39:10Z"},{"alias_kind":"pith_short_12","alias_value":"NPIGUIGPHUDL","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"NPIGUIGPHUDLEKHX","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"NPIGUIGP","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:1c9aeb13ae36adf2526c6da3401c8dd310b2ba30679774a52dc9f02bde41dd50","target":"graph","created_at":"2026-05-17T23:39:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"Mathar's order-5 operator, applied to each summand separately, reduces to a polynomial identity that simplifies to zero after a brief calculation."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The closed form a(n) = 1/2 (binomial(2n,n) + [n even] binomial(n,n/2)) correctly counts the orbits under the reversal group action, which rests on the standard application of Burnside's lemma to the two-element group."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"The conjectured recurrence for a(n) holds because the order-5 operator annihilates both the central binomial coefficient and the even-n middle binomial term in the closed form derived from Burnside's lemma."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Mathar's conjectured order-5 recurrence holds for the sequence counting binary strings up to reversal."}],"snapshot_sha256":"9d6bc1ddc51ef01e7399af09c71887d4957c80f9ac395a25771b2956caf29f56"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"For the OEIS sequence A032123, the number of length-$2n$ black-and-white strings with $n$ black beads, considered up to reversal, R. J. Mathar contributed in November 2013 the conjectured order-5 P-recursive recurrence \\[ \\begin{aligned} &n(n-1)\\,a(n) - 2(n-1)(3n-4)\\,a(n-1) + 4(2n^{2}-14n+19)\\,a(n-2) &\\qquad + 8(n^{2}+5n-19)\\,a(n-3) - 16(n-3)(3n-10)\\,a(n-4) &\\qquad + 32(n-4)(2n-9)\\,a(n-5) \\;=\\; 0, \\qquad n \\ge 6. \\end{aligned} \\] We give a short proof. Burnside's lemma applied to the reversal action gives the closed form $a(n) = \\tfrac{1}{2}\\bigl(\\binom{2n}{n} + [n \\text{ even}]\\binom{n}{n/2}\\","authors_text":"Tong Niu","cross_cats":[],"headline":"Mathar's conjectured order-5 recurrence holds for the sequence counting binary strings up to reversal.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T00:14:56Z","title":"A short proof of Mathar's 2013 recurrence conjecture for the reversible-binary-string sequence A032123"},"references":{"count":11,"internal_anchors":1,"resolved_work":11,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"Burnside,Theory of Groups of Finite Order, Cambridge University Press, 1897","work_id":"0109df99-6c74-40ae-8b96-1fc5e07fbce9","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"S. Chen, M. Kauers, C. Koutschan, X. Li, R.-H. Wang and Y. Wang,Non-minimality of minimal telescopers explained by residues, arXiv:2502.03757, 2025","work_id":"0b8abde8-788b-48a1-93ee-3e02d9581456","year":2025},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"Fried,Proofs of some conjectures from the OEIS","work_id":"5fb607ef-406b-4aff-ae7a-71aa50508bf2","year":2024},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"S. Fried,Proofs of several conjectures from the OEIS, J. Integer Seq.28(2025), Article 25.4.3. 10 TONG NIU","work_id":"fdc55654-e319-47dc-9ff2-c6deee479c49","year":2025},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"M. Kauers and C. Koutschan,A list of guessed but unproven holonomic recurrences in the OEIS, arXiv:2303.02793, 2023","work_id":"6d053669-aa63-4eeb-b76d-b87c3501d616","year":2023}],"snapshot_sha256":"5522a88d941f37fff786548c40e88afb8f28301ab1e08c1020d80037bf7d9de4"},"source":{"id":"2605.14213","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-15T02:51:24.114460Z","id":"bc5cb04f-e5d6-4e61-ab91-a0bc6eacda5d","model_set":{"reader":"grok-4.3"},"one_line_summary":"The conjectured recurrence for a(n) holds because the order-5 operator annihilates both the central binomial coefficient and the even-n middle binomial term in the closed form derived from Burnside's lemma.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Mathar's conjectured order-5 recurrence holds for the sequence counting binary strings up to reversal.","strongest_claim":"Mathar's order-5 operator, applied to each summand separately, reduces to a polynomial identity that simplifies to zero after a brief calculation.","weakest_assumption":"The closed form a(n) = 1/2 (binomial(2n,n) + [n even] binomial(n,n/2)) correctly counts the orbits under the reversal group action, which rests on the standard application of Burnside's lemma to the two-element group."}},"verdict_id":"bc5cb04f-e5d6-4e61-ab91-a0bc6eacda5d"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:5f0449ab7e1021ab01fc5ced953f7c497f8ed1242141090decd3db52e2ebd9d5","target":"record","created_at":"2026-05-17T23:39:10Z","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":"4cb981462f3abdffd4857adc50ac897ea08c06eecab7281f8c033ace3513f589","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T00:14:56Z","title_canon_sha256":"859203e54228cffbd31679dcbce8fea0291e9b17e26885afb42f63123d1bb18c"},"schema_version":"1.0","source":{"id":"2605.14213","kind":"arxiv","version":1}},"canonical_sha256":"6bd06a20cf3d06b228f7fef1cb17145ca42bff53c82ba3529c0713c6bc717db0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6bd06a20cf3d06b228f7fef1cb17145ca42bff53c82ba3529c0713c6bc717db0","first_computed_at":"2026-05-17T23:39:10.910373Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:39:10.910373Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ghONUcpAudfGUD1bvfumEo7VFG4/Q6EntcMoJRZPxbQQV3wrlY5IeWlEztX3e5RlYczJrzjyTlhg54vNGevJDQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:39:10.910833Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.14213","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5f0449ab7e1021ab01fc5ced953f7c497f8ed1242141090decd3db52e2ebd9d5","sha256:1c9aeb13ae36adf2526c6da3401c8dd310b2ba30679774a52dc9f02bde41dd50"],"state_sha256":"d60b420a0f0ade1fad54142a6283743a8b62a9597330deb0c202f65b29dd948a"}