{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:SCRT43WWAZLLPZ6IVWNLBZHDOR","short_pith_number":"pith:SCRT43WW","canonical_record":{"source":{"id":"2404.08139","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2024-04-11T21:54:20Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"cd265d47c9efe40aa6a726c158763ec852c8bc26aff24b53cf0a97ec4911ca63","abstract_canon_sha256":"2d7a6c11c018dc95c96f9bf5c43a3f8281dde778b011e7d76a7d6693269ec7bd"},"schema_version":"1.0"},"canonical_sha256":"90a33e6ed60656b7e7c8ad9ab0e4e3744b1d068b15981166943fded2aa2dafa2","source":{"kind":"arxiv","id":"2404.08139","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.08139","created_at":"2026-07-05T08:07:18Z"},{"alias_kind":"arxiv_version","alias_value":"2404.08139v1","created_at":"2026-07-05T08:07:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.08139","created_at":"2026-07-05T08:07:18Z"},{"alias_kind":"pith_short_12","alias_value":"SCRT43WWAZLL","created_at":"2026-07-05T08:07:18Z"},{"alias_kind":"pith_short_16","alias_value":"SCRT43WWAZLLPZ6I","created_at":"2026-07-05T08:07:18Z"},{"alias_kind":"pith_short_8","alias_value":"SCRT43WW","created_at":"2026-07-05T08:07:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:SCRT43WWAZLLPZ6IVWNLBZHDOR","target":"record","payload":{"canonical_record":{"source":{"id":"2404.08139","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2024-04-11T21:54:20Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"cd265d47c9efe40aa6a726c158763ec852c8bc26aff24b53cf0a97ec4911ca63","abstract_canon_sha256":"2d7a6c11c018dc95c96f9bf5c43a3f8281dde778b011e7d76a7d6693269ec7bd"},"schema_version":"1.0"},"canonical_sha256":"90a33e6ed60656b7e7c8ad9ab0e4e3744b1d068b15981166943fded2aa2dafa2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:07:18.354428Z","signature_b64":"MCaSbywkpZ0Z+/kTAuefzZdhzDv1h8OALcr1q4G0IOxPuGtalXWqLEl8xTHIudA4bvxp2Cz5ciLqfXTpf1YUDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"90a33e6ed60656b7e7c8ad9ab0e4e3744b1d068b15981166943fded2aa2dafa2","last_reissued_at":"2026-07-05T08:07:18.353936Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:07:18.353936Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2404.08139","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":"2026-07-05T08:07:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jaJ/WrRKh2Ek2mfBa5JB2c6sihTlv5+Hz9ASxl+7c9haOt1np8JJ2DwGPwZ9rfylDUgMY+cBViWDxFVKdav1BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T08:36:32.214539Z"},"content_sha256":"bc5d60f8941f3b727faaba76c25746a33e485310172aab4683a7dd86babb0ccd","schema_version":"1.0","event_id":"sha256:bc5d60f8941f3b727faaba76c25746a33e485310172aab4683a7dd86babb0ccd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:SCRT43WWAZLLPZ6IVWNLBZHDOR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Towards a General Theory of Dependent Sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.CT","authors_text":"Peter Bonart","submitted_at":"2024-04-11T21:54:20Z","abstract_excerpt":"We introduce dependent adders. A dependent adder $A$ has for every $x \\in A$ a way of adding together $x$ many elements of $A$. We provide examples from many disparate branches of mathematics. Examples include the field with one element $\\mathbb{F}_1$, the real numbers with integrals as sums, the category of categories with oplax colimits as sums. We also consider modules over dependent adders and provide examples."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.08139","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/2404.08139/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":"2026-07-05T08:07:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BZ0lygze1z4oei+g9Q0JH83PWvMoewq3qmCT11RZrxmjQyvGOi7VbShD+Sp9KxaDSpoANrQZLfxIypJ4Gkj+Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T08:36:32.215036Z"},"content_sha256":"7f9a617765f819b49f711da5afd421141b207ead0ec2bbfb56c4d43c5ecbdeb9","schema_version":"1.0","event_id":"sha256:7f9a617765f819b49f711da5afd421141b207ead0ec2bbfb56c4d43c5ecbdeb9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SCRT43WWAZLLPZ6IVWNLBZHDOR/bundle.json","state_url":"https://pith.science/pith/SCRT43WWAZLLPZ6IVWNLBZHDOR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SCRT43WWAZLLPZ6IVWNLBZHDOR/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":"2026-08-11T08:36:32Z","links":{"resolver":"https://pith.science/pith/SCRT43WWAZLLPZ6IVWNLBZHDOR","bundle":"https://pith.science/pith/SCRT43WWAZLLPZ6IVWNLBZHDOR/bundle.json","state":"https://pith.science/pith/SCRT43WWAZLLPZ6IVWNLBZHDOR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SCRT43WWAZLLPZ6IVWNLBZHDOR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:SCRT43WWAZLLPZ6IVWNLBZHDOR","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":"2d7a6c11c018dc95c96f9bf5c43a3f8281dde778b011e7d76a7d6693269ec7bd","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2024-04-11T21:54:20Z","title_canon_sha256":"cd265d47c9efe40aa6a726c158763ec852c8bc26aff24b53cf0a97ec4911ca63"},"schema_version":"1.0","source":{"id":"2404.08139","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.08139","created_at":"2026-07-05T08:07:18Z"},{"alias_kind":"arxiv_version","alias_value":"2404.08139v1","created_at":"2026-07-05T08:07:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.08139","created_at":"2026-07-05T08:07:18Z"},{"alias_kind":"pith_short_12","alias_value":"SCRT43WWAZLL","created_at":"2026-07-05T08:07:18Z"},{"alias_kind":"pith_short_16","alias_value":"SCRT43WWAZLLPZ6I","created_at":"2026-07-05T08:07:18Z"},{"alias_kind":"pith_short_8","alias_value":"SCRT43WW","created_at":"2026-07-05T08:07:18Z"}],"graph_snapshots":[{"event_id":"sha256:7f9a617765f819b49f711da5afd421141b207ead0ec2bbfb56c4d43c5ecbdeb9","target":"graph","created_at":"2026-07-05T08:07:18Z","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/2404.08139/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce dependent adders. A dependent adder $A$ has for every $x \\in A$ a way of adding together $x$ many elements of $A$. We provide examples from many disparate branches of mathematics. Examples include the field with one element $\\mathbb{F}_1$, the real numbers with integrals as sums, the category of categories with oplax colimits as sums. We also consider modules over dependent adders and provide examples.","authors_text":"Peter Bonart","cross_cats":["math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2024-04-11T21:54:20Z","title":"Towards a General Theory of Dependent Sums"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.08139","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:bc5d60f8941f3b727faaba76c25746a33e485310172aab4683a7dd86babb0ccd","target":"record","created_at":"2026-07-05T08:07:18Z","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":"2d7a6c11c018dc95c96f9bf5c43a3f8281dde778b011e7d76a7d6693269ec7bd","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2024-04-11T21:54:20Z","title_canon_sha256":"cd265d47c9efe40aa6a726c158763ec852c8bc26aff24b53cf0a97ec4911ca63"},"schema_version":"1.0","source":{"id":"2404.08139","kind":"arxiv","version":1}},"canonical_sha256":"90a33e6ed60656b7e7c8ad9ab0e4e3744b1d068b15981166943fded2aa2dafa2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"90a33e6ed60656b7e7c8ad9ab0e4e3744b1d068b15981166943fded2aa2dafa2","first_computed_at":"2026-07-05T08:07:18.353936Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:07:18.353936Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MCaSbywkpZ0Z+/kTAuefzZdhzDv1h8OALcr1q4G0IOxPuGtalXWqLEl8xTHIudA4bvxp2Cz5ciLqfXTpf1YUDA==","signature_status":"signed_v1","signed_at":"2026-07-05T08:07:18.354428Z","signed_message":"canonical_sha256_bytes"},"source_id":"2404.08139","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bc5d60f8941f3b727faaba76c25746a33e485310172aab4683a7dd86babb0ccd","sha256:7f9a617765f819b49f711da5afd421141b207ead0ec2bbfb56c4d43c5ecbdeb9"],"state_sha256":"bcd8a76fdbea058e4e069eca756e6b5cf351ca8d983cb2ea9cb8f0c5295150b9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vQhAJKur+nb/btKm5QHWfpu/WpgYZjPzdNGlv45f4p4FLcgdfgr7vog+v0TE55coy6vK7kykIKqPeODsz+8LDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T08:36:32.218440Z","bundle_sha256":"4bc3757cab72c34a8f999469ff31231a153f9634e981fbcf1921f39555c5c249"}}