{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:5HGXRUOGHC7ZDY2TVLWLD2XT5Y","short_pith_number":"pith:5HGXRUOG","schema_version":"1.0","canonical_sha256":"e9cd78d1c638bf91e353aaecb1eaf3ee0f11a9e497e51d52caf397a1cdd2b716","source":{"kind":"arxiv","id":"2607.22347","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.22347","created_at":"2026-07-27T01:21:15.872278+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.22347v1","created_at":"2026-07-27T01:21:15.872278+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.22347","created_at":"2026-07-27T01:21:15.872278+00:00"},{"alias_kind":"pith_short_12","alias_value":"5HGXRUOGHC7Z","created_at":"2026-07-27T01:21:15.872278+00:00"},{"alias_kind":"pith_short_16","alias_value":"5HGXRUOGHC7ZDY2T","created_at":"2026-07-27T01:21:15.872278+00:00"},{"alias_kind":"pith_short_8","alias_value":"5HGXRUOG","created_at":"2026-07-27T01:21:15.872278+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5HGXRUOGHC7ZDY2TVLWLD2XT5Y","json":"https://pith.science/pith/5HGXRUOGHC7ZDY2TVLWLD2XT5Y.json","graph_json":"https://pith.science/api/pith-number/5HGXRUOGHC7ZDY2TVLWLD2XT5Y/graph.json","events_json":"https://pith.science/api/pith-number/5HGXRUOGHC7ZDY2TVLWLD2XT5Y/events.json","paper":"https://pith.science/paper/5HGXRUOG"},"agent_actions":{"view_html":"https://pith.science/pith/5HGXRUOGHC7ZDY2TVLWLD2XT5Y","download_json":"https://pith.science/pith/5HGXRUOGHC7ZDY2TVLWLD2XT5Y.json","view_paper":"https://pith.science/paper/5HGXRUOG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.22347&json=true","fetch_graph":"https://pith.science/api/pith-number/5HGXRUOGHC7ZDY2TVLWLD2XT5Y/graph.json","fetch_events":"https://pith.science/api/pith-number/5HGXRUOGHC7ZDY2TVLWLD2XT5Y/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5HGXRUOGHC7ZDY2TVLWLD2XT5Y/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5HGXRUOGHC7ZDY2TVLWLD2XT5Y/action/storage_attestation","attest_author":"https://pith.science/pith/5HGXRUOGHC7ZDY2TVLWLD2XT5Y/action/author_attestation","sign_citation":"https://pith.science/pith/5HGXRUOGHC7ZDY2TVLWLD2XT5Y/action/citation_signature","submit_replication":"https://pith.science/pith/5HGXRUOGHC7ZDY2TVLWLD2XT5Y/action/replication_record"}},"created_at":"2026-07-27T01:21:15.872278+00:00","updated_at":"2026-07-27T01:21:15.872278+00:00"}