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The h-mass of a k-polyhedral chain P =$\\sum$j $\\theta$j$\\sigma$j in R n (0 $\\le$ k $\\le$ n) is defined as M h (P) := j h($\\theta$j) H k ($\\sigma$j). If T = $\\tau$ (M, $\\theta$, $\\xi$) is a k-rectifiable chain, the definition extends to M h (T) := M h($\\theta$) dH k. Given such a rectifiable flat chain T with M h (T) < $\\infty$ and $\\partial$T polyhedral, we prove that for every $\\eta$ > 0, it decomposes as T = P + $\\partial$V wi"},"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":"1806.05046","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-06-13T13:57:23Z","cross_cats_sorted":["math.FA","math.OC"],"title_canon_sha256":"425e7af6076aa225c2a70dd682a2e1ae74ec11c5e83688bce15df45d7118c0dd","abstract_canon_sha256":"5a6e2645b704ab5572b1d5bdec6bb3749f29d58bd68aaced3b77c2446a01e8dd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:20.266931Z","signature_b64":"cZvM7OUx+LxngZ1vwLlcKyESm5eEiJMV4DOhwjJ8MnTJOeJgnPCOUUrBpcnjs/o6TXiyDqohpnLQQzDyI0fsBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0c722f3df14829f670037684748652d25682f082a02dd78ede990465eb5ca28","last_reissued_at":"2026-05-18T00:13:20.266396Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:20.266396Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Strong approximation in h-mass of rectifiable currents under homological constraint","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.OC"],"primary_cat":"math.AP","authors_text":"Antonin Chambolle (CMAP), Beno\\\"it Merlet (RAPSODI), Luca Alberto Davide Ferrari (CMAP)","submitted_at":"2018-06-13T13:57:23Z","abstract_excerpt":"Let h : R $\\rightarrow$ R+ be a lower semi-continuous subbadditive and even function such that h(0) = 0 and h($\\theta$) $\\ge$ $\\alpha$|$\\theta$| for some $\\alpha$ > 0. 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Given such a rectifiable flat chain T with M h (T) < $\\infty$ and $\\partial$T polyhedral, we prove that for every $\\eta$ > 0, it decomposes as T = P + $\\partial$V wi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.05046","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":""},"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":"1806.05046","created_at":"2026-05-18T00:13:20.266464+00:00"},{"alias_kind":"arxiv_version","alias_value":"1806.05046v1","created_at":"2026-05-18T00:13:20.266464+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.05046","created_at":"2026-05-18T00:13:20.266464+00:00"},{"alias_kind":"pith_short_12","alias_value":"WDDSF467CSBJ","created_at":"2026-05-18T12:32:59.047623+00:00"},{"alias_kind":"pith_short_16","alias_value":"WDDSF467CSBJ6ZYA","created_at":"2026-05-18T12:32:59.047623+00:00"},{"alias_kind":"pith_short_8","alias_value":"WDDSF467","created_at":"2026-05-18T12:32:59.047623+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/WDDSF467CSBJ6ZYAG5UEOSDFFU","json":"https://pith.science/pith/WDDSF467CSBJ6ZYAG5UEOSDFFU.json","graph_json":"https://pith.science/api/pith-number/WDDSF467CSBJ6ZYAG5UEOSDFFU/graph.json","events_json":"https://pith.science/api/pith-number/WDDSF467CSBJ6ZYAG5UEOSDFFU/events.json","paper":"https://pith.science/paper/WDDSF467"},"agent_actions":{"view_html":"https://pith.science/pith/WDDSF467CSBJ6ZYAG5UEOSDFFU","download_json":"https://pith.science/pith/WDDSF467CSBJ6ZYAG5UEOSDFFU.json","view_paper":"https://pith.science/paper/WDDSF467","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1806.05046&json=true","fetch_graph":"https://pith.science/api/pith-number/WDDSF467CSBJ6ZYAG5UEOSDFFU/graph.json","fetch_events":"https://pith.science/api/pith-number/WDDSF467CSBJ6ZYAG5UEOSDFFU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WDDSF467CSBJ6ZYAG5UEOSDFFU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WDDSF467CSBJ6ZYAG5UEOSDFFU/action/storage_attestation","attest_author":"https://pith.science/pith/WDDSF467CSBJ6ZYAG5UEOSDFFU/action/author_attestation","sign_citation":"https://pith.science/pith/WDDSF467CSBJ6ZYAG5UEOSDFFU/action/citation_signature","submit_replication":"https://pith.science/pith/WDDSF467CSBJ6ZYAG5UEOSDFFU/action/replication_record"}},"created_at":"2026-05-18T00:13:20.266464+00:00","updated_at":"2026-05-18T00:13:20.266464+00:00"}