{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:CLCA577POWFJWQGQKLWQXW6KGA","short_pith_number":"pith:CLCA577P","canonical_record":{"source":{"id":"2606.00381","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-05-29T21:45:39Z","cross_cats_sorted":[],"title_canon_sha256":"c49433f34192b4f2597f431c9dd03d47dab2afe5a0c28df39a06e03449b770db","abstract_canon_sha256":"cef4ce8a0e0098cdf7c66f12591e9318a3cbb8e0ada949ce2e173c810c6f3537"},"schema_version":"1.0"},"canonical_sha256":"12c40effef758a9b40d052ed0bdbca30312b60fdc75f9c9e6982ca5c3abe86f0","source":{"kind":"arxiv","id":"2606.00381","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.00381","created_at":"2026-06-02T01:03:53Z"},{"alias_kind":"arxiv_version","alias_value":"2606.00381v1","created_at":"2026-06-02T01:03:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.00381","created_at":"2026-06-02T01:03:53Z"},{"alias_kind":"pith_short_12","alias_value":"CLCA577POWFJ","created_at":"2026-06-02T01:03:53Z"},{"alias_kind":"pith_short_16","alias_value":"CLCA577POWFJWQGQ","created_at":"2026-06-02T01:03:53Z"},{"alias_kind":"pith_short_8","alias_value":"CLCA577P","created_at":"2026-06-02T01:03:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:CLCA577POWFJWQGQKLWQXW6KGA","target":"record","payload":{"canonical_record":{"source":{"id":"2606.00381","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-05-29T21:45:39Z","cross_cats_sorted":[],"title_canon_sha256":"c49433f34192b4f2597f431c9dd03d47dab2afe5a0c28df39a06e03449b770db","abstract_canon_sha256":"cef4ce8a0e0098cdf7c66f12591e9318a3cbb8e0ada949ce2e173c810c6f3537"},"schema_version":"1.0"},"canonical_sha256":"12c40effef758a9b40d052ed0bdbca30312b60fdc75f9c9e6982ca5c3abe86f0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T01:03:53.087465Z","signature_b64":"E9OtlPhxZKELuKkZXhIxlg4sVgUJ2CG9QOSiu9563H3h6QHcL948nR2nN5/ItmdFcRfI2x5pDTyKrZLNkhe1BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"12c40effef758a9b40d052ed0bdbca30312b60fdc75f9c9e6982ca5c3abe86f0","last_reissued_at":"2026-06-02T01:03:53.087068Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T01:03:53.087068Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.00381","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":"2026-06-02T01:03:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XzdZBqqJYKZCdvyoDjxK8ORmlBPeG3uOaeMgqDPxKKi7sIXrcQgrxwQvuW548fsI5ZlFgjyShMyqjyGlUFcDDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T04:11:11.675725Z"},"content_sha256":"3c520eb2469de8a6a9bb4042c21d0e1601588a3fcd9420f6aa36a3251d5896de","schema_version":"1.0","event_id":"sha256:3c520eb2469de8a6a9bb4042c21d0e1601588a3fcd9420f6aa36a3251d5896de"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:CLCA577POWFJWQGQKLWQXW6KGA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Quantified Two-projection Theorem for Nonlinear Projections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Krystal Taylor, Zhangze Li","submitted_at":"2026-05-29T21:45:39Z","abstract_excerpt":"The classic Besicovitch projection theorem asserts that if a set is purely $1$-unrectifiable with finite length in $\\mathbb{R}^2$, its orthogonal projection has Lebesgue measure zero in almost every direction. In the opposite direction, the two-projection theorem states that if a Borel set has zero measure under orthogonal projections onto two distinct non-antipodal directions, it must be purely $1$-unrectifiable. We extend the two-projection theorem to certain families nonlinear projections and consider applications to pinned distance sets, radial projections, and curve projection operators. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.00381","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/2606.00381/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":"2026-06-02T01:03:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wDXt355KAl09XzFxZ3ETnHhDCZG5kuSaRa3IGRdFxNssiAIjNgzW9OwHTrhjJQUK201EERD0tEf/eD6wljW4DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T04:11:11.676561Z"},"content_sha256":"25a6732bcc15a7cbd9c80ecde44b99b05fd4b818f8827fd91ef608b6a3b4562e","schema_version":"1.0","event_id":"sha256:25a6732bcc15a7cbd9c80ecde44b99b05fd4b818f8827fd91ef608b6a3b4562e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CLCA577POWFJWQGQKLWQXW6KGA/bundle.json","state_url":"https://pith.science/pith/CLCA577POWFJWQGQKLWQXW6KGA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CLCA577POWFJWQGQKLWQXW6KGA/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":"2026-06-10T04:11:11Z","links":{"resolver":"https://pith.science/pith/CLCA577POWFJWQGQKLWQXW6KGA","bundle":"https://pith.science/pith/CLCA577POWFJWQGQKLWQXW6KGA/bundle.json","state":"https://pith.science/pith/CLCA577POWFJWQGQKLWQXW6KGA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CLCA577POWFJWQGQKLWQXW6KGA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:CLCA577POWFJWQGQKLWQXW6KGA","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":"cef4ce8a0e0098cdf7c66f12591e9318a3cbb8e0ada949ce2e173c810c6f3537","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-05-29T21:45:39Z","title_canon_sha256":"c49433f34192b4f2597f431c9dd03d47dab2afe5a0c28df39a06e03449b770db"},"schema_version":"1.0","source":{"id":"2606.00381","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.00381","created_at":"2026-06-02T01:03:53Z"},{"alias_kind":"arxiv_version","alias_value":"2606.00381v1","created_at":"2026-06-02T01:03:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.00381","created_at":"2026-06-02T01:03:53Z"},{"alias_kind":"pith_short_12","alias_value":"CLCA577POWFJ","created_at":"2026-06-02T01:03:53Z"},{"alias_kind":"pith_short_16","alias_value":"CLCA577POWFJWQGQ","created_at":"2026-06-02T01:03:53Z"},{"alias_kind":"pith_short_8","alias_value":"CLCA577P","created_at":"2026-06-02T01:03:53Z"}],"graph_snapshots":[{"event_id":"sha256:25a6732bcc15a7cbd9c80ecde44b99b05fd4b818f8827fd91ef608b6a3b4562e","target":"graph","created_at":"2026-06-02T01:03:53Z","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/2606.00381/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The classic Besicovitch projection theorem asserts that if a set is purely $1$-unrectifiable with finite length in $\\mathbb{R}^2$, its orthogonal projection has Lebesgue measure zero in almost every direction. In the opposite direction, the two-projection theorem states that if a Borel set has zero measure under orthogonal projections onto two distinct non-antipodal directions, it must be purely $1$-unrectifiable. We extend the two-projection theorem to certain families nonlinear projections and consider applications to pinned distance sets, radial projections, and curve projection operators. ","authors_text":"Krystal Taylor, Zhangze Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-05-29T21:45:39Z","title":"A Quantified Two-projection Theorem for Nonlinear Projections"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.00381","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":"sha256:3c520eb2469de8a6a9bb4042c21d0e1601588a3fcd9420f6aa36a3251d5896de","target":"record","created_at":"2026-06-02T01:03:53Z","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":"cef4ce8a0e0098cdf7c66f12591e9318a3cbb8e0ada949ce2e173c810c6f3537","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-05-29T21:45:39Z","title_canon_sha256":"c49433f34192b4f2597f431c9dd03d47dab2afe5a0c28df39a06e03449b770db"},"schema_version":"1.0","source":{"id":"2606.00381","kind":"arxiv","version":1}},"canonical_sha256":"12c40effef758a9b40d052ed0bdbca30312b60fdc75f9c9e6982ca5c3abe86f0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"12c40effef758a9b40d052ed0bdbca30312b60fdc75f9c9e6982ca5c3abe86f0","first_computed_at":"2026-06-02T01:03:53.087068Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T01:03:53.087068Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"E9OtlPhxZKELuKkZXhIxlg4sVgUJ2CG9QOSiu9563H3h6QHcL948nR2nN5/ItmdFcRfI2x5pDTyKrZLNkhe1BA==","signature_status":"signed_v1","signed_at":"2026-06-02T01:03:53.087465Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.00381","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3c520eb2469de8a6a9bb4042c21d0e1601588a3fcd9420f6aa36a3251d5896de","sha256:25a6732bcc15a7cbd9c80ecde44b99b05fd4b818f8827fd91ef608b6a3b4562e"],"state_sha256":"a6f9dda5a309b01e0c956d3a8fd54bb762a3be57e296be6903387eed57b19bb1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UwHsuNNQ8korPcdGVE1IJLVELs1okl3tRKwdXZPDJYZSyaLdHI7/GcocMWHomuw5Ik9QjS/ZijmfqZ4tm5b9AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T04:11:11.684201Z","bundle_sha256":"561b606302ebcb2a1f3dc0fbd1431fa2448bc1f15cbef17f511ce5d498ed8bea"}}