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Under the Bernstein transform, mass-conserving solutions correspond to solutions of a singular Hamilton--Jacobi equation studied by Tran and Van (Comm.Pure Appl.Math.75 (2022), no.6, 1292--1331). Through this correspondence they proved that mass-conserving solutions are unique on the full critical range \\(0<m\\le1\\), but could establish their existence only for \\(0<m<\\tfrac12\\). We identify a one-sided, convexity-type invar"},"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":"2607.02293","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-02T15:11:24Z","cross_cats_sorted":[],"title_canon_sha256":"fcecf6a4af3ff656245547dd8c392fad16a22ddbd3bcc329bedbb56baa8cdf43","abstract_canon_sha256":"265c351902b4475b64cf51ea7749502012a52f56ba8b5ef23d6629e483b32820"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T01:17:47.729854Z","signature_b64":"kdFIDqF9Z3jFBLunj1+RvPK/UQhY/RxEMdujnCcTy3TZsFYc4SEKplo11pU0i+B/KYSPztvNIcBEfDtOWVMUCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"de896d4996e6ec8ef6d21de5bdeb64fc8b14b9539d05b14eb47e001f664488c2","last_reissued_at":"2026-07-03T01:17:47.729421Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T01:17:47.729421Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A convexity-type invariant for the critical coagulation--fragmentation Hamilton--Jacobi equation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Truong-Son P. 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