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We prove the exact identity \\[ P_k = N_k - S_k + E_k \\] where $P_k = \\#\\{\\text{primes in } I_k\\}$, $N_k = 4k$ counts the odd integers in $I_k$, $S_k = \\sum_{n \\in I_k \\text{ odd}} r(n)$ is the total matrix multiplicity, and $E_k = \\sum_{n \\in I_k \\text{ odd}} (r(n)-1)$ measures the excess multiplicity of non-semiprime odd "},"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":"2605.21529","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-19T17:23:39Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"44218ec4fae9553379d8272141ed3f5911628fe14e73127b52573defa99edd46","abstract_canon_sha256":"d20d44e6b50c0f8a4513109a930b70a0b2245af6a0f960c248283aa0d3898e0c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T00:02:27.730818Z","signature_b64":"YZHWNMAHBqchqPLecnC7bgycx04/nOuzlLd/B6Fhh9wEH9KFwIEia4FEY3xv6r/UwolTtYQb9J0M+dN/wCWHDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ca4e370bf3e8ac2c1f65fe4a44531984169191d06813c3258e96e8a51dce38c","last_reissued_at":"2026-05-22T00:02:27.730362Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T00:02:27.730362Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Matrix-Theoretic Exact Formula for Counting Primes in Intervals Between Consecutive Odd Squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Wujie Shi","submitted_at":"2026-05-19T17:23:39Z","abstract_excerpt":"Let $I_k = [(2k-1)^2, (2k+1)^2)$ for $k \\geq 1$. Starting from the odd-composite matrix $(b_{ij})$ with $b_{ij} = (2i-1)(2j-1)$, introduced by the author in [1], we define for each odd integer $n$ the \\emph{matrix multiplicity} $r(n)$, the number of times $n$ appears in $B$. We prove the exact identity \\[ P_k = N_k - S_k + E_k \\] where $P_k = \\#\\{\\text{primes in } I_k\\}$, $N_k = 4k$ counts the odd integers in $I_k$, $S_k = \\sum_{n \\in I_k \\text{ odd}} r(n)$ is the total matrix multiplicity, and $E_k = \\sum_{n \\in I_k \\text{ odd}} (r(n)-1)$ measures the excess multiplicity of non-semiprime odd "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21529","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/2605.21529/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":"2605.21529","created_at":"2026-05-22T00:02:27.730436+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.21529v1","created_at":"2026-05-22T00:02:27.730436+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.21529","created_at":"2026-05-22T00:02:27.730436+00:00"},{"alias_kind":"pith_short_12","alias_value":"PSSOG4F7H2FM","created_at":"2026-05-22T00:02:27.730436+00:00"},{"alias_kind":"pith_short_16","alias_value":"PSSOG4F7H2FMFQPW","created_at":"2026-05-22T00:02:27.730436+00:00"},{"alias_kind":"pith_short_8","alias_value":"PSSOG4F7","created_at":"2026-05-22T00:02:27.730436+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/PSSOG4F7H2FMFQPWL7SKIRJRTB","json":"https://pith.science/pith/PSSOG4F7H2FMFQPWL7SKIRJRTB.json","graph_json":"https://pith.science/api/pith-number/PSSOG4F7H2FMFQPWL7SKIRJRTB/graph.json","events_json":"https://pith.science/api/pith-number/PSSOG4F7H2FMFQPWL7SKIRJRTB/events.json","paper":"https://pith.science/paper/PSSOG4F7"},"agent_actions":{"view_html":"https://pith.science/pith/PSSOG4F7H2FMFQPWL7SKIRJRTB","download_json":"https://pith.science/pith/PSSOG4F7H2FMFQPWL7SKIRJRTB.json","view_paper":"https://pith.science/paper/PSSOG4F7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.21529&json=true","fetch_graph":"https://pith.science/api/pith-number/PSSOG4F7H2FMFQPWL7SKIRJRTB/graph.json","fetch_events":"https://pith.science/api/pith-number/PSSOG4F7H2FMFQPWL7SKIRJRTB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PSSOG4F7H2FMFQPWL7SKIRJRTB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PSSOG4F7H2FMFQPWL7SKIRJRTB/action/storage_attestation","attest_author":"https://pith.science/pith/PSSOG4F7H2FMFQPWL7SKIRJRTB/action/author_attestation","sign_citation":"https://pith.science/pith/PSSOG4F7H2FMFQPWL7SKIRJRTB/action/citation_signature","submit_replication":"https://pith.science/pith/PSSOG4F7H2FMFQPWL7SKIRJRTB/action/replication_record"}},"created_at":"2026-05-22T00:02:27.730436+00:00","updated_at":"2026-05-22T00:02:27.730436+00:00"}