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However, the mechanisms governing the irregularities observed in their sequence and linking them to physical systems remained unclear. Here, it is shown that prime gaps at different separation distances follow a function depending on that distance and can be described by an iterative map which predicts the primary growth of successive primes. On the other hand, the analysis of remaining fluctuations reveals the existence of a well-defined cohomological structure, where the deterministic ","authors_text":"Marzena Ciszak","cross_cats":["math.NT"],"headline":"Prime gaps follow a distance-dependent iterative map whose cohomological equation is solved by the logarithmic integral function.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2026-05-17T19:39:50Z","title":"Iterative maps emerging from cohomological structure of primes"},"references":{"count":28,"internal_anchors":0,"resolved_work":28,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"ln(p(n+τ)) = (1 + 1 24)p(n)− 1 24(p(n) + 1","work_id":"7adefb3b-d4e0-4a3d-8b6d-5414dee6e76f","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"ln(p(n)) +τ[ln(p(n) + 2πτ)− 1 24(ln(p(n)))2] +ε(16) 9 It is clearly seen that the functional relation holds: f(p(n+τ)) =f(p(n)) +g(p(n);τ) +ε(17) wheref(x) = 1 24(25x−(x+ 1","work_id":"9e4ba953-a803-4178-a546-2e48687635e2","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"Note that the functiongis positive and monotonically decreasing","work_id":"44aca53f-3d79-4e21-a43f-80991fb00331","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Solving this equation forπ c gives the local number of primes between two numbersx1 andx 2 (see Fig","work_id":"d3194ceb-91e0-405b-9b3a-33aadcbc952f","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Narkiewicz, The Development of Prime Number Theory","work_id":"d3300dc7-d7eb-46ae-9066-397b13ea74df","year":2000}],"snapshot_sha256":"e864b63abac9301f40ac5f723554f3eb902c4f3675af03dcae6ef91a8a859b7c"},"source":{"id":"2605.17622","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T22:20:58.737896Z","id":"d75d9f2b-126d-4e41-a90f-cb8ce36edae7","model_set":{"reader":"grok-4.3"},"one_line_summary":"Prime gaps are described by an iterative map, and fluctuations encode a cohomological structure whose solution is the logarithmic integral.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Prime gaps follow a distance-dependent iterative map whose cohomological equation is solved by the logarithmic integral function.","strongest_claim":"the solution to this cohomological equation turns out to be the logarithmic integral function.","weakest_assumption":"The analysis of remaining fluctuations reveals the existence of a well-defined cohomological structure, where the deterministic functional relation holds for primes up to small decaying fluctuations."}},"verdict_id":"d75d9f2b-126d-4e41-a90f-cb8ce36edae7"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1d80b63aeb0c48c0730374fcddb989e80b0cbd5d923d84fe1b1fa4dd3596f172","target":"record","created_at":"2026-05-20T00:04:49Z","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":"aa26a16193484365d0faa7d42f10da27a37973a92adc42d73d905cf83063a23f","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2026-05-17T19:39:50Z","title_canon_sha256":"7ed34c51a0e73041cb900326e729d83dadbeebfc64d179402ef896bb54b277c5"},"schema_version":"1.0","source":{"id":"2605.17622","kind":"arxiv","version":1}},"canonical_sha256":"4d64545e912586400af3b882b22dabe4678309310f3a89da0f9ec2a98be9acbd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4d64545e912586400af3b882b22dabe4678309310f3a89da0f9ec2a98be9acbd","first_computed_at":"2026-05-20T00:04:49.209959Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:04:49.209959Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ai/Nh3YJR8UBdHcqoi/9/8AzPrUUrHaN4hWp+JM5XkAVu8WIv29W+JZXXzlQb4MH6NPOihp7BhsfiU524Z41AA==","signature_status":"signed_v1","signed_at":"2026-05-20T00:04:49.210872Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.17622","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1d80b63aeb0c48c0730374fcddb989e80b0cbd5d923d84fe1b1fa4dd3596f172","sha256:59b2136fde22e5ff97209242aba3e69acb11488a72500073abd913d1e376bb5a"],"state_sha256":"a5e10bf492f7dce390e4e4f8ec1a7432c5ce07967e9a6769c0b870b159259c02"}