{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:7DAQQ3JP5MMSW64A3MSHWBRUPQ","short_pith_number":"pith:7DAQQ3JP","schema_version":"1.0","canonical_sha256":"f8c1086d2feb192b7b80db247b06347c15a1217259c9b4f746ce473efd99d4aa","source":{"kind":"arxiv","id":"2608.13310","version":1},"attestation_state":"computed","paper":{"title":"On the Structure of $(\\min,+)$ Convolution","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"cs.CC","authors_text":"Huanyi Zhou","submitted_at":"2026-08-13T14:37:04Z","abstract_excerpt":"The $(\\min,+)$ convolution is a central problem in fine-grained complexity, and whether it admits a truly subquadratic algorithm remains open. We study it through tropical polynomials, where $(\\min,+)$ convolution is exactly polynomial multiplication.\n  We introduce tropical decomposition width, a parameter measuring how finely a tropical polynomial can be decomposed into low-degree factors. We prove modular convexity theorems showing that bounded tropical decomposition width forces strong convexity on arithmetic subpolynomials. This yields deterministic algorithms for computing $a\\otimes b$ i"},"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":"2608.13310","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CC","submitted_at":"2026-08-13T14:37:04Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"2c30fa933594336055aaddc16e27d95445b88a8382050e318a56390c842e5649","abstract_canon_sha256":"3ff279c54d7eb8d795144707043e0273a41ba410f3170b97b1cfacac47e355a6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-14T01:04:11.118895Z","signature_b64":"mx5HE5xXTcAYI84deQgSviV0Ua4WFtq0zsTA9742SJkyfYfyD6emSqmL8ozdtfDr2e6/fPX9P40ri+dG8mlMBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8c1086d2feb192b7b80db247b06347c15a1217259c9b4f746ce473efd99d4aa","last_reissued_at":"2026-08-14T01:04:11.117103Z","signature_status":"signed_v1","first_computed_at":"2026-08-14T01:04:11.117103Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Structure of $(\\min,+)$ Convolution","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"cs.CC","authors_text":"Huanyi Zhou","submitted_at":"2026-08-13T14:37:04Z","abstract_excerpt":"The $(\\min,+)$ convolution is a central problem in fine-grained complexity, and whether it admits a truly subquadratic algorithm remains open. We study it through tropical polynomials, where $(\\min,+)$ convolution is exactly polynomial multiplication.\n  We introduce tropical decomposition width, a parameter measuring how finely a tropical polynomial can be decomposed into low-degree factors. We prove modular convexity theorems showing that bounded tropical decomposition width forces strong convexity on arithmetic subpolynomials. This yields deterministic algorithms for computing $a\\otimes b$ i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.13310","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/2608.13310/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":"2608.13310","created_at":"2026-08-14T01:04:11.117993+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.13310v1","created_at":"2026-08-14T01:04:11.117993+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.13310","created_at":"2026-08-14T01:04:11.117993+00:00"},{"alias_kind":"pith_short_12","alias_value":"7DAQQ3JP5MMS","created_at":"2026-08-14T01:04:11.117993+00:00"},{"alias_kind":"pith_short_16","alias_value":"7DAQQ3JP5MMSW64A","created_at":"2026-08-14T01:04:11.117993+00:00"},{"alias_kind":"pith_short_8","alias_value":"7DAQQ3JP","created_at":"2026-08-14T01:04:11.117993+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/7DAQQ3JP5MMSW64A3MSHWBRUPQ","json":"https://pith.science/pith/7DAQQ3JP5MMSW64A3MSHWBRUPQ.json","graph_json":"https://pith.science/api/pith-number/7DAQQ3JP5MMSW64A3MSHWBRUPQ/graph.json","events_json":"https://pith.science/api/pith-number/7DAQQ3JP5MMSW64A3MSHWBRUPQ/events.json","paper":"https://pith.science/paper/7DAQQ3JP"},"agent_actions":{"view_html":"https://pith.science/pith/7DAQQ3JP5MMSW64A3MSHWBRUPQ","download_json":"https://pith.science/pith/7DAQQ3JP5MMSW64A3MSHWBRUPQ.json","view_paper":"https://pith.science/paper/7DAQQ3JP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.13310&json=true","fetch_graph":"https://pith.science/api/pith-number/7DAQQ3JP5MMSW64A3MSHWBRUPQ/graph.json","fetch_events":"https://pith.science/api/pith-number/7DAQQ3JP5MMSW64A3MSHWBRUPQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7DAQQ3JP5MMSW64A3MSHWBRUPQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7DAQQ3JP5MMSW64A3MSHWBRUPQ/action/storage_attestation","attest_author":"https://pith.science/pith/7DAQQ3JP5MMSW64A3MSHWBRUPQ/action/author_attestation","sign_citation":"https://pith.science/pith/7DAQQ3JP5MMSW64A3MSHWBRUPQ/action/citation_signature","submit_replication":"https://pith.science/pith/7DAQQ3JP5MMSW64A3MSHWBRUPQ/action/replication_record"}},"created_at":"2026-08-14T01:04:11.117993+00:00","updated_at":"2026-08-14T01:04:11.117993+00:00"}