{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1994:CBBEXMBZMGPOHAGGSLWK326P5Q","short_pith_number":"pith:CBBEXMBZ","canonical_record":{"source":{"id":"math/9408213","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.LO","submitted_at":"1994-08-15T00:00:00Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"755d1ded9f3b8e7226a782c96c97ce09b36812d5868749046e3e9d50aab31f9e","abstract_canon_sha256":"a9012969cdbe992be6c7bd0481d8ce6e6cc2359dbf51fe65b3cab3eaa5f35b31"},"schema_version":"1.0"},"canonical_sha256":"10424bb039619ee380c692ecadebcfec0a5e4ad941ad502bbfc7136037d76cb5","source":{"kind":"arxiv","id":"math/9408213","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9408213","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"arxiv_version","alias_value":"math/9408213v1","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9408213","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"pith_short_12","alias_value":"CBBEXMBZMGPO","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"CBBEXMBZMGPOHAGG","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"CBBEXMBZ","created_at":"2026-05-18T12:25:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1994:CBBEXMBZMGPOHAGGSLWK326P5Q","target":"record","payload":{"canonical_record":{"source":{"id":"math/9408213","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.LO","submitted_at":"1994-08-15T00:00:00Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"755d1ded9f3b8e7226a782c96c97ce09b36812d5868749046e3e9d50aab31f9e","abstract_canon_sha256":"a9012969cdbe992be6c7bd0481d8ce6e6cc2359dbf51fe65b3cab3eaa5f35b31"},"schema_version":"1.0"},"canonical_sha256":"10424bb039619ee380c692ecadebcfec0a5e4ad941ad502bbfc7136037d76cb5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:51.163205Z","signature_b64":"CI2Se9O4KRJk68ugfPU9xizlpmvpGcU8Zx74uAKHAu44D0BlWhxSqmNgwMJD51t+zKhEsmy8gqBvcjOFAhC7Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"10424bb039619ee380c692ecadebcfec0a5e4ad941ad502bbfc7136037d76cb5","last_reissued_at":"2026-05-18T01:05:51.162483Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:51.162483Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/9408213","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":"2026-05-18T01:05:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mk/da3Q9BwGiJ4Wdr9Naq0+ssN7lseHwbT9/xNUa9kvSZ3NDAAcVGKNR1Qzl6HqTJCFFKXQvO4+HFFO350htBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T02:58:38.015450Z"},"content_sha256":"1101e7625949d298bd4df7ec98d9a233bb91b6d93fc59a3ba401c09c1d46c0ba","schema_version":"1.0","event_id":"sha256:1101e7625949d298bd4df7ec98d9a233bb91b6d93fc59a3ba401c09c1d46c0ba"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1994:CBBEXMBZMGPOHAGGSLWK326P5Q","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Almost free algebras","license":"","headline":"","cross_cats":["math.RA"],"primary_cat":"math.LO","authors_text":"Alan H. Mekler, Saharon Shelah","submitted_at":"1994-08-15T00:00:00Z","abstract_excerpt":"The essentially non-free spectrum is the class of uncountable cardinals kappa in which there is an essentially non-free algebra of cardinality kappa which is almost free. In L, the essentially non-free spectrum of a variety is entirely determined by whether or not the construction principle holds. In ZFC may be more complicated. For some varieties, such as groups, abelian groups or any variety of modules over a non-left perfect ring, the essentially non-free spectrum contains not only aleph_1 but aleph_n for all n>0. The reason for this being true in ZFC (rather than under some special set the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9408213","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":"2026-05-18T01:05:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R/Rtc9ldZUukGbfAv+If2gD/4M4sTlQokxUSaZtEw49WDMD04Eke/WbhQBUZEtUgshXV1OL0MjJpSsTS+ttmCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T02:58:38.015785Z"},"content_sha256":"a1cbd3654c40b4a19e20aa0883e4d663b164262af0e861f1361d0746ebbee280","schema_version":"1.0","event_id":"sha256:a1cbd3654c40b4a19e20aa0883e4d663b164262af0e861f1361d0746ebbee280"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CBBEXMBZMGPOHAGGSLWK326P5Q/bundle.json","state_url":"https://pith.science/pith/CBBEXMBZMGPOHAGGSLWK326P5Q/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CBBEXMBZMGPOHAGGSLWK326P5Q/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":"2026-06-29T02:58:38Z","links":{"resolver":"https://pith.science/pith/CBBEXMBZMGPOHAGGSLWK326P5Q","bundle":"https://pith.science/pith/CBBEXMBZMGPOHAGGSLWK326P5Q/bundle.json","state":"https://pith.science/pith/CBBEXMBZMGPOHAGGSLWK326P5Q/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CBBEXMBZMGPOHAGGSLWK326P5Q/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1994:CBBEXMBZMGPOHAGGSLWK326P5Q","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":"a9012969cdbe992be6c7bd0481d8ce6e6cc2359dbf51fe65b3cab3eaa5f35b31","cross_cats_sorted":["math.RA"],"license":"","primary_cat":"math.LO","submitted_at":"1994-08-15T00:00:00Z","title_canon_sha256":"755d1ded9f3b8e7226a782c96c97ce09b36812d5868749046e3e9d50aab31f9e"},"schema_version":"1.0","source":{"id":"math/9408213","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9408213","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"arxiv_version","alias_value":"math/9408213v1","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9408213","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"pith_short_12","alias_value":"CBBEXMBZMGPO","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"CBBEXMBZMGPOHAGG","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"CBBEXMBZ","created_at":"2026-05-18T12:25:47Z"}],"graph_snapshots":[{"event_id":"sha256:a1cbd3654c40b4a19e20aa0883e4d663b164262af0e861f1361d0746ebbee280","target":"graph","created_at":"2026-05-18T01:05:51Z","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":"The essentially non-free spectrum is the class of uncountable cardinals kappa in which there is an essentially non-free algebra of cardinality kappa which is almost free. In L, the essentially non-free spectrum of a variety is entirely determined by whether or not the construction principle holds. In ZFC may be more complicated. For some varieties, such as groups, abelian groups or any variety of modules over a non-left perfect ring, the essentially non-free spectrum contains not only aleph_1 but aleph_n for all n>0. The reason for this being true in ZFC (rather than under some special set the","authors_text":"Alan H. Mekler, Saharon Shelah","cross_cats":["math.RA"],"headline":"","license":"","primary_cat":"math.LO","submitted_at":"1994-08-15T00:00:00Z","title":"Almost free algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9408213","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":"sha256:1101e7625949d298bd4df7ec98d9a233bb91b6d93fc59a3ba401c09c1d46c0ba","target":"record","created_at":"2026-05-18T01:05:51Z","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":"a9012969cdbe992be6c7bd0481d8ce6e6cc2359dbf51fe65b3cab3eaa5f35b31","cross_cats_sorted":["math.RA"],"license":"","primary_cat":"math.LO","submitted_at":"1994-08-15T00:00:00Z","title_canon_sha256":"755d1ded9f3b8e7226a782c96c97ce09b36812d5868749046e3e9d50aab31f9e"},"schema_version":"1.0","source":{"id":"math/9408213","kind":"arxiv","version":1}},"canonical_sha256":"10424bb039619ee380c692ecadebcfec0a5e4ad941ad502bbfc7136037d76cb5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"10424bb039619ee380c692ecadebcfec0a5e4ad941ad502bbfc7136037d76cb5","first_computed_at":"2026-05-18T01:05:51.162483Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:51.162483Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CI2Se9O4KRJk68ugfPU9xizlpmvpGcU8Zx74uAKHAu44D0BlWhxSqmNgwMJD51t+zKhEsmy8gqBvcjOFAhC7Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:51.163205Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9408213","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1101e7625949d298bd4df7ec98d9a233bb91b6d93fc59a3ba401c09c1d46c0ba","sha256:a1cbd3654c40b4a19e20aa0883e4d663b164262af0e861f1361d0746ebbee280"],"state_sha256":"e75e7494246834b0b744cc81f52b007585bdb3b18617f3659485b1e5b1fdd2ec"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ua45JQ/+BuMfoEFEiM4c9umFCiRW8hY4k66pTdTiou2T8j++vgUSXUlFQjpWlQ2FvDBMPk+F4CNnRbpGGA7sDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T02:58:38.017645Z","bundle_sha256":"eb08d510a8eccab0dedfe0d22b0ad9879808d35e62aed04aa84d66033a492fa6"}}