{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:LXGCCA3QWHBAEGUPYOW3J5KC24","short_pith_number":"pith:LXGCCA3Q","canonical_record":{"source":{"id":"1908.09642","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"cs.IT","submitted_at":"2019-08-23T17:05:49Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"8a2bb948f3942cc4ce3a592a504a4480afa3af14cd9e44cbb90fedacdd7bdc4a","abstract_canon_sha256":"71b433661bbe6e0819fcd2e90be35c3b595f644c0b394db9f77a41b381cc4cdb"},"schema_version":"1.0"},"canonical_sha256":"5dcc210370b1c2021a8fc3adb4f542d73241c627d999ad0ab6fd3b4554638b08","source":{"kind":"arxiv","id":"1908.09642","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.09642","created_at":"2026-07-05T00:21:14Z"},{"alias_kind":"arxiv_version","alias_value":"1908.09642v2","created_at":"2026-07-05T00:21:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.09642","created_at":"2026-07-05T00:21:14Z"},{"alias_kind":"pith_short_12","alias_value":"LXGCCA3QWHBA","created_at":"2026-07-05T00:21:14Z"},{"alias_kind":"pith_short_16","alias_value":"LXGCCA3QWHBAEGUP","created_at":"2026-07-05T00:21:14Z"},{"alias_kind":"pith_short_8","alias_value":"LXGCCA3Q","created_at":"2026-07-05T00:21:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:LXGCCA3QWHBAEGUPYOW3J5KC24","target":"record","payload":{"canonical_record":{"source":{"id":"1908.09642","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"cs.IT","submitted_at":"2019-08-23T17:05:49Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"8a2bb948f3942cc4ce3a592a504a4480afa3af14cd9e44cbb90fedacdd7bdc4a","abstract_canon_sha256":"71b433661bbe6e0819fcd2e90be35c3b595f644c0b394db9f77a41b381cc4cdb"},"schema_version":"1.0"},"canonical_sha256":"5dcc210370b1c2021a8fc3adb4f542d73241c627d999ad0ab6fd3b4554638b08","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:21:14.592147Z","signature_b64":"8YD+WLCGNiTFtAH031098S/NEUnOTtxQUnQRr37/wirF2KenDXEDnvKO0H16ZejzZqqk9xjCNYtFJ0MHQ10bBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5dcc210370b1c2021a8fc3adb4f542d73241c627d999ad0ab6fd3b4554638b08","last_reissued_at":"2026-07-05T00:21:14.591671Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:21:14.591671Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.09642","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:21:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bT+pNu7qwco/WGEf03E4rXDwSGciEg0qXIeg4JWRnZEX/Qn5QvmQ1CHIbhWQldkrPIpt0fCb0GKpwmfVNxuICQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T08:40:50.009857Z"},"content_sha256":"02c1a31b485100095a740872fb791907420ee6f04edba9611f208d8301a1ba9d","schema_version":"1.0","event_id":"sha256:02c1a31b485100095a740872fb791907420ee6f04edba9611f208d8301a1ba9d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:LXGCCA3QWHBAEGUPYOW3J5KC24","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Group Theoretic Roots of Information: permutations, symmetry, and entropy","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"David J. Galas","submitted_at":"2019-08-23T17:05:49Z","abstract_excerpt":"We propose a new interpretation of measures of information and disorder by connecting these concepts to group theory in a new way. Entropy and group theory are connected here by their common relation to sets of permutations. A combinatorial measure of information and disorder is proposed, in terms of integers and discrete functions, that we call the integer entropy. The Shannon measure of information is the limiting case of a richer, more general conceptual structure that reveals relations among finite groups, information, and symmetries. It is shown that the integer entropy converges uniforml"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.09642","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.09642/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T00:21:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2EPctzMJjpOrLChIdlnd9RYUTKpTBFVJ9VlAaDydb5gjdQOYMDFOAr680vg6FwNRJHdM+tlLDvlpnFCd+WpvBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T08:40:50.010351Z"},"content_sha256":"9beb75a40c0945480f2fe712525d6d618828020835fd382ec1949fac8a5c88ce","schema_version":"1.0","event_id":"sha256:9beb75a40c0945480f2fe712525d6d618828020835fd382ec1949fac8a5c88ce"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LXGCCA3QWHBAEGUPYOW3J5KC24/bundle.json","state_url":"https://pith.science/pith/LXGCCA3QWHBAEGUPYOW3J5KC24/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LXGCCA3QWHBAEGUPYOW3J5KC24/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":"2026-08-16T08:40:50Z","links":{"resolver":"https://pith.science/pith/LXGCCA3QWHBAEGUPYOW3J5KC24","bundle":"https://pith.science/pith/LXGCCA3QWHBAEGUPYOW3J5KC24/bundle.json","state":"https://pith.science/pith/LXGCCA3QWHBAEGUPYOW3J5KC24/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LXGCCA3QWHBAEGUPYOW3J5KC24/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:LXGCCA3QWHBAEGUPYOW3J5KC24","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":"71b433661bbe6e0819fcd2e90be35c3b595f644c0b394db9f77a41b381cc4cdb","cross_cats_sorted":["math.IT"],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"cs.IT","submitted_at":"2019-08-23T17:05:49Z","title_canon_sha256":"8a2bb948f3942cc4ce3a592a504a4480afa3af14cd9e44cbb90fedacdd7bdc4a"},"schema_version":"1.0","source":{"id":"1908.09642","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.09642","created_at":"2026-07-05T00:21:14Z"},{"alias_kind":"arxiv_version","alias_value":"1908.09642v2","created_at":"2026-07-05T00:21:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.09642","created_at":"2026-07-05T00:21:14Z"},{"alias_kind":"pith_short_12","alias_value":"LXGCCA3QWHBA","created_at":"2026-07-05T00:21:14Z"},{"alias_kind":"pith_short_16","alias_value":"LXGCCA3QWHBAEGUP","created_at":"2026-07-05T00:21:14Z"},{"alias_kind":"pith_short_8","alias_value":"LXGCCA3Q","created_at":"2026-07-05T00:21:14Z"}],"graph_snapshots":[{"event_id":"sha256:9beb75a40c0945480f2fe712525d6d618828020835fd382ec1949fac8a5c88ce","target":"graph","created_at":"2026-07-05T00:21:14Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.09642/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We propose a new interpretation of measures of information and disorder by connecting these concepts to group theory in a new way. Entropy and group theory are connected here by their common relation to sets of permutations. A combinatorial measure of information and disorder is proposed, in terms of integers and discrete functions, that we call the integer entropy. The Shannon measure of information is the limiting case of a richer, more general conceptual structure that reveals relations among finite groups, information, and symmetries. It is shown that the integer entropy converges uniforml","authors_text":"David J. Galas","cross_cats":["math.IT"],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"cs.IT","submitted_at":"2019-08-23T17:05:49Z","title":"The Group Theoretic Roots of Information: permutations, symmetry, and entropy"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.09642","kind":"arxiv","version":2},"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":"sha256:02c1a31b485100095a740872fb791907420ee6f04edba9611f208d8301a1ba9d","target":"record","created_at":"2026-07-05T00:21:14Z","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":"71b433661bbe6e0819fcd2e90be35c3b595f644c0b394db9f77a41b381cc4cdb","cross_cats_sorted":["math.IT"],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"cs.IT","submitted_at":"2019-08-23T17:05:49Z","title_canon_sha256":"8a2bb948f3942cc4ce3a592a504a4480afa3af14cd9e44cbb90fedacdd7bdc4a"},"schema_version":"1.0","source":{"id":"1908.09642","kind":"arxiv","version":2}},"canonical_sha256":"5dcc210370b1c2021a8fc3adb4f542d73241c627d999ad0ab6fd3b4554638b08","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5dcc210370b1c2021a8fc3adb4f542d73241c627d999ad0ab6fd3b4554638b08","first_computed_at":"2026-07-05T00:21:14.591671Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:21:14.591671Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8YD+WLCGNiTFtAH031098S/NEUnOTtxQUnQRr37/wirF2KenDXEDnvKO0H16ZejzZqqk9xjCNYtFJ0MHQ10bBg==","signature_status":"signed_v1","signed_at":"2026-07-05T00:21:14.592147Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.09642","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:02c1a31b485100095a740872fb791907420ee6f04edba9611f208d8301a1ba9d","sha256:9beb75a40c0945480f2fe712525d6d618828020835fd382ec1949fac8a5c88ce"],"state_sha256":"8cd3d27f937635d7d4b73bde3ac7a167db8b5453af353d20a276446227d3ea34"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZD24y/NDMmsgi9z/7xr5zO6SIVdMuSVUGkv9sN6sXrYRcAMO8EIH6VwSNfz8VbcFeeWM2f7z0rzn/sRUL6NOAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T08:40:50.015413Z","bundle_sha256":"3a37fc2095e1626099cccf02f9398bb6a583c6c4ea05c971e51b36a022ff1e07"}}