{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:NTTF4Z3AOBAR2A3TUBVZXQKO5M","short_pith_number":"pith:NTTF4Z3A","canonical_record":{"source":{"id":"2605.14954","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T15:25:02Z","cross_cats_sorted":[],"title_canon_sha256":"c503f365e8e9fd6a1eb0080731ad670406367cbda2e1fe7c132d7e1698eb76d1","abstract_canon_sha256":"434b1584765c23e10b729ac46d391bda8adb39d32bf7ef49c4883b55be99946a"},"schema_version":"1.0"},"canonical_sha256":"6ce65e676070411d0373a06b9bc14eeb02233ced53f67f44413f14c52f7a423d","source":{"kind":"arxiv","id":"2605.14954","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.14954","created_at":"2026-05-17T23:38:55Z"},{"alias_kind":"arxiv_version","alias_value":"2605.14954v1","created_at":"2026-05-17T23:38:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.14954","created_at":"2026-05-17T23:38:55Z"},{"alias_kind":"pith_short_12","alias_value":"NTTF4Z3AOBAR","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"NTTF4Z3AOBAR2A3T","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"NTTF4Z3A","created_at":"2026-05-18T12:33:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:NTTF4Z3AOBAR2A3TUBVZXQKO5M","target":"record","payload":{"canonical_record":{"source":{"id":"2605.14954","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T15:25:02Z","cross_cats_sorted":[],"title_canon_sha256":"c503f365e8e9fd6a1eb0080731ad670406367cbda2e1fe7c132d7e1698eb76d1","abstract_canon_sha256":"434b1584765c23e10b729ac46d391bda8adb39d32bf7ef49c4883b55be99946a"},"schema_version":"1.0"},"canonical_sha256":"6ce65e676070411d0373a06b9bc14eeb02233ced53f67f44413f14c52f7a423d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:38:55.357567Z","signature_b64":"r4pg8XgggvaDbBoxJbKo3F2yKkt4a535aZldJvxyQz55DRf7jm4rX6rX6DMpaeXM6dCqWhTcXf7BsUKhlqDHDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6ce65e676070411d0373a06b9bc14eeb02233ced53f67f44413f14c52f7a423d","last_reissued_at":"2026-05-17T23:38:55.356961Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:38:55.356961Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.14954","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:38:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HbnQ6yqSjxpLSvhM52eLzS5Aj3RUlNDibQVmaMDSzyDDk06sa28LmkikUvXUXieNKyUpKswIw1tCCoxMME78CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T19:56:52.503295Z"},"content_sha256":"80f0fb6c27f56efed0d4c1440500a2596204ca05ddbbd7859061be9de4d2fcf2","schema_version":"1.0","event_id":"sha256:80f0fb6c27f56efed0d4c1440500a2596204ca05ddbbd7859061be9de4d2fcf2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:NTTF4Z3AOBAR2A3TUBVZXQKO5M","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On zero-sum Ramsey numbers of cycles and wheels","license":"http://creativecommons.org/licenses/by/4.0/","headline":"R(C_qk, Z_q) equals qk + q - 1 exactly for odd q ≥ 3 and k ≥ 35q, with matching exact results for q=3 cycles and wheels W_3k.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cheng Chi, Jialin He","submitted_at":"2026-05-14T15:25:02Z","abstract_excerpt":"For an integer $q\\ge 2$ and a graph $F$ with $q\\mid e(F)$, let $R(F,\\Z_q)$ be the least integer $n$ such that every edge-labeling $w\\colon E(K_n)\\to \\Z_q$ contains a copy of $F$ whose edge-label sum is zero in $\\Z_q$. Write $C_{qk}$ for the cycle on $qk$ vertices. We prove that $R(C_{qk},\\Z_q)\\le \\max\\{R(C_{2q},\\Z_q),qk+q-1\\}$ via an insertion argument rooted in the classic Erd\\H{o}s-Ginzburg-Ziv theorem. Combined with Pikhurko's result, we obtain $R(C_{qk},\\Z_q)\\le \\max\\{35q^2,qk+q-1\\}$ for every $q\\ge 3$. We also show that $R(C_{qk},\\Z_q)\\ge qk+q-1$ for odd $q\\ge 3$. Hence, for every fixed o"},"claims":{"count":3,"items":[{"kind":"strongest_claim","text":"For every fixed odd q≥3 and every k≥35q, we obtain the exact value R(C_qk,Z_q)=qk+q-1. For q=3, R(C_3k,Z_3)=3k+2 for k≥2 and R(W_3k,Z_3)=3k+1 for k≥2.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The upper bound for general q relies on Pikhurko's external result that R(C_2q,Z_q)≤35q²; if that bound is loose or inapplicable in the insertion step, the claimed exact value for large k would not hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"R(C_qk, Z_q) equals qk + q - 1 exactly for odd q ≥ 3 and k ≥ 35q, with matching exact results for q=3 cycles and wheels W_3k.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"}],"snapshot_sha256":"6059f9b986b475e9cb2c5cb98940c8b0c781ff098592495219e0f68bd9738db5"},"source":{"id":"2605.14954","kind":"arxiv","version":1},"verdict":{"id":"460bad04-3323-4951-9a65-82fb62a3cf08","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T03:20:31.295043Z","strongest_claim":"For every fixed odd q≥3 and every k≥35q, we obtain the exact value R(C_qk,Z_q)=qk+q-1. For q=3, R(C_3k,Z_3)=3k+2 for k≥2 and R(W_3k,Z_3)=3k+1 for k≥2.","one_line_summary":"R(C_qk, Z_q) equals qk + q - 1 exactly for odd q ≥ 3 and k ≥ 35q, with matching exact results for q=3 cycles and wheels W_3k.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The upper bound for general q relies on Pikhurko's external result that R(C_2q,Z_q)≤35q²; if that bound is loose or inapplicable in the insertion step, the claimed exact value for large k would not hold.","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":"460bad04-3323-4951-9a65-82fb62a3cf08"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:38:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FbrxlgvEzsGzR3YpOLBfvvlpe1BPXrhIEZtNPMju57C976Hi/WNw5cU+E3mlEVfKxLQGsY1Lu407AddkMRJxBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T19:56:52.504225Z"},"content_sha256":"ccd8ecd4dbf8af7d88b0038f95cf01c0318659d6f8ee50171d428f6a9164cb07","schema_version":"1.0","event_id":"sha256:ccd8ecd4dbf8af7d88b0038f95cf01c0318659d6f8ee50171d428f6a9164cb07"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NTTF4Z3AOBAR2A3TUBVZXQKO5M/bundle.json","state_url":"https://pith.science/pith/NTTF4Z3AOBAR2A3TUBVZXQKO5M/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NTTF4Z3AOBAR2A3TUBVZXQKO5M/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-27T19:56:52Z","links":{"resolver":"https://pith.science/pith/NTTF4Z3AOBAR2A3TUBVZXQKO5M","bundle":"https://pith.science/pith/NTTF4Z3AOBAR2A3TUBVZXQKO5M/bundle.json","state":"https://pith.science/pith/NTTF4Z3AOBAR2A3TUBVZXQKO5M/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NTTF4Z3AOBAR2A3TUBVZXQKO5M/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:NTTF4Z3AOBAR2A3TUBVZXQKO5M","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":"434b1584765c23e10b729ac46d391bda8adb39d32bf7ef49c4883b55be99946a","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T15:25:02Z","title_canon_sha256":"c503f365e8e9fd6a1eb0080731ad670406367cbda2e1fe7c132d7e1698eb76d1"},"schema_version":"1.0","source":{"id":"2605.14954","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.14954","created_at":"2026-05-17T23:38:55Z"},{"alias_kind":"arxiv_version","alias_value":"2605.14954v1","created_at":"2026-05-17T23:38:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.14954","created_at":"2026-05-17T23:38:55Z"},{"alias_kind":"pith_short_12","alias_value":"NTTF4Z3AOBAR","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"NTTF4Z3AOBAR2A3T","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"NTTF4Z3A","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:ccd8ecd4dbf8af7d88b0038f95cf01c0318659d6f8ee50171d428f6a9164cb07","target":"graph","created_at":"2026-05-17T23:38:55Z","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":3,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"For every fixed odd q≥3 and every k≥35q, we obtain the exact value R(C_qk,Z_q)=qk+q-1. For q=3, R(C_3k,Z_3)=3k+2 for k≥2 and R(W_3k,Z_3)=3k+1 for k≥2."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The upper bound for general q relies on Pikhurko's external result that R(C_2q,Z_q)≤35q²; if that bound is loose or inapplicable in the insertion step, the claimed exact value for large k would not hold."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"R(C_qk, Z_q) equals qk + q - 1 exactly for odd q ≥ 3 and k ≥ 35q, with matching exact results for q=3 cycles and wheels W_3k."}],"snapshot_sha256":"6059f9b986b475e9cb2c5cb98940c8b0c781ff098592495219e0f68bd9738db5"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"For an integer $q\\ge 2$ and a graph $F$ with $q\\mid e(F)$, let $R(F,\\Z_q)$ be the least integer $n$ such that every edge-labeling $w\\colon E(K_n)\\to \\Z_q$ contains a copy of $F$ whose edge-label sum is zero in $\\Z_q$. Write $C_{qk}$ for the cycle on $qk$ vertices. We prove that $R(C_{qk},\\Z_q)\\le \\max\\{R(C_{2q},\\Z_q),qk+q-1\\}$ via an insertion argument rooted in the classic Erd\\H{o}s-Ginzburg-Ziv theorem. Combined with Pikhurko's result, we obtain $R(C_{qk},\\Z_q)\\le \\max\\{35q^2,qk+q-1\\}$ for every $q\\ge 3$. We also show that $R(C_{qk},\\Z_q)\\ge qk+q-1$ for odd $q\\ge 3$. Hence, for every fixed o","authors_text":"Cheng Chi, Jialin He","cross_cats":[],"headline":"R(C_qk, Z_q) equals qk + q - 1 exactly for odd q ≥ 3 and k ≥ 35q, with matching exact results for q=3 cycles and wheels W_3k.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T15:25:02Z","title":"On zero-sum Ramsey numbers of cycles and wheels"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.14954","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-15T03:20:31.295043Z","id":"460bad04-3323-4951-9a65-82fb62a3cf08","model_set":{"reader":"grok-4.3"},"one_line_summary":"R(C_qk, Z_q) equals qk + q - 1 exactly for odd q ≥ 3 and k ≥ 35q, with matching exact results for q=3 cycles and wheels W_3k.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"","strongest_claim":"For every fixed odd q≥3 and every k≥35q, we obtain the exact value R(C_qk,Z_q)=qk+q-1. For q=3, R(C_3k,Z_3)=3k+2 for k≥2 and R(W_3k,Z_3)=3k+1 for k≥2.","weakest_assumption":"The upper bound for general q relies on Pikhurko's external result that R(C_2q,Z_q)≤35q²; if that bound is loose or inapplicable in the insertion step, the claimed exact value for large k would not hold."}},"verdict_id":"460bad04-3323-4951-9a65-82fb62a3cf08"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:80f0fb6c27f56efed0d4c1440500a2596204ca05ddbbd7859061be9de4d2fcf2","target":"record","created_at":"2026-05-17T23:38:55Z","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":"434b1584765c23e10b729ac46d391bda8adb39d32bf7ef49c4883b55be99946a","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T15:25:02Z","title_canon_sha256":"c503f365e8e9fd6a1eb0080731ad670406367cbda2e1fe7c132d7e1698eb76d1"},"schema_version":"1.0","source":{"id":"2605.14954","kind":"arxiv","version":1}},"canonical_sha256":"6ce65e676070411d0373a06b9bc14eeb02233ced53f67f44413f14c52f7a423d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6ce65e676070411d0373a06b9bc14eeb02233ced53f67f44413f14c52f7a423d","first_computed_at":"2026-05-17T23:38:55.356961Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:38:55.356961Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"r4pg8XgggvaDbBoxJbKo3F2yKkt4a535aZldJvxyQz55DRf7jm4rX6rX6DMpaeXM6dCqWhTcXf7BsUKhlqDHDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:38:55.357567Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.14954","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:80f0fb6c27f56efed0d4c1440500a2596204ca05ddbbd7859061be9de4d2fcf2","sha256:ccd8ecd4dbf8af7d88b0038f95cf01c0318659d6f8ee50171d428f6a9164cb07"],"state_sha256":"ccdb03f3113465f0808b8e3f7e74107ae9ac54ed47b52ba4f6e782399de6cc16"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IjAE+kp8OBIB0Pj6gl315nyl9IOaXI24/rGr1Xsh340T5eWevFVqI+zw7xcKdql+aNSyNN53DpGbPri0KGiGDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T19:56:52.508748Z","bundle_sha256":"a80b79dd996f42f66e0b09027b0e66dc530f4aa541a74b0e2b98e786e553c54b"}}