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Math.443(2024), Paper No. 109597","work_id":"44966e4f-0ff0-4698-812e-21d4f18e0cfd","year":2024}],"snapshot_sha256":"9060309db29dc2bbf98539939ce2b6722abafe253ad30d1e83b413b737e49ef9"},"source":{"id":"2605.17529","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T22:48:50.318803Z","id":"1a9e31fa-0e7f-418c-a8d5-33754bd649f8","model_set":{"reader":"grok-4.3"},"one_line_summary":"Counterexamples demonstrate that integer-coefficient derivative-span conditions fail to imply thickness or non-emptiness of common return-time sets for recurrence along Hardy field functions.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Integer-coefficient conditions on Hardy field functions fail to guarantee thick common return-time sets.","strongest_claim":"For the pair f1(t)=t^{3/2} and f2(t)=λ t^{3/2}+t where λ is irrational, every F in nabla_Z(f1,f2) satisfies lim |F(t)| in {0,∞}, yet there exists E subset N of positive density such that R_f1(E) cap R_f2(E) is piecewise syndetic but not thick; even under the full integer derivative-span condition the common return-time set may be empty.","weakest_assumption":"The constructions assume that elementary Bohr sets can be chosen to simultaneously achieve positive natural density, make the intersection piecewise syndetic but not thick, and satisfy the recurrence relations for the given Hardy field functions without hidden constraints from the field structure."}},"verdict_id":"1a9e31fa-0e7f-418c-a8d5-33754bd649f8"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:bb00a1c209909bce78362f4ad52bc54108ad1eb0f3ccbdb4f02882cd8855425e","target":"record","created_at":"2026-05-20T00:04:44Z","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":"f9ce8328df0b9455387b0d1edb9d77662405faefbbf6cdc3e3985278af5488bf","cross_cats_sorted":["math.CO","math.DS"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-17T16:26:45Z","title_canon_sha256":"2d082696a5ad1feaf047ba4dc11580df499983a765d75f5980d60c92796d1433"},"schema_version":"1.0","source":{"id":"2605.17529","kind":"arxiv","version":1}},"canonical_sha256":"7c8576bacd68ae076aaa165402a21503a0f9cbd7dde3e0d45761312f95684360","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7c8576bacd68ae076aaa165402a21503a0f9cbd7dde3e0d45761312f95684360","first_computed_at":"2026-05-20T00:04:44.224356Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:04:44.224356Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GD57FMekMMF74hpas1EK0qLnKdJckCAEwIlp93nnWl48tfmiQi01IgrJSLtiZTTR/r5C32fSMW1XrfeJ8PTtDQ==","signature_status":"signed_v1","signed_at":"2026-05-20T00:04:44.225198Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.17529","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bb00a1c209909bce78362f4ad52bc54108ad1eb0f3ccbdb4f02882cd8855425e","sha256:a7b1c1bcd8c8169e6e76cc056dac45b720c5ba7d5835f74e31fc4a02256cd24b"],"state_sha256":"199c4ee0262add18f539c4245b2f2a2c4a75a1f062c1bad40716b67e5797ed51"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"12dMcXGyniUNb6ddTmdI4rAHoIsVYkFzq0x209x+5xFz+Ct6LAgl1g73VZD6IlChCtGZ61/aXQmmgcJxPJD8CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T13:19:20.483953Z","bundle_sha256":"ec281eae67c156c99846025671fdb73c9daaa8024a985472aef181b665c0dd22"}}