{"paper":{"title":"Divergence is Uncertainty: A Closed-Form Posterior Covariance for Flow Matching","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For any pre-trained flow matching velocity field the trace of the posterior covariance equals the divergence of that field up to a time-dependent factor.","cross_cats":["cs.CV"],"primary_cat":"cs.LG","authors_text":"Jian Wang, Jiarui Xing, Song Wang","submitted_at":"2026-05-01T04:25:00Z","abstract_excerpt":"Flow matching has become a leading framework for generative modeling, but quantifying the uncertainty of its samples remains an open problem. Existing approaches retrain the model with auxiliary variance heads, maintain costly ensembles, or propagate approximate covariance through many integration steps, trading off training cost, inference cost, or accuracy. We show that none of these trade-offs is necessary. By extending Tweedie's formula from the denoising setting to the flow matching interpolant, we derive an exact, closed-form expression for the posterior covariance at every point along t"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For any pre-trained flow matching velocity field, the trace of the posterior covariance over the clean data given the current state equals, in closed form, the divergence of the velocity field, up to a known time-dependent prefactor and an additive constant.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The flow matching velocity field is assumed to be the exact conditional expectation under the posterior; any deviation from this (e.g., due to optimization error or model misspecification) would break the exact equality.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"In flow matching, the uncertainty of the clean data given the current state is exactly the divergence of the velocity field (up to a known scalar).","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For any pre-trained flow matching velocity field the trace of the posterior covariance equals the divergence of that field up to a time-dependent factor.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"76f5d6a38954e794c82b0982785fd533b4423f3ed09846e162e90f9458b85c66"},"source":{"id":"2605.00941","kind":"arxiv","version":3},"verdict":{"id":"0ac2f697-b512-40c7-bfe1-c8f7ba0412e1","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-09T20:30:57.124801Z","strongest_claim":"For any pre-trained flow matching velocity field, the trace of the posterior covariance over the clean data given the current state equals, in closed form, the divergence of the velocity field, up to a known time-dependent prefactor and an additive constant.","one_line_summary":"In flow matching, the uncertainty of the clean data given the current state is exactly the divergence of the velocity field (up to a known scalar).","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The flow matching velocity field is assumed to be the exact conditional expectation under the posterior; any deviation from this (e.g., due to optimization error or model misspecification) would break the exact equality.","pith_extraction_headline":"For any pre-trained flow matching velocity field the trace of the posterior covariance equals the divergence of that field up to a time-dependent factor."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.00941/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T19:43:32.161009Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T18:12:39.527351Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"e12b99dc86399c298cfcdc686ca1e3607517506443de689e2d1f6afd7e15f7b9"},"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"}