{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:5RJ45H6GBZZR5QZYBWSXDEVNNA","short_pith_number":"pith:5RJ45H6G","canonical_record":{"source":{"id":"1406.0781","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2014-06-03T17:04:11Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b4106f1888265bb444ef3a8fbba3988ec51888a7b051505d52039d9fcacfe299","abstract_canon_sha256":"312bc536f5f668a5e4457ca52bc45632b3b1ea2058e8ae0606954fae4b0bb1bb"},"schema_version":"1.0"},"canonical_sha256":"ec53ce9fc60e731ec3380da57192ad68174145bb3244428dd406336b75b0f9ff","source":{"kind":"arxiv","id":"1406.0781","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.0781","created_at":"2026-05-18T01:53:09Z"},{"alias_kind":"arxiv_version","alias_value":"1406.0781v3","created_at":"2026-05-18T01:53:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.0781","created_at":"2026-05-18T01:53:09Z"},{"alias_kind":"pith_short_12","alias_value":"5RJ45H6GBZZR","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"5RJ45H6GBZZR5QZY","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"5RJ45H6G","created_at":"2026-05-18T12:28:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:5RJ45H6GBZZR5QZYBWSXDEVNNA","target":"record","payload":{"canonical_record":{"source":{"id":"1406.0781","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2014-06-03T17:04:11Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b4106f1888265bb444ef3a8fbba3988ec51888a7b051505d52039d9fcacfe299","abstract_canon_sha256":"312bc536f5f668a5e4457ca52bc45632b3b1ea2058e8ae0606954fae4b0bb1bb"},"schema_version":"1.0"},"canonical_sha256":"ec53ce9fc60e731ec3380da57192ad68174145bb3244428dd406336b75b0f9ff","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:53:09.328882Z","signature_b64":"EGXV+xrAdpigFfujFlJxHWtY1K2yRJhymO8K2EyX6rDMIuncJbf56iNDX7NOYZYeIJdme0+mYqaGsp83etM7Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec53ce9fc60e731ec3380da57192ad68174145bb3244428dd406336b75b0f9ff","last_reissued_at":"2026-05-18T01:53:09.328398Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:53:09.328398Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1406.0781","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:53:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"esW+41J2Ohsf5Nb9VT8dh2sc4v18/4rxcB1kuu50ockQBv+AlLu5S/Q0Czoryv3RIYT6LpzzCtcd8PhjezY9Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T05:40:27.286627Z"},"content_sha256":"83dbf8429f922cfe7b97a6db6367c9282d1367ccbac0cbbeb57722bb68bbd380","schema_version":"1.0","event_id":"sha256:83dbf8429f922cfe7b97a6db6367c9282d1367ccbac0cbbeb57722bb68bbd380"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:5RJ45H6GBZZR5QZYBWSXDEVNNA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The smallest sets of points not determined by their X-rays","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Andreas Alpers, David G. Larman","submitted_at":"2014-06-03T17:04:11Z","abstract_excerpt":"Let $F$ be an $n$-point set in $\\mathbb{K}^d$ with $\\mathbb{K}\\in\\{\\mathbb{R},\\mathbb{Z}\\}$ and $d\\geq 2$. A (discrete) X-ray of $F$ in direction $s$ gives the number of points of $F$ on each line parallel to $s$. We define $\\psi_{\\mathbb{K}^d}(m)$ as the minimum number $n$ for which there exist $m$ directions $s_1,...,s_m$ (pairwise linearly independent and spanning $\\mathbb{R}^d$) such that two $n$-point sets in $\\mathbb{K}^d$ exist that have the same X-rays in these directions. The bound $\\psi_{\\mathbb{Z}^d}(m)\\leq 2^{m-1}$ has been observed many times in the literature. In this note we sho"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.0781","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T01:53:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SzVqTNzxtzxQNRzEckqpyz5SeRKukZLS+K97pujq7wFME2sNAHFDosrQkb8NP7mW9OEp74reABj55/xqoQimDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T05:40:27.287309Z"},"content_sha256":"1a4de76d522f8db3407a84c74d54b3c2d324e219d61b2b5e1acb4d6a4c5affdb","schema_version":"1.0","event_id":"sha256:1a4de76d522f8db3407a84c74d54b3c2d324e219d61b2b5e1acb4d6a4c5affdb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5RJ45H6GBZZR5QZYBWSXDEVNNA/bundle.json","state_url":"https://pith.science/pith/5RJ45H6GBZZR5QZYBWSXDEVNNA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5RJ45H6GBZZR5QZYBWSXDEVNNA/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":"2026-05-26T05:40:27Z","links":{"resolver":"https://pith.science/pith/5RJ45H6GBZZR5QZYBWSXDEVNNA","bundle":"https://pith.science/pith/5RJ45H6GBZZR5QZYBWSXDEVNNA/bundle.json","state":"https://pith.science/pith/5RJ45H6GBZZR5QZYBWSXDEVNNA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5RJ45H6GBZZR5QZYBWSXDEVNNA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:5RJ45H6GBZZR5QZYBWSXDEVNNA","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":"312bc536f5f668a5e4457ca52bc45632b3b1ea2058e8ae0606954fae4b0bb1bb","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2014-06-03T17:04:11Z","title_canon_sha256":"b4106f1888265bb444ef3a8fbba3988ec51888a7b051505d52039d9fcacfe299"},"schema_version":"1.0","source":{"id":"1406.0781","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.0781","created_at":"2026-05-18T01:53:09Z"},{"alias_kind":"arxiv_version","alias_value":"1406.0781v3","created_at":"2026-05-18T01:53:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.0781","created_at":"2026-05-18T01:53:09Z"},{"alias_kind":"pith_short_12","alias_value":"5RJ45H6GBZZR","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"5RJ45H6GBZZR5QZY","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"5RJ45H6G","created_at":"2026-05-18T12:28:14Z"}],"graph_snapshots":[{"event_id":"sha256:1a4de76d522f8db3407a84c74d54b3c2d324e219d61b2b5e1acb4d6a4c5affdb","target":"graph","created_at":"2026-05-18T01:53:09Z","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"},"paper":{"abstract_excerpt":"Let $F$ be an $n$-point set in $\\mathbb{K}^d$ with $\\mathbb{K}\\in\\{\\mathbb{R},\\mathbb{Z}\\}$ and $d\\geq 2$. A (discrete) X-ray of $F$ in direction $s$ gives the number of points of $F$ on each line parallel to $s$. We define $\\psi_{\\mathbb{K}^d}(m)$ as the minimum number $n$ for which there exist $m$ directions $s_1,...,s_m$ (pairwise linearly independent and spanning $\\mathbb{R}^d$) such that two $n$-point sets in $\\mathbb{K}^d$ exist that have the same X-rays in these directions. The bound $\\psi_{\\mathbb{Z}^d}(m)\\leq 2^{m-1}$ has been observed many times in the literature. In this note we sho","authors_text":"Andreas Alpers, David G. Larman","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2014-06-03T17:04:11Z","title":"The smallest sets of points not determined by their X-rays"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.0781","kind":"arxiv","version":3},"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":"sha256:83dbf8429f922cfe7b97a6db6367c9282d1367ccbac0cbbeb57722bb68bbd380","target":"record","created_at":"2026-05-18T01:53:09Z","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":"312bc536f5f668a5e4457ca52bc45632b3b1ea2058e8ae0606954fae4b0bb1bb","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2014-06-03T17:04:11Z","title_canon_sha256":"b4106f1888265bb444ef3a8fbba3988ec51888a7b051505d52039d9fcacfe299"},"schema_version":"1.0","source":{"id":"1406.0781","kind":"arxiv","version":3}},"canonical_sha256":"ec53ce9fc60e731ec3380da57192ad68174145bb3244428dd406336b75b0f9ff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ec53ce9fc60e731ec3380da57192ad68174145bb3244428dd406336b75b0f9ff","first_computed_at":"2026-05-18T01:53:09.328398Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:53:09.328398Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EGXV+xrAdpigFfujFlJxHWtY1K2yRJhymO8K2EyX6rDMIuncJbf56iNDX7NOYZYeIJdme0+mYqaGsp83etM7Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T01:53:09.328882Z","signed_message":"canonical_sha256_bytes"},"source_id":"1406.0781","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:83dbf8429f922cfe7b97a6db6367c9282d1367ccbac0cbbeb57722bb68bbd380","sha256:1a4de76d522f8db3407a84c74d54b3c2d324e219d61b2b5e1acb4d6a4c5affdb"],"state_sha256":"bfb171661608db6019db46029a3ba0fc9dcdcd3e0b79212da852011ca297e90d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nYCySxnHAZ2dI5jfEhWpbVD/o75uiHtZz43+Gvbo0lNTi/wWCssODMrooAwVSKlVSuMjA5wwwDA4m4YC79mnDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T05:40:27.291004Z","bundle_sha256":"4a450a583e0662311329ac80e9df797ac9c55884e17dab0f632d98584bc15c0a"}}