{"paper":{"title":"From Witness-Space Sharpness To Family-Pointwise Exactness For The Solvability Complexity Index","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"Witness-space sharpness coincides with worst-case exactness but is strictly weaker than family-pointwise exactness for SCI families","cross_cats":["cs.CC","math.DS","math.FA"],"primary_cat":"math.LO","authors_text":"Christopher Sorg","submitted_at":"2026-04-14T13:59:35Z","abstract_excerpt":"We study how exact Solvability Complexity Index (SCI) statements should be formulated for families of computational problems rather than for single problems. While the equality \\(\\mathrm{SCI}_G (\\mathcal P)=k\\) is unambiguous for an individual computational problem \\(\\mathcal P\\), the family setting requires one to distinguish family-pointwise exactness, witness-space sharpness, and worst-case exactness. We formalize this trichotomy, prove that witness-space sharpness coincides with worst-case exactness but is, in general, strictly weaker than family-pointwise exactness, and give a canonical s"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that witness-space sharpness coincides with worst-case exactness but is, in general, strictly weaker than family-pointwise exactness, and give a canonical source-family example witnessing the strictness. We then establish two positive upgrade theorems: an abstract pullback principle and a concrete finite-query criterion guaranteeing that witness-space sharpness upgrades to family-pointwise exactness.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The upgrade theorems and transport results rest on the assumption that the families admit decoder-regular finite-query transports and that the decoder classes satisfy the regularity conditions needed for the pullback principle to apply; this is invoked when moving from the abstract preorder to concrete families such as integration and spectral decision.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Formalizes a trichotomy of exactness notions for SCI on families, proves witness-space sharpness equals worst-case exactness but is strictly weaker than family-pointwise exactness, supplies upgrade theorems, and illustrates with integration and spectral families.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Witness-space sharpness coincides with worst-case exactness but is strictly weaker than family-pointwise exactness for SCI families","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"1494f3253d017b8061a5ebf8d28a23caad6f7b207eb37f0f30550c863bcedcd5"},"source":{"id":"2604.12750","kind":"arxiv","version":2},"verdict":{"id":"25de2953-c52f-4420-9ef3-716e2c8c8baa","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T17:45:21.701451Z","strongest_claim":"We prove that witness-space sharpness coincides with worst-case exactness but is, in general, strictly weaker than family-pointwise exactness, and give a canonical source-family example witnessing the strictness. We then establish two positive upgrade theorems: an abstract pullback principle and a concrete finite-query criterion guaranteeing that witness-space sharpness upgrades to family-pointwise exactness.","one_line_summary":"Formalizes a trichotomy of exactness notions for SCI on families, proves witness-space sharpness equals worst-case exactness but is strictly weaker than family-pointwise exactness, supplies upgrade theorems, and illustrates with integration and spectral families.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The upgrade theorems and transport results rest on the assumption that the families admit decoder-regular finite-query transports and that the decoder classes satisfy the regularity conditions needed for the pullback principle to apply; this is invoked when moving from the abstract preorder to concrete families such as integration and spectral decision.","pith_extraction_headline":"Witness-space sharpness coincides with worst-case exactness but is strictly weaker than family-pointwise exactness for SCI families"},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.12750/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":11,"sample":[{"doi":"","year":2015,"title":"Computing spectra--on the solvability complexity index hierarchy and towers of algorithms","work_id":"73652e4a-a445-4a7e-819e-a2e123411869","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"Colbrook and Anders C","work_id":"1c782bcf-f11c-40d3-86d9-6f36b1706081","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1984,"title":"Davis and Philip Rabinowitz","work_id":"83a3a204-0419-4f3c-aae9-03f1255756a0","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2011,"title":"On the solvability complexity index, the -pseudospectrum and approximations of spectra of operators","work_id":"7be73212-eeb9-4d2b-b86e-4e320003990e","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1995,"title":"Perturbation theory for linear operators","work_id":"b27acd16-892a-4118-8328-359994702800","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":11,"snapshot_sha256":"4451e256e9319650b764fbd1c8eb8e20ac2cf1c388112562e9f2d87159111375","internal_anchors":2},"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"}