{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:4BQH35IHMWHNXKYGTRTW6ZZA5S","short_pith_number":"pith:4BQH35IH","canonical_record":{"source":{"id":"2112.13103","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2021-12-24T16:47:33Z","cross_cats_sorted":["astro-ph.GA","hep-th"],"title_canon_sha256":"e2c05f4d11a2df489fa27fba85050d3311468908afd34a4ff50710e5b109bbaa","abstract_canon_sha256":"509039e2763d927c24f579d4759dd7bf837957a7ce2279df4a8fddd27625d74c"},"schema_version":"1.0"},"canonical_sha256":"e0607df507658edbab069c676f6720ec8b76aba64a78c5c1d883adb668dda290","source":{"kind":"arxiv","id":"2112.13103","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2112.13103","created_at":"2026-07-05T04:46:34Z"},{"alias_kind":"arxiv_version","alias_value":"2112.13103v2","created_at":"2026-07-05T04:46:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2112.13103","created_at":"2026-07-05T04:46:34Z"},{"alias_kind":"pith_short_12","alias_value":"4BQH35IHMWHN","created_at":"2026-07-05T04:46:34Z"},{"alias_kind":"pith_short_16","alias_value":"4BQH35IHMWHNXKYG","created_at":"2026-07-05T04:46:34Z"},{"alias_kind":"pith_short_8","alias_value":"4BQH35IH","created_at":"2026-07-05T04:46:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:4BQH35IHMWHNXKYGTRTW6ZZA5S","target":"record","payload":{"canonical_record":{"source":{"id":"2112.13103","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2021-12-24T16:47:33Z","cross_cats_sorted":["astro-ph.GA","hep-th"],"title_canon_sha256":"e2c05f4d11a2df489fa27fba85050d3311468908afd34a4ff50710e5b109bbaa","abstract_canon_sha256":"509039e2763d927c24f579d4759dd7bf837957a7ce2279df4a8fddd27625d74c"},"schema_version":"1.0"},"canonical_sha256":"e0607df507658edbab069c676f6720ec8b76aba64a78c5c1d883adb668dda290","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:46:34.458545Z","signature_b64":"EucmM/mqQOEh1dmLhkf3N4l/oUwFo8W4U2kGDNq7cEpvgFHGAH2ycXwcBUogsktsiEgAGyahw42pq96oRqPwAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e0607df507658edbab069c676f6720ec8b76aba64a78c5c1d883adb668dda290","last_reissued_at":"2026-07-05T04:46:34.458096Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:46:34.458096Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2112.13103","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":"2026-07-05T04:46:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"srlgzdYs8lYJYTrFJZmgasL92KMJyxhVq13UeNqL8nEOtj7Ci9lj7m7Q2+V5N6OimTAaSA0AWpJjFhkyRRXNDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T05:05:39.085946Z"},"content_sha256":"504874bca741e32019deda23b4d66c486b897e6f23969edf72f777f0501e5e38","schema_version":"1.0","event_id":"sha256:504874bca741e32019deda23b4d66c486b897e6f23969edf72f777f0501e5e38"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:4BQH35IHMWHNXKYGTRTW6ZZA5S","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Gravitational potential and galaxy rotation curves in multi-fractional spacetimes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["astro-ph.GA","hep-th"],"primary_cat":"gr-qc","authors_text":"Gabriele U. Varieschi, Gianluca Calcagni","submitted_at":"2021-12-24T16:47:33Z","abstract_excerpt":"Multi-fractional theories with integer-order derivatives are models of gravitational and matter fields living in spacetimes with variable Hausdorff and spectral dimension, originally proposed as descriptions of geometries arising in quantum gravity. We derive the Poisson equation and the Newtonian potential of these theories starting from their covariant modified Einstein's equations. In particular, in the case of the theory $T_v$ with weighted derivatives with small fractional corrections, we find a gravitational potential that grows logarithmically at large radii when the fractional exponent"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.13103","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/2112.13103/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":"2026-07-05T04:46:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2C4f0XKhp01O/b/5XvGmAtVDgYWqcewMPJYRNp+XyMsVWQXH+axxYJdRNTieoJKo04rW/tUvP0tlobjishevAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T05:05:39.086330Z"},"content_sha256":"9702792071d23d3b79d986fc5cca76a11821894909b15866ab1e067b8dd6e4e8","schema_version":"1.0","event_id":"sha256:9702792071d23d3b79d986fc5cca76a11821894909b15866ab1e067b8dd6e4e8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4BQH35IHMWHNXKYGTRTW6ZZA5S/bundle.json","state_url":"https://pith.science/pith/4BQH35IHMWHNXKYGTRTW6ZZA5S/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4BQH35IHMWHNXKYGTRTW6ZZA5S/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":"2026-07-06T05:05:39Z","links":{"resolver":"https://pith.science/pith/4BQH35IHMWHNXKYGTRTW6ZZA5S","bundle":"https://pith.science/pith/4BQH35IHMWHNXKYGTRTW6ZZA5S/bundle.json","state":"https://pith.science/pith/4BQH35IHMWHNXKYGTRTW6ZZA5S/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4BQH35IHMWHNXKYGTRTW6ZZA5S/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:4BQH35IHMWHNXKYGTRTW6ZZA5S","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":"509039e2763d927c24f579d4759dd7bf837957a7ce2279df4a8fddd27625d74c","cross_cats_sorted":["astro-ph.GA","hep-th"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2021-12-24T16:47:33Z","title_canon_sha256":"e2c05f4d11a2df489fa27fba85050d3311468908afd34a4ff50710e5b109bbaa"},"schema_version":"1.0","source":{"id":"2112.13103","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2112.13103","created_at":"2026-07-05T04:46:34Z"},{"alias_kind":"arxiv_version","alias_value":"2112.13103v2","created_at":"2026-07-05T04:46:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2112.13103","created_at":"2026-07-05T04:46:34Z"},{"alias_kind":"pith_short_12","alias_value":"4BQH35IHMWHN","created_at":"2026-07-05T04:46:34Z"},{"alias_kind":"pith_short_16","alias_value":"4BQH35IHMWHNXKYG","created_at":"2026-07-05T04:46:34Z"},{"alias_kind":"pith_short_8","alias_value":"4BQH35IH","created_at":"2026-07-05T04:46:34Z"}],"graph_snapshots":[{"event_id":"sha256:9702792071d23d3b79d986fc5cca76a11821894909b15866ab1e067b8dd6e4e8","target":"graph","created_at":"2026-07-05T04:46:34Z","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/2112.13103/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Multi-fractional theories with integer-order derivatives are models of gravitational and matter fields living in spacetimes with variable Hausdorff and spectral dimension, originally proposed as descriptions of geometries arising in quantum gravity. We derive the Poisson equation and the Newtonian potential of these theories starting from their covariant modified Einstein's equations. In particular, in the case of the theory $T_v$ with weighted derivatives with small fractional corrections, we find a gravitational potential that grows logarithmically at large radii when the fractional exponent","authors_text":"Gabriele U. Varieschi, Gianluca Calcagni","cross_cats":["astro-ph.GA","hep-th"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2021-12-24T16:47:33Z","title":"Gravitational potential and galaxy rotation curves in multi-fractional spacetimes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.13103","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":"sha256:504874bca741e32019deda23b4d66c486b897e6f23969edf72f777f0501e5e38","target":"record","created_at":"2026-07-05T04:46:34Z","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":"509039e2763d927c24f579d4759dd7bf837957a7ce2279df4a8fddd27625d74c","cross_cats_sorted":["astro-ph.GA","hep-th"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2021-12-24T16:47:33Z","title_canon_sha256":"e2c05f4d11a2df489fa27fba85050d3311468908afd34a4ff50710e5b109bbaa"},"schema_version":"1.0","source":{"id":"2112.13103","kind":"arxiv","version":2}},"canonical_sha256":"e0607df507658edbab069c676f6720ec8b76aba64a78c5c1d883adb668dda290","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e0607df507658edbab069c676f6720ec8b76aba64a78c5c1d883adb668dda290","first_computed_at":"2026-07-05T04:46:34.458096Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:46:34.458096Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EucmM/mqQOEh1dmLhkf3N4l/oUwFo8W4U2kGDNq7cEpvgFHGAH2ycXwcBUogsktsiEgAGyahw42pq96oRqPwAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T04:46:34.458545Z","signed_message":"canonical_sha256_bytes"},"source_id":"2112.13103","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:504874bca741e32019deda23b4d66c486b897e6f23969edf72f777f0501e5e38","sha256:9702792071d23d3b79d986fc5cca76a11821894909b15866ab1e067b8dd6e4e8"],"state_sha256":"fa1a2db9252fb7a397891a7f21ebca5c295b2e98c6be62ae00caf1341fa92ba5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lz9L1qP7BxwuQsyfleLAxVn7+8mhzTwttAqS1pbgeD3acwNOqvPd7LuzidqbiSjkBH8kRuI1yw44+CjKKV9QDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T05:05:39.088399Z","bundle_sha256":"1d3c9a4fac047cdd82b017c6f8692d33763fdb4560914bd8f0e20d08abe9f08f"}}