{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:5HGXRUOGHC7ZDY2TVLWLD2XT5Y","short_pith_number":"pith:5HGXRUOG","canonical_record":{"source":{"id":"2607.22347","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-24T14:25:24Z","cross_cats_sorted":[],"title_canon_sha256":"f430138ac935a42f8c47e3b8a88cb4f18f93481270b0c12b625566fe1eb97a6a","abstract_canon_sha256":"b196aa2cccfa326a6d939007c4e6d8f7c42d3bb4c5528673ba76b3e158fed3ac"},"schema_version":"1.0"},"canonical_sha256":"e9cd78d1c638bf91e353aaecb1eaf3ee0f11a9e497e51d52caf397a1cdd2b716","source":{"kind":"arxiv","id":"2607.22347","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.22347","created_at":"2026-07-27T01:21:15Z"},{"alias_kind":"arxiv_version","alias_value":"2607.22347v1","created_at":"2026-07-27T01:21:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.22347","created_at":"2026-07-27T01:21:15Z"},{"alias_kind":"pith_short_12","alias_value":"5HGXRUOGHC7Z","created_at":"2026-07-27T01:21:15Z"},{"alias_kind":"pith_short_16","alias_value":"5HGXRUOGHC7ZDY2T","created_at":"2026-07-27T01:21:15Z"},{"alias_kind":"pith_short_8","alias_value":"5HGXRUOG","created_at":"2026-07-27T01:21:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:5HGXRUOGHC7ZDY2TVLWLD2XT5Y","target":"record","payload":{"canonical_record":{"source":{"id":"2607.22347","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-24T14:25:24Z","cross_cats_sorted":[],"title_canon_sha256":"f430138ac935a42f8c47e3b8a88cb4f18f93481270b0c12b625566fe1eb97a6a","abstract_canon_sha256":"b196aa2cccfa326a6d939007c4e6d8f7c42d3bb4c5528673ba76b3e158fed3ac"},"schema_version":"1.0"},"canonical_sha256":"e9cd78d1c638bf91e353aaecb1eaf3ee0f11a9e497e51d52caf397a1cdd2b716","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-27T01:21:15.872696Z","signature_b64":"aDP99RHbp9DGqcPs0V+O9MaHiXtihzu5c+YFxrRTyE6Yw1gQ08Hk29q64LeTLNn8Q1/GksOi5LHomwBJcQvxAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e9cd78d1c638bf91e353aaecb1eaf3ee0f11a9e497e51d52caf397a1cdd2b716","last_reissued_at":"2026-07-27T01:21:15.871832Z","signature_status":"signed_v1","first_computed_at":"2026-07-27T01:21:15.871832Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.22347","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":"2026-07-27T01:21:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lOPeZim8s7NEO3J7/kuCkuJooK0tR7cx2z53j/jBBoCbggDsMeS8F46/JDhKkbNQRi3KIPPx8PlZWzWPu3TsBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T22:54:21.826780Z"},"content_sha256":"6f2fc963d840d24bb70ffe8430661a64ff886dac854c8880ec4f0b2cfa8b71d1","schema_version":"1.0","event_id":"sha256:6f2fc963d840d24bb70ffe8430661a64ff886dac854c8880ec4f0b2cfa8b71d1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:5HGXRUOGHC7ZDY2TVLWLD2XT5Y","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Prime-Interval Algebras","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Joseph M. Shunia","submitted_at":"2026-07-24T14:25:24Z","abstract_excerpt":"Starting from a positive integer $n$ and no a priori information about the primes above it, we construct a polynomial quotient ring that recovers exactly the primes in $(n,2n]$ from a single modular exponentiation. The primes occur simultaneously as the nonzero monomial degrees of the resulting polynomial remainder, and each coefficient independently certifies its corresponding prime through its additive order.\n  When $n=p_k$ is prime, the least nonzero degree is $p_{k+1}$. Thus the next prime is recovered from the preceding prime alone, without using the index $k$, the prime-counting function"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.22347","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/2607.22347/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":"2026-07-27T01:21:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1Qa8YXtTS8g1RNVkWJIJD3bxtXzEyB/hMHrHb8gagOxipwkP/F0bvYI1cVf9ug2Ic0cuz7ZC4xJh/81iZkysDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T22:54:21.827291Z"},"content_sha256":"63149d0b05fa08a42b3d2ba5a02a3ccae566b88c36a7f7f751f5b16b2353ca89","schema_version":"1.0","event_id":"sha256:63149d0b05fa08a42b3d2ba5a02a3ccae566b88c36a7f7f751f5b16b2353ca89"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5HGXRUOGHC7ZDY2TVLWLD2XT5Y/bundle.json","state_url":"https://pith.science/pith/5HGXRUOGHC7ZDY2TVLWLD2XT5Y/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5HGXRUOGHC7ZDY2TVLWLD2XT5Y/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":"2026-08-15T22:54:21Z","links":{"resolver":"https://pith.science/pith/5HGXRUOGHC7ZDY2TVLWLD2XT5Y","bundle":"https://pith.science/pith/5HGXRUOGHC7ZDY2TVLWLD2XT5Y/bundle.json","state":"https://pith.science/pith/5HGXRUOGHC7ZDY2TVLWLD2XT5Y/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5HGXRUOGHC7ZDY2TVLWLD2XT5Y/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:5HGXRUOGHC7ZDY2TVLWLD2XT5Y","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":"b196aa2cccfa326a6d939007c4e6d8f7c42d3bb4c5528673ba76b3e158fed3ac","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-24T14:25:24Z","title_canon_sha256":"f430138ac935a42f8c47e3b8a88cb4f18f93481270b0c12b625566fe1eb97a6a"},"schema_version":"1.0","source":{"id":"2607.22347","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.22347","created_at":"2026-07-27T01:21:15Z"},{"alias_kind":"arxiv_version","alias_value":"2607.22347v1","created_at":"2026-07-27T01:21:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.22347","created_at":"2026-07-27T01:21:15Z"},{"alias_kind":"pith_short_12","alias_value":"5HGXRUOGHC7Z","created_at":"2026-07-27T01:21:15Z"},{"alias_kind":"pith_short_16","alias_value":"5HGXRUOGHC7ZDY2T","created_at":"2026-07-27T01:21:15Z"},{"alias_kind":"pith_short_8","alias_value":"5HGXRUOG","created_at":"2026-07-27T01:21:15Z"}],"graph_snapshots":[{"event_id":"sha256:63149d0b05fa08a42b3d2ba5a02a3ccae566b88c36a7f7f751f5b16b2353ca89","target":"graph","created_at":"2026-07-27T01:21:15Z","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/2607.22347/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Starting from a positive integer $n$ and no a priori information about the primes above it, we construct a polynomial quotient ring that recovers exactly the primes in $(n,2n]$ from a single modular exponentiation. The primes occur simultaneously as the nonzero monomial degrees of the resulting polynomial remainder, and each coefficient independently certifies its corresponding prime through its additive order.\n  When $n=p_k$ is prime, the least nonzero degree is $p_{k+1}$. Thus the next prime is recovered from the preceding prime alone, without using the index $k$, the prime-counting function","authors_text":"Joseph M. Shunia","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-24T14:25:24Z","title":"Prime-Interval Algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.22347","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":"sha256:6f2fc963d840d24bb70ffe8430661a64ff886dac854c8880ec4f0b2cfa8b71d1","target":"record","created_at":"2026-07-27T01:21:15Z","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":"b196aa2cccfa326a6d939007c4e6d8f7c42d3bb4c5528673ba76b3e158fed3ac","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-24T14:25:24Z","title_canon_sha256":"f430138ac935a42f8c47e3b8a88cb4f18f93481270b0c12b625566fe1eb97a6a"},"schema_version":"1.0","source":{"id":"2607.22347","kind":"arxiv","version":1}},"canonical_sha256":"e9cd78d1c638bf91e353aaecb1eaf3ee0f11a9e497e51d52caf397a1cdd2b716","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e9cd78d1c638bf91e353aaecb1eaf3ee0f11a9e497e51d52caf397a1cdd2b716","first_computed_at":"2026-07-27T01:21:15.871832Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-27T01:21:15.871832Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aDP99RHbp9DGqcPs0V+O9MaHiXtihzu5c+YFxrRTyE6Yw1gQ08Hk29q64LeTLNn8Q1/GksOi5LHomwBJcQvxAA==","signature_status":"signed_v1","signed_at":"2026-07-27T01:21:15.872696Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.22347","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6f2fc963d840d24bb70ffe8430661a64ff886dac854c8880ec4f0b2cfa8b71d1","sha256:63149d0b05fa08a42b3d2ba5a02a3ccae566b88c36a7f7f751f5b16b2353ca89"],"state_sha256":"339be70febc362ed19ef6a6776f1910c9e38d4107e8972159ee0a65eeaa713be"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OOvM5I7QeXZq+S/PX+oRjtzSbMr1sljT9viEaOn7R6S3GuO6B6IBtc9tV4GELHKbHuLqe5llsdXG365pzk0+CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-15T22:54:21.830733Z","bundle_sha256":"62b81c3fe77f2638f5d043e4d37170660cacf0b91029efee1481da487fd15290"}}