{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:4L6SLHOVJNVPNUCZDGPTAYNUY2","short_pith_number":"pith:4L6SLHOV","canonical_record":{"source":{"id":"2606.01218","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2026-05-31T13:11:16Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"734d60f1492d2cb2fc7898838bf5893257f1f1a3f86dceeffbac9b3e9228948a","abstract_canon_sha256":"0c21b28690024560b3bf7fb8dc885957c502c761dfd84a5558fab8fac4a36ab1"},"schema_version":"1.0"},"canonical_sha256":"e2fd259dd54b6af6d059199f3061b4c691a873698bdee504d7bb21468b81441b","source":{"kind":"arxiv","id":"2606.01218","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.01218","created_at":"2026-06-02T02:04:27Z"},{"alias_kind":"arxiv_version","alias_value":"2606.01218v1","created_at":"2026-06-02T02:04:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.01218","created_at":"2026-06-02T02:04:27Z"},{"alias_kind":"pith_short_12","alias_value":"4L6SLHOVJNVP","created_at":"2026-06-02T02:04:27Z"},{"alias_kind":"pith_short_16","alias_value":"4L6SLHOVJNVPNUCZ","created_at":"2026-06-02T02:04:27Z"},{"alias_kind":"pith_short_8","alias_value":"4L6SLHOV","created_at":"2026-06-02T02:04:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:4L6SLHOVJNVPNUCZDGPTAYNUY2","target":"record","payload":{"canonical_record":{"source":{"id":"2606.01218","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2026-05-31T13:11:16Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"734d60f1492d2cb2fc7898838bf5893257f1f1a3f86dceeffbac9b3e9228948a","abstract_canon_sha256":"0c21b28690024560b3bf7fb8dc885957c502c761dfd84a5558fab8fac4a36ab1"},"schema_version":"1.0"},"canonical_sha256":"e2fd259dd54b6af6d059199f3061b4c691a873698bdee504d7bb21468b81441b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T02:04:27.086273Z","signature_b64":"sT6VyNT0Wal6/49eH9EoBqBK51rRAy3g0A/bNHl74vX6+X2WOT4a+LIe8ai88J7sd+qQVdRDGARszdxLqIZiBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e2fd259dd54b6af6d059199f3061b4c691a873698bdee504d7bb21468b81441b","last_reissued_at":"2026-06-02T02:04:27.085839Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T02:04:27.085839Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.01218","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":"2026-06-02T02:04:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"D4MCzzfzIbGB76AQl2Q6O1po6lLYU/ehFCpikZRSq9NFycLPf+J1daMMvOYITDhHnlpqeyoS8cSjegf+020kDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T06:06:44.503554Z"},"content_sha256":"e42a9ac44b4d716934730a46e3eb53e75dd491fb8ec3862ca1908753c8b9f75f","schema_version":"1.0","event_id":"sha256:e42a9ac44b4d716934730a46e3eb53e75dd491fb8ec3862ca1908753c8b9f75f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:4L6SLHOVJNVPNUCZDGPTAYNUY2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Finite-Resolution Information from Collision Statistics","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Alexander J. Gates","submitted_at":"2026-05-31T13:11:16Z","abstract_excerpt":"Collision statistics provide a finite-resolution view of information by measuring how often a fixed number of independent samples fall on the same state. These directly countable quantities form the basis of integer-order R\\'enyi entropies. Here, we use low-order R\\'enyi entropies to approximate Shannon entropy and mutual information, while characterizing what is necessarily lost when only finitely many collision moments are used. We derive interpolation-error bounds showing that approximation error is controlled by the shape of the R\\'enyi entropy path near the Shannon point. We also separate"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.01218","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.01218/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":"2026-06-02T02:04:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4lKYjBtFDa7BIZhfeHjVdYTV+5LjQfETm0G1NcDK+ak6rdIW/rBrn7Fttea4IZ9GOwDFQKcurL4K+ULXby4/Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T06:06:44.503956Z"},"content_sha256":"98f5a390024a8cd922b86e6170719490b7c03367ffcd2dcb969afded1424fc98","schema_version":"1.0","event_id":"sha256:98f5a390024a8cd922b86e6170719490b7c03367ffcd2dcb969afded1424fc98"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4L6SLHOVJNVPNUCZDGPTAYNUY2/bundle.json","state_url":"https://pith.science/pith/4L6SLHOVJNVPNUCZDGPTAYNUY2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4L6SLHOVJNVPNUCZDGPTAYNUY2/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":"2026-06-30T06:06:44Z","links":{"resolver":"https://pith.science/pith/4L6SLHOVJNVPNUCZDGPTAYNUY2","bundle":"https://pith.science/pith/4L6SLHOVJNVPNUCZDGPTAYNUY2/bundle.json","state":"https://pith.science/pith/4L6SLHOVJNVPNUCZDGPTAYNUY2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4L6SLHOVJNVPNUCZDGPTAYNUY2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:4L6SLHOVJNVPNUCZDGPTAYNUY2","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":"0c21b28690024560b3bf7fb8dc885957c502c761dfd84a5558fab8fac4a36ab1","cross_cats_sorted":["math.IT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2026-05-31T13:11:16Z","title_canon_sha256":"734d60f1492d2cb2fc7898838bf5893257f1f1a3f86dceeffbac9b3e9228948a"},"schema_version":"1.0","source":{"id":"2606.01218","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.01218","created_at":"2026-06-02T02:04:27Z"},{"alias_kind":"arxiv_version","alias_value":"2606.01218v1","created_at":"2026-06-02T02:04:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.01218","created_at":"2026-06-02T02:04:27Z"},{"alias_kind":"pith_short_12","alias_value":"4L6SLHOVJNVP","created_at":"2026-06-02T02:04:27Z"},{"alias_kind":"pith_short_16","alias_value":"4L6SLHOVJNVPNUCZ","created_at":"2026-06-02T02:04:27Z"},{"alias_kind":"pith_short_8","alias_value":"4L6SLHOV","created_at":"2026-06-02T02:04:27Z"}],"graph_snapshots":[{"event_id":"sha256:98f5a390024a8cd922b86e6170719490b7c03367ffcd2dcb969afded1424fc98","target":"graph","created_at":"2026-06-02T02:04:27Z","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/2606.01218/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Collision statistics provide a finite-resolution view of information by measuring how often a fixed number of independent samples fall on the same state. These directly countable quantities form the basis of integer-order R\\'enyi entropies. Here, we use low-order R\\'enyi entropies to approximate Shannon entropy and mutual information, while characterizing what is necessarily lost when only finitely many collision moments are used. We derive interpolation-error bounds showing that approximation error is controlled by the shape of the R\\'enyi entropy path near the Shannon point. We also separate","authors_text":"Alexander J. Gates","cross_cats":["math.IT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2026-05-31T13:11:16Z","title":"Finite-Resolution Information from Collision Statistics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.01218","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:e42a9ac44b4d716934730a46e3eb53e75dd491fb8ec3862ca1908753c8b9f75f","target":"record","created_at":"2026-06-02T02:04:27Z","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":"0c21b28690024560b3bf7fb8dc885957c502c761dfd84a5558fab8fac4a36ab1","cross_cats_sorted":["math.IT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2026-05-31T13:11:16Z","title_canon_sha256":"734d60f1492d2cb2fc7898838bf5893257f1f1a3f86dceeffbac9b3e9228948a"},"schema_version":"1.0","source":{"id":"2606.01218","kind":"arxiv","version":1}},"canonical_sha256":"e2fd259dd54b6af6d059199f3061b4c691a873698bdee504d7bb21468b81441b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e2fd259dd54b6af6d059199f3061b4c691a873698bdee504d7bb21468b81441b","first_computed_at":"2026-06-02T02:04:27.085839Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T02:04:27.085839Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sT6VyNT0Wal6/49eH9EoBqBK51rRAy3g0A/bNHl74vX6+X2WOT4a+LIe8ai88J7sd+qQVdRDGARszdxLqIZiBQ==","signature_status":"signed_v1","signed_at":"2026-06-02T02:04:27.086273Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.01218","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e42a9ac44b4d716934730a46e3eb53e75dd491fb8ec3862ca1908753c8b9f75f","sha256:98f5a390024a8cd922b86e6170719490b7c03367ffcd2dcb969afded1424fc98"],"state_sha256":"d321fb6eb6d9c1b5a6534577537489ad9b3335cda04ba1fd1050b6dcab568b0c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Fh8DkZQQuKlDK9lwXEzFGVTqi+kwFL/NND828OJ47ncPE+nY9ltX/oGxLpA/S33x5l0ARyXodWgVG9Eoo5fwCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T06:06:44.506099Z","bundle_sha256":"37dfcd83de3e18dfd2334a66232f3b9f7a811b600e6f41720d44b52d337b5b05"}}