{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:5QL4XY3RD5L4CMLGD3KJZ7YIKB","short_pith_number":"pith:5QL4XY3R","canonical_record":{"source":{"id":"1601.03803","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-01-15T03:23:14Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"dee6b7ffb3764daa95e882e7e5bfd7836aba140bb140deb35d52871f5fb6a392","abstract_canon_sha256":"cae75f5e0bbd4779f19ec8e28e1c96b755a643a05f7ff6a025d3cca163266c6d"},"schema_version":"1.0"},"canonical_sha256":"ec17cbe3711f57c131661ed49cff08505aee9fd8860c13616757a9a1f8713e2e","source":{"kind":"arxiv","id":"1601.03803","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.03803","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"arxiv_version","alias_value":"1601.03803v1","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.03803","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"pith_short_12","alias_value":"5QL4XY3RD5L4","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_16","alias_value":"5QL4XY3RD5L4CMLG","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_8","alias_value":"5QL4XY3R","created_at":"2026-05-18T12:30:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:5QL4XY3RD5L4CMLGD3KJZ7YIKB","target":"record","payload":{"canonical_record":{"source":{"id":"1601.03803","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-01-15T03:23:14Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"dee6b7ffb3764daa95e882e7e5bfd7836aba140bb140deb35d52871f5fb6a392","abstract_canon_sha256":"cae75f5e0bbd4779f19ec8e28e1c96b755a643a05f7ff6a025d3cca163266c6d"},"schema_version":"1.0"},"canonical_sha256":"ec17cbe3711f57c131661ed49cff08505aee9fd8860c13616757a9a1f8713e2e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:54.534790Z","signature_b64":"c9hKA/7FyoZji/zv7rscGJqqgtUUbVHiWExcteP6GFJ5l69/24r7jsxLfCL9cFrNLBY94ryznWzDoyHtQvKSAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec17cbe3711f57c131661ed49cff08505aee9fd8860c13616757a9a1f8713e2e","last_reissued_at":"2026-05-18T00:24:54.534070Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:54.534070Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.03803","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:24:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cfxzSart7t1pePIvW4V/HrctxQ48Xjfmsmd3uQcK2jCkGmB7aQHCSNDTPMkLqmJXkyt2Vp3n3my3/RhQfCIhDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T21:18:42.417997Z"},"content_sha256":"021e1b22330203cdae0fb50ec788975bd21c38416dfa9abcecd9e1cee302bdd2","schema_version":"1.0","event_id":"sha256:021e1b22330203cdae0fb50ec788975bd21c38416dfa9abcecd9e1cee302bdd2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:5QL4XY3RD5L4CMLGD3KJZ7YIKB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Class of Non-Linearly Solvable Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Joseph Connelly, Kenneth Zeger","submitted_at":"2016-01-15T03:23:14Z","abstract_excerpt":"For each integer $m \\geq 2$, a network is constructed which is solvable over an alphabet of size $m$ but is not solvable over any smaller alphabets. If $m$ is composite, then the network has no vector linear solution over any $R$-module alphabet and is not asymptotically linear solvable over any finite-field alphabet. The network's capacity is shown to equal one, and when $m$ is composite, its linear capacity is shown to be bounded away from one for all finite-field alphabets."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03803","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:24:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kr66APuv5o7KgG4QDd05GF+YBZM7tHCkEopyhLvhLJiyVBGxuzHgMkE+DLUb89uQucJLVD/LRhhAvQQdliRLBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T21:18:42.418347Z"},"content_sha256":"153aa839adf8db51f815b9165fe814ec9bf44bf97623c4a671d030d7cc0838dc","schema_version":"1.0","event_id":"sha256:153aa839adf8db51f815b9165fe814ec9bf44bf97623c4a671d030d7cc0838dc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5QL4XY3RD5L4CMLGD3KJZ7YIKB/bundle.json","state_url":"https://pith.science/pith/5QL4XY3RD5L4CMLGD3KJZ7YIKB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5QL4XY3RD5L4CMLGD3KJZ7YIKB/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-03T21:18:42Z","links":{"resolver":"https://pith.science/pith/5QL4XY3RD5L4CMLGD3KJZ7YIKB","bundle":"https://pith.science/pith/5QL4XY3RD5L4CMLGD3KJZ7YIKB/bundle.json","state":"https://pith.science/pith/5QL4XY3RD5L4CMLGD3KJZ7YIKB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5QL4XY3RD5L4CMLGD3KJZ7YIKB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:5QL4XY3RD5L4CMLGD3KJZ7YIKB","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"cae75f5e0bbd4779f19ec8e28e1c96b755a643a05f7ff6a025d3cca163266c6d","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-01-15T03:23:14Z","title_canon_sha256":"dee6b7ffb3764daa95e882e7e5bfd7836aba140bb140deb35d52871f5fb6a392"},"schema_version":"1.0","source":{"id":"1601.03803","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.03803","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"arxiv_version","alias_value":"1601.03803v1","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.03803","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"pith_short_12","alias_value":"5QL4XY3RD5L4","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_16","alias_value":"5QL4XY3RD5L4CMLG","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_8","alias_value":"5QL4XY3R","created_at":"2026-05-18T12:30:01Z"}],"graph_snapshots":[{"event_id":"sha256:153aa839adf8db51f815b9165fe814ec9bf44bf97623c4a671d030d7cc0838dc","target":"graph","created_at":"2026-05-18T00:24:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"For each integer $m \\geq 2$, a network is constructed which is solvable over an alphabet of size $m$ but is not solvable over any smaller alphabets. If $m$ is composite, then the network has no vector linear solution over any $R$-module alphabet and is not asymptotically linear solvable over any finite-field alphabet. The network's capacity is shown to equal one, and when $m$ is composite, its linear capacity is shown to be bounded away from one for all finite-field alphabets.","authors_text":"Joseph Connelly, Kenneth Zeger","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-01-15T03:23:14Z","title":"A Class of Non-Linearly Solvable Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03803","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:021e1b22330203cdae0fb50ec788975bd21c38416dfa9abcecd9e1cee302bdd2","target":"record","created_at":"2026-05-18T00:24:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"cae75f5e0bbd4779f19ec8e28e1c96b755a643a05f7ff6a025d3cca163266c6d","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-01-15T03:23:14Z","title_canon_sha256":"dee6b7ffb3764daa95e882e7e5bfd7836aba140bb140deb35d52871f5fb6a392"},"schema_version":"1.0","source":{"id":"1601.03803","kind":"arxiv","version":1}},"canonical_sha256":"ec17cbe3711f57c131661ed49cff08505aee9fd8860c13616757a9a1f8713e2e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ec17cbe3711f57c131661ed49cff08505aee9fd8860c13616757a9a1f8713e2e","first_computed_at":"2026-05-18T00:24:54.534070Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:24:54.534070Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"c9hKA/7FyoZji/zv7rscGJqqgtUUbVHiWExcteP6GFJ5l69/24r7jsxLfCL9cFrNLBY94ryznWzDoyHtQvKSAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:24:54.534790Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.03803","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:021e1b22330203cdae0fb50ec788975bd21c38416dfa9abcecd9e1cee302bdd2","sha256:153aa839adf8db51f815b9165fe814ec9bf44bf97623c4a671d030d7cc0838dc"],"state_sha256":"fffa4ddd5209d537988f4df0e73ff5533f94187f982cdab8d583a1dee029dcd2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OIYdOJsHj7CTXLI5uEu1fa43iiw5Dt4trYBBe0HoH6eVtRSIF1IGU5ZDtRb6yW1d1bT7bBwo/IE/+jxX9C38Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T21:18:42.420323Z","bundle_sha256":"eabf823c6644edf2ad69a0a959a7ec229a070c0bcd5823c0d675a36a7f20d9a3"}}