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[npj Quantum Inf. 11, 82 (2025)] proved the exact quantum maximum of the doubly-tilted CHSH functional $B_{\\alpha\\beta}=\\alpha\\langle A_0\\rangle+\\beta\\langle B_0\\rangle+\\mathrm{CHSH}$ and observed that the NPA level required to reach it grows without evident bound toward the critical line $\\alpha+\\beta=2$. We quantify the mechanism on the symmetric slice $s=2-\\alpha-\\beta$: (i) the quantum value leaves the local bound cubically, $c_Q=4-s+s^3/6-s^4/36+O(s^5)$; (ii) each NPA level overshoots quadratically, $c_k(s)=4-s+a_k s^2+O(s^3)$, with the almost-quantum coefficient computed ex"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.13774","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2026-07-15T12:34:25Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"f64fd537b36c0617a7da2d6d0af08105b22d1589ff55fd31bb13599d17be0438","abstract_canon_sha256":"5dd48bf37c7a8b6bbb394e861e9b35cf8e83439228bca25330999cac320c7720"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-16T01:23:05.903748Z","signature_b64":"LxDGfmHGBT8nLy+n/GRbeUgryU1vyFK8Vv8iJuugmoNqTPldNo2jzEYoaGt3XNZIYJ6+8gCROVBGDBzsQdfhCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a540b82a132785ed270e33a7d9cc0355f379f92124cc26211610b567c274ea42","last_reissued_at":"2026-07-16T01:23:05.902567Z","signature_status":"signed_v1","first_computed_at":"2026-07-16T01:23:05.902567Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A phase transition in the exactness of the NPA hierarchy at the critical doubly-tilted CHSH functional","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"quant-ph","authors_text":"Anton Pakhunov","submitted_at":"2026-07-15T12:34:25Z","abstract_excerpt":"Gigena et al. [npj Quantum Inf. 11, 82 (2025)] proved the exact quantum maximum of the doubly-tilted CHSH functional $B_{\\alpha\\beta}=\\alpha\\langle A_0\\rangle+\\beta\\langle B_0\\rangle+\\mathrm{CHSH}$ and observed that the NPA level required to reach it grows without evident bound toward the critical line $\\alpha+\\beta=2$. We quantify the mechanism on the symmetric slice $s=2-\\alpha-\\beta$: (i) the quantum value leaves the local bound cubically, $c_Q=4-s+s^3/6-s^4/36+O(s^5)$; (ii) each NPA level overshoots quadratically, $c_k(s)=4-s+a_k s^2+O(s^3)$, with the almost-quantum coefficient computed ex"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.13774","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.13774/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":"2607.13774","created_at":"2026-07-16T01:23:05.903342+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.13774v1","created_at":"2026-07-16T01:23:05.903342+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.13774","created_at":"2026-07-16T01:23:05.903342+00:00"},{"alias_kind":"pith_short_12","alias_value":"UVALQKQTE6C6","created_at":"2026-07-16T01:23:05.903342+00:00"},{"alias_kind":"pith_short_16","alias_value":"UVALQKQTE6C62JYO","created_at":"2026-07-16T01:23:05.903342+00:00"},{"alias_kind":"pith_short_8","alias_value":"UVALQKQT","created_at":"2026-07-16T01:23:05.903342+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UVALQKQTE6C62JYOGOT5TTADKX","json":"https://pith.science/pith/UVALQKQTE6C62JYOGOT5TTADKX.json","graph_json":"https://pith.science/api/pith-number/UVALQKQTE6C62JYOGOT5TTADKX/graph.json","events_json":"https://pith.science/api/pith-number/UVALQKQTE6C62JYOGOT5TTADKX/events.json","paper":"https://pith.science/paper/UVALQKQT"},"agent_actions":{"view_html":"https://pith.science/pith/UVALQKQTE6C62JYOGOT5TTADKX","download_json":"https://pith.science/pith/UVALQKQTE6C62JYOGOT5TTADKX.json","view_paper":"https://pith.science/paper/UVALQKQT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.13774&json=true","fetch_graph":"https://pith.science/api/pith-number/UVALQKQTE6C62JYOGOT5TTADKX/graph.json","fetch_events":"https://pith.science/api/pith-number/UVALQKQTE6C62JYOGOT5TTADKX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UVALQKQTE6C62JYOGOT5TTADKX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UVALQKQTE6C62JYOGOT5TTADKX/action/storage_attestation","attest_author":"https://pith.science/pith/UVALQKQTE6C62JYOGOT5TTADKX/action/author_attestation","sign_citation":"https://pith.science/pith/UVALQKQTE6C62JYOGOT5TTADKX/action/citation_signature","submit_replication":"https://pith.science/pith/UVALQKQTE6C62JYOGOT5TTADKX/action/replication_record"}},"created_at":"2026-07-16T01:23:05.903342+00:00","updated_at":"2026-07-16T01:23:05.903342+00:00"}