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We sharpen this to a quantitative rate: when the boundaries are $\\mathcal{C}^2$ near the limit point $\\bar z$, \\cCRM\\ converges Q-quadratically, with an asymptotic constant \\(\n  2\\max(\\kappa_X,\\kappa_Y)/\\omega \\) governed by the boundary curvatures $\\kappa_X,\\kappa_Y$ at $\\bar z$ and the local error-b"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that cCRM converges superlinearly when aff(X)=aff(Y), ri(X)∩ri(Y)≠∅, and the relative boundaries are C^1 of appropriate relative dimension; and Q-quadratically when the relative boundaries are C^2, with explicit asymptotic constant expressed in terms of the boundary curvatures at the limit point and the local error-bound constant.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that aff(X)=aff(Y) and ri(X)∩ri(Y)≠∅ together with the relative boundaries being C^1 (or C^2) hypersurfaces of appropriate dimension; 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and Q-quadratically when the relative boundaries are C^2, with explicit asymptotic constant expressed in terms of the boundary curvatures at the limit point and the local error-bound constant.","one_line_summary":"cCRM achieves Q-quadratic convergence to solutions of find z in X cap Y when aff(X)=aff(Y), ri(X) cap ri(Y) nonempty, and relative boundaries are C^2, with explicit rate constant from curvatures and local error bound.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that aff(X)=aff(Y) and ri(X)∩ri(Y)≠∅ together with the relative boundaries being C^1 (or C^2) hypersurfaces of appropriate dimension; the paper explicitly leaves the case aff(X)≠aff(Y) as open.","pith_extraction_headline":"The centralized circumcentered-reflection method converges Q-quadratically when sets share an affine hull, their relative interiors intersect, and relative boundaries are twice differentiable."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.11450/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":"2604.11450","created_at":"2026-06-09T01:05:17.226612+00:00"},{"alias_kind":"arxiv_version","alias_value":"2604.11450v2","created_at":"2026-06-09T01:05:17.226612+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.11450","created_at":"2026-06-09T01:05:17.226612+00:00"},{"alias_kind":"pith_short_12","alias_value":"RI5J4L4AMRF7","created_at":"2026-06-09T01:05:17.226612+00:00"},{"alias_kind":"pith_short_16","alias_value":"RI5J4L4AMRF7D4GT","created_at":"2026-06-09T01:05:17.226612+00:00"},{"alias_kind":"pith_short_8","alias_value":"RI5J4L4A","created_at":"2026-06-09T01:05:17.226612+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2604.26228","citing_title":"On the geometry of circumcentric directions of cones","ref_index":16,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RI5J4L4AMRF7D4GTJADPWYHOPR","json":"https://pith.science/pith/RI5J4L4AMRF7D4GTJADPWYHOPR.json","graph_json":"https://pith.science/api/pith-number/RI5J4L4AMRF7D4GTJADPWYHOPR/graph.json","events_json":"https://pith.science/api/pith-number/RI5J4L4AMRF7D4GTJADPWYHOPR/events.json","paper":"https://pith.science/paper/RI5J4L4A"},"agent_actions":{"view_html":"https://pith.science/pith/RI5J4L4AMRF7D4GTJADPWYHOPR","download_json":"https://pith.science/pith/RI5J4L4AMRF7D4GTJADPWYHOPR.json","view_paper":"https://pith.science/paper/RI5J4L4A","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2604.11450&json=true","fetch_graph":"https://pith.science/api/pith-number/RI5J4L4AMRF7D4GTJADPWYHOPR/graph.json","fetch_events":"https://pith.science/api/pith-number/RI5J4L4AMRF7D4GTJADPWYHOPR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RI5J4L4AMRF7D4GTJADPWYHOPR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RI5J4L4AMRF7D4GTJADPWYHOPR/action/storage_attestation","attest_author":"https://pith.science/pith/RI5J4L4AMRF7D4GTJADPWYHOPR/action/author_attestation","sign_citation":"https://pith.science/pith/RI5J4L4AMRF7D4GTJADPWYHOPR/action/citation_signature","submit_replication":"https://pith.science/pith/RI5J4L4AMRF7D4GTJADPWYHOPR/action/replication_record"}},"created_at":"2026-06-09T01:05:17.226612+00:00","updated_at":"2026-06-09T01:05:17.226612+00:00"}