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The first class of exact solutions is described by a fragmentation rate $a({x_1},...,{x_d}) = 1$ and daughter distribution function $b({x_1},..,{x_d} | {{x_{1}^{\\p}}},...,{{x_{d}^{\\p}}})= {{(\\a_1 + 2)x_1^{\\a_1}}\\over{x_1^{\\p(\\a_1+1)}}}...{{(\\a_d+2)x_d^{\\a_d}}\\over {x_d^{\\p(\\a_d+1)}}}$. The second class of exact solutions is described by a fragmentation rate $ a({x_1},...,{x_d}) = {{{x_1}^{\\a_1}}...{{x_d}^{\\a_d}}/{2^d}}$ and a daughter distr"},"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":"cond-mat/9607027","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"cond-mat","submitted_at":"1996-07-04T11:10:25Z","cross_cats_sorted":[],"title_canon_sha256":"37a7478c952e34ef76eb47b2f28af8c973578106f172343869a151d45e9ebb51","abstract_canon_sha256":"3188e2a1b65aa5a44a9d22942dbf88631c1df105202d9e32813398c3328059b5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T16:00:49.511165Z","signature_b64":"1vFDWrvqu9fPVtzZlqvKoveGiTgjQgxt/ZyC9nBFQX2BfqB14icPSn8lBkFuxt5xMEL4n8QZEekul/3MN4caCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ce2771480e6739a07129576a4ff644411c673d1744ef3d3b9394495ce5ef01a","last_reissued_at":"2026-07-04T16:00:49.510791Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T16:00:49.510791Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Kinetics of Multi-dimensional Fragmentation","license":"","headline":"","cross_cats":[],"primary_cat":"cond-mat","authors_text":"M. 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