{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:I3XGNGSSX4NDEF7IEWOZEYCEL2","short_pith_number":"pith:I3XGNGSS","canonical_record":{"source":{"id":"2604.05648","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.RO","submitted_at":"2026-04-07T09:53:48Z","cross_cats_sorted":["cs.SY","eess.SY"],"title_canon_sha256":"6e85d941bebb227f0bd46bb69bd979c71e2ec333272a4990289b25b0c07b3a6b","abstract_canon_sha256":"8b1d2254620f0ec068416c1a12053556f5a34f25da1638fc773e998d49d5f484"},"schema_version":"1.0"},"canonical_sha256":"46ee669a52bf1a3217e8259d9260445e9773f2c9d8bbb66f7deb5f6b68f18b28","source":{"kind":"arxiv","id":"2604.05648","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.05648","created_at":"2026-06-23T01:12:04Z"},{"alias_kind":"arxiv_version","alias_value":"2604.05648v2","created_at":"2026-06-23T01:12:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.05648","created_at":"2026-06-23T01:12:04Z"},{"alias_kind":"pith_short_12","alias_value":"I3XGNGSSX4ND","created_at":"2026-06-23T01:12:04Z"},{"alias_kind":"pith_short_16","alias_value":"I3XGNGSSX4NDEF7I","created_at":"2026-06-23T01:12:04Z"},{"alias_kind":"pith_short_8","alias_value":"I3XGNGSS","created_at":"2026-06-23T01:12:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:I3XGNGSSX4NDEF7IEWOZEYCEL2","target":"record","payload":{"canonical_record":{"source":{"id":"2604.05648","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.RO","submitted_at":"2026-04-07T09:53:48Z","cross_cats_sorted":["cs.SY","eess.SY"],"title_canon_sha256":"6e85d941bebb227f0bd46bb69bd979c71e2ec333272a4990289b25b0c07b3a6b","abstract_canon_sha256":"8b1d2254620f0ec068416c1a12053556f5a34f25da1638fc773e998d49d5f484"},"schema_version":"1.0"},"canonical_sha256":"46ee669a52bf1a3217e8259d9260445e9773f2c9d8bbb66f7deb5f6b68f18b28","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T01:12:04.437066Z","signature_b64":"haFtZU4MS2DB7fpXYFS8kmWwqwojBr9n2kD2OeR4y5TNrmwVFhdicyeJVh6OnOYK1grxx8Plqyp4A2lMtoqUAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"46ee669a52bf1a3217e8259d9260445e9773f2c9d8bbb66f7deb5f6b68f18b28","last_reissued_at":"2026-06-23T01:12:04.436630Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T01:12:04.436630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2604.05648","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":"2026-06-23T01:12:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e2NXtDxJCxozJxaoUYzBCiqjtBnQlSvwJQiMBbIr2x8W27Je4bCkExTGfJK6JrJV1vo8YhN0l2nCA+3I5LcoBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T21:52:08.964696Z"},"content_sha256":"b7b6ccf1fa4b5ad119d2aa4ea92da427ec7f23a6427c81e4d3121b6372c73d62","schema_version":"1.0","event_id":"sha256:b7b6ccf1fa4b5ad119d2aa4ea92da427ec7f23a6427c81e4d3121b6372c73d62"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:I3XGNGSSX4NDEF7IEWOZEYCEL2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Leaderless Collective Motion in Affine Formation Control over the Complex Plane","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"Modifying the Laplacian weights allows leaderless swarms to perform any affine transformation of a reference shape in the plane.","cross_cats":["cs.SY","eess.SY"],"primary_cat":"cs.RO","authors_text":"Enric Morella, Hector Garcia de Marina, Jesus Bautista, Lili Wang","submitted_at":"2026-04-07T09:53:48Z","abstract_excerpt":"We propose a method for the collective maneuvering of affine formations in the plane by modifying the original weights of the Laplacian matrix used to achieve static formations of robot swarms. Specifically, the resulting collective motion is characterized as a time-varying affine transformation of a reference configuration, or shape. Unlike the traditional leader-follower strategy, our leaderless scheme allows agents to maintain distinct and possibly time-varying velocities, enabling a broader range of collective motions, including all the linear combinations of translations, rotations, scali"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The resulting collective motion is characterized as a time-varying affine transformation of a reference configuration, or shape, with an analytic solution governing the motion via explicitly designed eigenvectors and eigenvalues of the modified Laplacian.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The analysis assumes that the modified weights produce a Laplacian matrix whose eigenvectors and eigenvalues can be designed to exactly realize any desired affine motion while preserving formation stability, without specifying how to choose weights for arbitrary graphs or under disturbances.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Modifying Laplacian weights produces leaderless affine formation motions in the plane, with an analytic solution via complex-number eigenvectors and eigenvalues.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Modifying the Laplacian weights allows leaderless swarms to perform any affine transformation of a reference shape in the plane.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"6668528f4745db0d1dd841086b20df77f961a5061cf825225c0177b9991b204c"},"source":{"id":"2604.05648","kind":"arxiv","version":2},"verdict":{"id":"15bb6339-fef4-42a9-88a8-da810e9d66d6","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T19:32:10.345010Z","strongest_claim":"The resulting collective motion is characterized as a time-varying affine transformation of a reference configuration, or shape, with an analytic solution governing the motion via explicitly designed eigenvectors and eigenvalues of the modified Laplacian.","one_line_summary":"Modifying Laplacian weights produces leaderless affine formation motions in the plane, with an analytic solution via complex-number eigenvectors and eigenvalues.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The analysis assumes that the modified weights produce a Laplacian matrix whose eigenvectors and eigenvalues can be designed to exactly realize any desired affine motion while preserving formation stability, without specifying how to choose weights for arbitrary graphs or under disturbances.","pith_extraction_headline":"Modifying the Laplacian weights allows leaderless swarms to perform any affine transformation of a reference shape in the plane."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.05648/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":"15bb6339-fef4-42a9-88a8-da810e9d66d6"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-23T01:12:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4TnkABxPEqbFc4GtlZdKEADxaWzndZ8xPPNUVFSLhjlMH0dA42dT3qqgkfG3VhGgNCJMJA5Qmr8FDE2MQ374AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T21:52:08.965170Z"},"content_sha256":"3640d676a8600e910070393078f0a8627c1c1b9334234c4221a687fcb522f48e","schema_version":"1.0","event_id":"sha256:3640d676a8600e910070393078f0a8627c1c1b9334234c4221a687fcb522f48e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I3XGNGSSX4NDEF7IEWOZEYCEL2/bundle.json","state_url":"https://pith.science/pith/I3XGNGSSX4NDEF7IEWOZEYCEL2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I3XGNGSSX4NDEF7IEWOZEYCEL2/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":"2026-06-26T21:52:08Z","links":{"resolver":"https://pith.science/pith/I3XGNGSSX4NDEF7IEWOZEYCEL2","bundle":"https://pith.science/pith/I3XGNGSSX4NDEF7IEWOZEYCEL2/bundle.json","state":"https://pith.science/pith/I3XGNGSSX4NDEF7IEWOZEYCEL2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I3XGNGSSX4NDEF7IEWOZEYCEL2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:I3XGNGSSX4NDEF7IEWOZEYCEL2","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":"8b1d2254620f0ec068416c1a12053556f5a34f25da1638fc773e998d49d5f484","cross_cats_sorted":["cs.SY","eess.SY"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.RO","submitted_at":"2026-04-07T09:53:48Z","title_canon_sha256":"6e85d941bebb227f0bd46bb69bd979c71e2ec333272a4990289b25b0c07b3a6b"},"schema_version":"1.0","source":{"id":"2604.05648","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.05648","created_at":"2026-06-23T01:12:04Z"},{"alias_kind":"arxiv_version","alias_value":"2604.05648v2","created_at":"2026-06-23T01:12:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.05648","created_at":"2026-06-23T01:12:04Z"},{"alias_kind":"pith_short_12","alias_value":"I3XGNGSSX4ND","created_at":"2026-06-23T01:12:04Z"},{"alias_kind":"pith_short_16","alias_value":"I3XGNGSSX4NDEF7I","created_at":"2026-06-23T01:12:04Z"},{"alias_kind":"pith_short_8","alias_value":"I3XGNGSS","created_at":"2026-06-23T01:12:04Z"}],"graph_snapshots":[{"event_id":"sha256:3640d676a8600e910070393078f0a8627c1c1b9334234c4221a687fcb522f48e","target":"graph","created_at":"2026-06-23T01:12:04Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"The resulting collective motion is characterized as a time-varying affine transformation of a reference configuration, or shape, with an analytic solution governing the motion via explicitly designed eigenvectors and eigenvalues of the modified Laplacian."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The analysis assumes that the modified weights produce a Laplacian matrix whose eigenvectors and eigenvalues can be designed to exactly realize any desired affine motion while preserving formation stability, without specifying how to choose weights for arbitrary graphs or under disturbances."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Modifying Laplacian weights produces leaderless affine formation motions in the plane, with an analytic solution via complex-number eigenvectors and eigenvalues."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Modifying the Laplacian weights allows leaderless swarms to perform any affine transformation of a reference shape in the plane."}],"snapshot_sha256":"6668528f4745db0d1dd841086b20df77f961a5061cf825225c0177b9991b204c"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2604.05648/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We propose a method for the collective maneuvering of affine formations in the plane by modifying the original weights of the Laplacian matrix used to achieve static formations of robot swarms. Specifically, the resulting collective motion is characterized as a time-varying affine transformation of a reference configuration, or shape. Unlike the traditional leader-follower strategy, our leaderless scheme allows agents to maintain distinct and possibly time-varying velocities, enabling a broader range of collective motions, including all the linear combinations of translations, rotations, scali","authors_text":"Enric Morella, Hector Garcia de Marina, Jesus Bautista, Lili Wang","cross_cats":["cs.SY","eess.SY"],"headline":"Modifying the Laplacian weights allows leaderless swarms to perform any affine transformation of a reference shape in the plane.","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.RO","submitted_at":"2026-04-07T09:53:48Z","title":"Leaderless Collective Motion in Affine Formation Control over the Complex Plane"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.05648","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-10T19:32:10.345010Z","id":"15bb6339-fef4-42a9-88a8-da810e9d66d6","model_set":{"reader":"grok-4.3"},"one_line_summary":"Modifying Laplacian weights produces leaderless affine formation motions in the plane, with an analytic solution via complex-number eigenvectors and eigenvalues.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Modifying the Laplacian weights allows leaderless swarms to perform any affine transformation of a reference shape in the plane.","strongest_claim":"The resulting collective motion is characterized as a time-varying affine transformation of a reference configuration, or shape, with an analytic solution governing the motion via explicitly designed eigenvectors and eigenvalues of the modified Laplacian.","weakest_assumption":"The analysis assumes that the modified weights produce a Laplacian matrix whose eigenvectors and eigenvalues can be designed to exactly realize any desired affine motion while preserving formation stability, without specifying how to choose weights for arbitrary graphs or under disturbances."}},"verdict_id":"15bb6339-fef4-42a9-88a8-da810e9d66d6"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b7b6ccf1fa4b5ad119d2aa4ea92da427ec7f23a6427c81e4d3121b6372c73d62","target":"record","created_at":"2026-06-23T01:12:04Z","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":"8b1d2254620f0ec068416c1a12053556f5a34f25da1638fc773e998d49d5f484","cross_cats_sorted":["cs.SY","eess.SY"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.RO","submitted_at":"2026-04-07T09:53:48Z","title_canon_sha256":"6e85d941bebb227f0bd46bb69bd979c71e2ec333272a4990289b25b0c07b3a6b"},"schema_version":"1.0","source":{"id":"2604.05648","kind":"arxiv","version":2}},"canonical_sha256":"46ee669a52bf1a3217e8259d9260445e9773f2c9d8bbb66f7deb5f6b68f18b28","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"46ee669a52bf1a3217e8259d9260445e9773f2c9d8bbb66f7deb5f6b68f18b28","first_computed_at":"2026-06-23T01:12:04.436630Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-23T01:12:04.436630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"haFtZU4MS2DB7fpXYFS8kmWwqwojBr9n2kD2OeR4y5TNrmwVFhdicyeJVh6OnOYK1grxx8Plqyp4A2lMtoqUAQ==","signature_status":"signed_v1","signed_at":"2026-06-23T01:12:04.437066Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.05648","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b7b6ccf1fa4b5ad119d2aa4ea92da427ec7f23a6427c81e4d3121b6372c73d62","sha256:3640d676a8600e910070393078f0a8627c1c1b9334234c4221a687fcb522f48e"],"state_sha256":"c98da8641f854955e8332c67205f5e77652afa70e852baed57a12c52dc52a343"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tt87lMTj3fHd8BKqgHFwlA/Q1LZf3PJ4jf3GdW+l5D7b37Sa7GjNFmnK9QUhOCMloW5E82+/R1+axO8fCLs+AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T21:52:08.967370Z","bundle_sha256":"b0b1bda304df9c3a87e23067778b3529e715dfc83153979872cff2452b8867e3"}}