{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:VXMG55XPP2I3ZVPE6LXQBMSY67","short_pith_number":"pith:VXMG55XP","canonical_record":{"source":{"id":"2505.01628","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-05-02T23:33:38Z","cross_cats_sorted":[],"title_canon_sha256":"cc4f1465f4c60b910a600c2fb8bfe55df3e5f0a596b0257e3127717b9a257536","abstract_canon_sha256":"9a62a5b30e0d96ad228f17340ab15956cbffbbea7d0bd9e0f034bc38e98b76fb"},"schema_version":"1.0"},"canonical_sha256":"add86ef6ef7e91bcd5e4f2ef00b258f7c20c8e87ea698839ca027c92726baef8","source":{"kind":"arxiv","id":"2505.01628","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2505.01628","created_at":"2026-07-05T10:58:01Z"},{"alias_kind":"arxiv_version","alias_value":"2505.01628v1","created_at":"2026-07-05T10:58:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.01628","created_at":"2026-07-05T10:58:01Z"},{"alias_kind":"pith_short_12","alias_value":"VXMG55XPP2I3","created_at":"2026-07-05T10:58:01Z"},{"alias_kind":"pith_short_16","alias_value":"VXMG55XPP2I3ZVPE","created_at":"2026-07-05T10:58:01Z"},{"alias_kind":"pith_short_8","alias_value":"VXMG55XP","created_at":"2026-07-05T10:58:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:VXMG55XPP2I3ZVPE6LXQBMSY67","target":"record","payload":{"canonical_record":{"source":{"id":"2505.01628","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-05-02T23:33:38Z","cross_cats_sorted":[],"title_canon_sha256":"cc4f1465f4c60b910a600c2fb8bfe55df3e5f0a596b0257e3127717b9a257536","abstract_canon_sha256":"9a62a5b30e0d96ad228f17340ab15956cbffbbea7d0bd9e0f034bc38e98b76fb"},"schema_version":"1.0"},"canonical_sha256":"add86ef6ef7e91bcd5e4f2ef00b258f7c20c8e87ea698839ca027c92726baef8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:58:01.035156Z","signature_b64":"Ei+TgD860/ea5qYGv5ttmj27+dn5wdSeMTXUte1bIejwOQveEZ+mH5cTgkyPBNYGKuH7MZXywMNzlD7b0KukCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"add86ef6ef7e91bcd5e4f2ef00b258f7c20c8e87ea698839ca027c92726baef8","last_reissued_at":"2026-07-05T10:58:01.034575Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:58:01.034575Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2505.01628","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":"2026-07-05T10:58:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OPwBre8twjCe0cUir4y2NsU3mN2O+l77DBj2QHzjfkQAc67OEfxOffOlCe1WaoYZVm6OOnfT8leLA+Dy9sgPBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T04:01:10.270856Z"},"content_sha256":"7187c15f4f2e68ac667472927695a1bc5c476cebcf14711c2479c840c34aa9e5","schema_version":"1.0","event_id":"sha256:7187c15f4f2e68ac667472927695a1bc5c476cebcf14711c2479c840c34aa9e5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:VXMG55XPP2I3ZVPE6LXQBMSY67","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Satisfiability Threshold for K-XOR Games","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jared A. Hughes, J. William Helton","submitted_at":"2025-05-02T23:33:38Z","abstract_excerpt":"A $K$-XORGAME system corresponds to a $K$-XORSAT system with the additional restriction that the variables divide uniformly into $K$ blocks. This forms a system of $m$ equations with $K n$ unknowns over $\\mathbb{Z}_2$, and a perfect strategy corresponds to a solution to these equations. Equivalently, such equations correspond to colorings of a $K$-uniform $K$-partite hypergraph. This paper proves that the satisfiability threshold of $m/n$ for $K$-XORGAME problems exists and equals the satisfiability threshold for $K$-XORSAT."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.01628","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.01628/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T10:58:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+SzuGNWESS5mGvDV5owPz9O0+JrusbusNEraHb/WLohJ+kXxlajFOxgXzZGprulARZrnwlUXDVglEZlyRkcdAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T04:01:10.271583Z"},"content_sha256":"2637baf01f38f8f468af6a02e869b3cb72649445369a2b96f71519220be08ed9","schema_version":"1.0","event_id":"sha256:2637baf01f38f8f468af6a02e869b3cb72649445369a2b96f71519220be08ed9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VXMG55XPP2I3ZVPE6LXQBMSY67/bundle.json","state_url":"https://pith.science/pith/VXMG55XPP2I3ZVPE6LXQBMSY67/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VXMG55XPP2I3ZVPE6LXQBMSY67/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":"2026-08-22T04:01:10Z","links":{"resolver":"https://pith.science/pith/VXMG55XPP2I3ZVPE6LXQBMSY67","bundle":"https://pith.science/pith/VXMG55XPP2I3ZVPE6LXQBMSY67/bundle.json","state":"https://pith.science/pith/VXMG55XPP2I3ZVPE6LXQBMSY67/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VXMG55XPP2I3ZVPE6LXQBMSY67/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:VXMG55XPP2I3ZVPE6LXQBMSY67","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":"9a62a5b30e0d96ad228f17340ab15956cbffbbea7d0bd9e0f034bc38e98b76fb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-05-02T23:33:38Z","title_canon_sha256":"cc4f1465f4c60b910a600c2fb8bfe55df3e5f0a596b0257e3127717b9a257536"},"schema_version":"1.0","source":{"id":"2505.01628","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2505.01628","created_at":"2026-07-05T10:58:01Z"},{"alias_kind":"arxiv_version","alias_value":"2505.01628v1","created_at":"2026-07-05T10:58:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.01628","created_at":"2026-07-05T10:58:01Z"},{"alias_kind":"pith_short_12","alias_value":"VXMG55XPP2I3","created_at":"2026-07-05T10:58:01Z"},{"alias_kind":"pith_short_16","alias_value":"VXMG55XPP2I3ZVPE","created_at":"2026-07-05T10:58:01Z"},{"alias_kind":"pith_short_8","alias_value":"VXMG55XP","created_at":"2026-07-05T10:58:01Z"}],"graph_snapshots":[{"event_id":"sha256:2637baf01f38f8f468af6a02e869b3cb72649445369a2b96f71519220be08ed9","target":"graph","created_at":"2026-07-05T10:58:01Z","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":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2505.01628/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A $K$-XORGAME system corresponds to a $K$-XORSAT system with the additional restriction that the variables divide uniformly into $K$ blocks. This forms a system of $m$ equations with $K n$ unknowns over $\\mathbb{Z}_2$, and a perfect strategy corresponds to a solution to these equations. Equivalently, such equations correspond to colorings of a $K$-uniform $K$-partite hypergraph. This paper proves that the satisfiability threshold of $m/n$ for $K$-XORGAME problems exists and equals the satisfiability threshold for $K$-XORSAT.","authors_text":"Jared A. Hughes, J. William Helton","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-05-02T23:33:38Z","title":"The Satisfiability Threshold for K-XOR Games"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.01628","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:7187c15f4f2e68ac667472927695a1bc5c476cebcf14711c2479c840c34aa9e5","target":"record","created_at":"2026-07-05T10:58:01Z","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":"9a62a5b30e0d96ad228f17340ab15956cbffbbea7d0bd9e0f034bc38e98b76fb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-05-02T23:33:38Z","title_canon_sha256":"cc4f1465f4c60b910a600c2fb8bfe55df3e5f0a596b0257e3127717b9a257536"},"schema_version":"1.0","source":{"id":"2505.01628","kind":"arxiv","version":1}},"canonical_sha256":"add86ef6ef7e91bcd5e4f2ef00b258f7c20c8e87ea698839ca027c92726baef8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"add86ef6ef7e91bcd5e4f2ef00b258f7c20c8e87ea698839ca027c92726baef8","first_computed_at":"2026-07-05T10:58:01.034575Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:58:01.034575Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ei+TgD860/ea5qYGv5ttmj27+dn5wdSeMTXUte1bIejwOQveEZ+mH5cTgkyPBNYGKuH7MZXywMNzlD7b0KukCw==","signature_status":"signed_v1","signed_at":"2026-07-05T10:58:01.035156Z","signed_message":"canonical_sha256_bytes"},"source_id":"2505.01628","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7187c15f4f2e68ac667472927695a1bc5c476cebcf14711c2479c840c34aa9e5","sha256:2637baf01f38f8f468af6a02e869b3cb72649445369a2b96f71519220be08ed9"],"state_sha256":"60fd5a690d503c6b787d5b78e057f843e7cbbffdef2ae7c65fa1ef27cae6f512"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CEWD0mIc9jIHXGh+mZP2lojwSVrcfQBwkZFPpozdpCemHcQxh7NvmF3fUusovRxrgmIIHtNPmKy/Y/W84UpeAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-22T04:01:10.277202Z","bundle_sha256":"bc9b0edd601f7212870c998789a233a432cf94a1216b6502d1d9288fd9e3ee99"}}