{"paper":{"title":"Rigidity and Structural Asymmetry of Bounded Solutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"If solutions for both s and 1-conjugate(s) are bounded on [1,∞) then Re(s) must equal 1/2.","cross_cats":[],"primary_cat":"math.DS","authors_text":"Walid Oukil","submitted_at":"2026-03-16T15:45:16Z","abstract_excerpt":"In this manuscript, we introduce a family of parametrized non-homogeneous linear complex differential equations on $[1,\\infty)$, depending on a complex parameter. We identify a \"Rotation number hypothesis\" on the non-homogeneous term, which establishes a structural asymmetry: if two solutions with the same initial condition equal to $1$, corresponding respectively to the parameters $s$ and $1-\\overline{s}$ lying in the critical strip, are both bounded on $[1,+\\infty)$, then $\\Re(s) = \\tfrac{1}{2}$."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"if two solutions with the same initial condition equal to 1, corresponding respectively to the parameters s and 1−s̄ lying in the critical strip, are both bounded on [1,+∞), then Re(s) = 1/2.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The rotation number hypothesis imposed on the non-homogeneous term, which is required to establish the structural asymmetry between s and 1-conjugate(s).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Bounded solutions for parameters s and 1-conjugate(s) imply Re(s) = 1/2 under the rotation number hypothesis on the non-homogeneous term.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"If solutions for both s and 1-conjugate(s) are bounded on [1,∞) then Re(s) must equal 1/2.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"89508cc9709bd3a7944d0d6abea5640b7fac5c7613ffb7d71ce6e2087b11aaac"},"source":{"id":"2603.15716","kind":"arxiv","version":4},"verdict":{"id":"8c8ea362-ce5e-49d5-921e-6e5a72208d2b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T10:05:16.888504Z","strongest_claim":"if two solutions with the same initial condition equal to 1, corresponding respectively to the parameters s and 1−s̄ lying in the critical strip, are both bounded on [1,+∞), then Re(s) = 1/2.","one_line_summary":"Bounded solutions for parameters s and 1-conjugate(s) imply Re(s) = 1/2 under the rotation number hypothesis on the non-homogeneous term.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The rotation number hypothesis imposed on the non-homogeneous term, which is required to establish the structural asymmetry between s and 1-conjugate(s).","pith_extraction_headline":"If solutions for both s and 1-conjugate(s) are bounded on [1,∞) then Re(s) must equal 1/2."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.15716/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"}