{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:5OPRPFEMFNHBK7SBTHI7OA6VY3","short_pith_number":"pith:5OPRPFEM","canonical_record":{"source":{"id":"2605.19063","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-05-18T19:36:08Z","cross_cats_sorted":[],"title_canon_sha256":"8e2d37aeb4035c0924d3a047e012f88c16ff61250924fcb45972752b4e501e13","abstract_canon_sha256":"32ef5e037594724eb9009a58b031392deb1e45904afbbef29734a319453fb6d0"},"schema_version":"1.0"},"canonical_sha256":"eb9f17948c2b4e157e4199d1f703d5c6cf5dbb34564476decb054cbada913843","source":{"kind":"arxiv","id":"2605.19063","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.19063","created_at":"2026-05-20T01:05:25Z"},{"alias_kind":"arxiv_version","alias_value":"2605.19063v1","created_at":"2026-05-20T01:05:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.19063","created_at":"2026-05-20T01:05:25Z"},{"alias_kind":"pith_short_12","alias_value":"5OPRPFEMFNHB","created_at":"2026-05-20T01:05:25Z"},{"alias_kind":"pith_short_16","alias_value":"5OPRPFEMFNHBK7SB","created_at":"2026-05-20T01:05:25Z"},{"alias_kind":"pith_short_8","alias_value":"5OPRPFEM","created_at":"2026-05-20T01:05:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:5OPRPFEMFNHBK7SBTHI7OA6VY3","target":"record","payload":{"canonical_record":{"source":{"id":"2605.19063","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-05-18T19:36:08Z","cross_cats_sorted":[],"title_canon_sha256":"8e2d37aeb4035c0924d3a047e012f88c16ff61250924fcb45972752b4e501e13","abstract_canon_sha256":"32ef5e037594724eb9009a58b031392deb1e45904afbbef29734a319453fb6d0"},"schema_version":"1.0"},"canonical_sha256":"eb9f17948c2b4e157e4199d1f703d5c6cf5dbb34564476decb054cbada913843","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T01:05:25.658894Z","signature_b64":"xUfOa96KV4vkGwgLLoxHeFnHyF2/ocN1WuPfFbYbOW25tZWhyPv5OCwZOWUtpwkvWpyqbG5QOh7froyX11iwDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eb9f17948c2b4e157e4199d1f703d5c6cf5dbb34564476decb054cbada913843","last_reissued_at":"2026-05-20T01:05:25.658175Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T01:05:25.658175Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.19063","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":"2026-05-20T01:05:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O9EB5uzqQfG3MJ5nYvXO5HT1qrSAJzZGx49ZrJriRF7fsf2eYPSIpsVuvP8LTvmXxD+POChIBlVzuUPzVzhyCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T03:30:22.268503Z"},"content_sha256":"46f14456bcc3db95e87ae84b2e159739b1f04373828786a4d3ff147c1273abff","schema_version":"1.0","event_id":"sha256:46f14456bcc3db95e87ae84b2e159739b1f04373828786a4d3ff147c1273abff"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:5OPRPFEMFNHBK7SBTHI7OA6VY3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Mapping Uncharted Symmetries: Machine Discovery in Combinatorics","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Alessandro Iraci, Eugenio Cainelli, Giovanni Paolini, Lorenzo Luccioli, Michele D'Adderio","submitted_at":"2026-05-18T19:36:08Z","abstract_excerpt":"Inspired by long-standing open problems in algebraic combinatorics, we show that modern machine learning can meaningfully contribute to verifiable mathematical discoveries. In particular, we focus on the construction of simple mathematical functions under exact distributional constraints, a setting we formalize as Simple Learning Under Rigid Proportions (SLURP). We tackle this problem by introducing two methods: MapSeek-Functional, which models the desired function alternating pseudo-labeling and supervised training steps; and MapSeek-Symbolic, designed to directly produce symbolic formulas. W"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.19063","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/2605.19063/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":"2026-05-20T01:05:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mnwgbCDMxhG1drH5P09r01YzwqfeL0dt0OB/Oa/3A8sTNczcJNsXnrQDyPhLsUcbHkG2LxSN0vB1NKi1wLJmAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T03:30:22.269199Z"},"content_sha256":"400cd853b606bbabecef8c4f664236f984b43939d0927e4ae51ebe268736f894","schema_version":"1.0","event_id":"sha256:400cd853b606bbabecef8c4f664236f984b43939d0927e4ae51ebe268736f894"},{"event_type":"integrity_finding","subject_pith_number":"pith:2026:5OPRPFEMFNHBK7SBTHI7OA6VY3","target":"integrity","payload":{"note":"DOI in the printed bibliography is fragmented by whitespace or line breaks. A longer candidate (10.48550/arXiv.2304.12602.13) was visible in the surrounding text but could not be confirmed against doi.org as printed.","snippet":"Geordie Williamson. Is deep learning a useful tool for the pure mathematician?CoRR, abs/2304.12602, 2023. doi: 10.48550/ARXIV .2304.12602. URL https://doi.org/10. 48550/arXiv.2304.12602. 13 Aq, t-Combinatorics Background & Narayana Models T","arxiv_id":"2605.19063","detector":"doi_compliance","evidence":{"ref_index":33,"verdict_class":"incontrovertible","resolved_title":null,"printed_excerpt":"Geordie Williamson. Is deep learning a useful tool for the pure mathematician?CoRR, abs/2304.12602, 2023. doi: 10.48550/ARXIV .2304.12602. URL https://doi.org/10. 48550/arXiv.2304.12602. 13 Aq, t-Combinatorics Background & Narayana Models T","reconstructed_doi":"10.48550/arXiv.2304.12602.13"},"severity":"advisory","ref_index":33,"audited_at":"2026-05-20T11:52:15.934464Z","event_type":"pith.integrity.v1","detected_doi":"10.48550/arXiv.2304.12602.13","detector_url":"https://pith.science/pith-integrity-protocol#doi_compliance","external_url":null,"finding_type":"recoverable_identifier","evidence_hash":"e257e12a0f880fbd21d6f409f8c43c1757c673e8ebac2fa8b14a842e66b074a1","paper_version":1,"verdict_class":"incontrovertible","resolved_title":null,"detector_version":"1.0.0","detected_arxiv_id":null,"integrity_event_id":5128,"payload_sha256":"99dc2736f05ca75e0e833f1bbb500b260a3a86deefd772552c95779a6e27aaa1","signature_b64":"fQXaLsT0uNX6pElS9Bv0RP5m97jBJYo7M/UXFs7BSNFy4mViMfosZ5sGnWC5X86u+IV0PZT7lTAc874YfBSjCg==","signing_key_id":"pith-v1-2026-05"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-20T11:52:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rxJsDn3Gf/tmGhs3W8hzAhn9ijnXdDUNyeP9FTCzaA1MhclLXKS94tNSaaqj4KvnWhhh1ZJ2dQiy5a/WRiRRAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T03:30:22.270543Z"},"content_sha256":"2bd8e87b7df587384f2493120c39d62b9906c7f5dcd22a7cb2b830dfb5e9f34a","schema_version":"1.0","event_id":"sha256:2bd8e87b7df587384f2493120c39d62b9906c7f5dcd22a7cb2b830dfb5e9f34a"},{"event_type":"integrity_finding","subject_pith_number":"pith:2026:5OPRPFEMFNHBK7SBTHI7OA6VY3","target":"integrity","payload":{"note":"DOI in the printed bibliography is fragmented by whitespace or line breaks. A longer candidate (10.1016/j.aim.2020.107447.Id/No107447) was visible in the surrounding text but could not be confirmed against doi.org as printed.","snippet":"Michele D’Adderio, Alessandro Iraci, and Anna Vanden Wyngaerd. Theta operators, refined delta conjectures, and coinvariants.Adv. Math., 376:60, 2021. ISSN 0001-8708. doi: 10.1016/j. aim.2020.107447. Id/No 107447","arxiv_id":"2605.19063","detector":"doi_compliance","evidence":{"ref_index":11,"verdict_class":"incontrovertible","resolved_title":null,"printed_excerpt":"Michele D’Adderio, Alessandro Iraci, and Anna Vanden Wyngaerd. Theta operators, refined delta conjectures, and coinvariants.Adv. Math., 376:60, 2021. ISSN 0001-8708. doi: 10.1016/j. aim.2020.107447. Id/No 107447","reconstructed_doi":"10.1016/j.aim.2020.107447.Id/No107447"},"severity":"advisory","ref_index":11,"audited_at":"2026-05-20T11:52:15.934464Z","event_type":"pith.integrity.v1","detected_doi":"10.1016/j.aim.2020.107447.Id/No107447","detector_url":"https://pith.science/pith-integrity-protocol#doi_compliance","external_url":null,"finding_type":"recoverable_identifier","evidence_hash":"9ebdb08723c116475502d22eaa2493cd67aa6de928b718a27eef457b917c50a4","paper_version":1,"verdict_class":"incontrovertible","resolved_title":null,"detector_version":"1.0.0","detected_arxiv_id":null,"integrity_event_id":5127,"payload_sha256":"1fb5837a7a0747f2a11c7fde98985f35d79b9d170c1d75903962c6710b9b9ec3","signature_b64":"TD4omKAWfeiZXNfNXeY8ZEUX+bNY+UtQtZ+OXQ9ExKel+tr03sqjYe1gFGFhlDzqvAW+kPBEZumgHACde4TlDA==","signing_key_id":"pith-v1-2026-05"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-20T11:52:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PxiMRaHQLbfwrmQhsu2EIJnZFdihgM8bTNqpVQH4VjpDAeN5xojr3euOZMsPjzbSXfZonqKMRUzHT6WiPKf2Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T03:30:22.271108Z"},"content_sha256":"4f94158fe8a60c705cda964a1da2a3139499958bcdefa0c4ec4ae93f790b794a","schema_version":"1.0","event_id":"sha256:4f94158fe8a60c705cda964a1da2a3139499958bcdefa0c4ec4ae93f790b794a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5OPRPFEMFNHBK7SBTHI7OA6VY3/bundle.json","state_url":"https://pith.science/pith/5OPRPFEMFNHBK7SBTHI7OA6VY3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5OPRPFEMFNHBK7SBTHI7OA6VY3/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":"2026-05-21T03:30:22Z","links":{"resolver":"https://pith.science/pith/5OPRPFEMFNHBK7SBTHI7OA6VY3","bundle":"https://pith.science/pith/5OPRPFEMFNHBK7SBTHI7OA6VY3/bundle.json","state":"https://pith.science/pith/5OPRPFEMFNHBK7SBTHI7OA6VY3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5OPRPFEMFNHBK7SBTHI7OA6VY3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:5OPRPFEMFNHBK7SBTHI7OA6VY3","merge_version":"pith-open-graph-merge-v1","event_count":4,"valid_event_count":4,"invalid_event_count":0,"equivocation_count":1,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"32ef5e037594724eb9009a58b031392deb1e45904afbbef29734a319453fb6d0","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-05-18T19:36:08Z","title_canon_sha256":"8e2d37aeb4035c0924d3a047e012f88c16ff61250924fcb45972752b4e501e13"},"schema_version":"1.0","source":{"id":"2605.19063","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.19063","created_at":"2026-05-20T01:05:25Z"},{"alias_kind":"arxiv_version","alias_value":"2605.19063v1","created_at":"2026-05-20T01:05:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.19063","created_at":"2026-05-20T01:05:25Z"},{"alias_kind":"pith_short_12","alias_value":"5OPRPFEMFNHB","created_at":"2026-05-20T01:05:25Z"},{"alias_kind":"pith_short_16","alias_value":"5OPRPFEMFNHBK7SB","created_at":"2026-05-20T01:05:25Z"},{"alias_kind":"pith_short_8","alias_value":"5OPRPFEM","created_at":"2026-05-20T01:05:25Z"}],"graph_snapshots":[{"event_id":"sha256:400cd853b606bbabecef8c4f664236f984b43939d0927e4ae51ebe268736f894","target":"graph","created_at":"2026-05-20T01:05:25Z","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/2605.19063/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Inspired by long-standing open problems in algebraic combinatorics, we show that modern machine learning can meaningfully contribute to verifiable mathematical discoveries. In particular, we focus on the construction of simple mathematical functions under exact distributional constraints, a setting we formalize as Simple Learning Under Rigid Proportions (SLURP). We tackle this problem by introducing two methods: MapSeek-Functional, which models the desired function alternating pseudo-labeling and supervised training steps; and MapSeek-Symbolic, designed to directly produce symbolic formulas. W","authors_text":"Alessandro Iraci, Eugenio Cainelli, Giovanni Paolini, Lorenzo Luccioli, Michele D'Adderio","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-05-18T19:36:08Z","title":"Mapping Uncharted Symmetries: Machine Discovery in Combinatorics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.19063","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":"sha256:46f14456bcc3db95e87ae84b2e159739b1f04373828786a4d3ff147c1273abff","target":"record","created_at":"2026-05-20T01:05:25Z","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":"32ef5e037594724eb9009a58b031392deb1e45904afbbef29734a319453fb6d0","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-05-18T19:36:08Z","title_canon_sha256":"8e2d37aeb4035c0924d3a047e012f88c16ff61250924fcb45972752b4e501e13"},"schema_version":"1.0","source":{"id":"2605.19063","kind":"arxiv","version":1}},"canonical_sha256":"eb9f17948c2b4e157e4199d1f703d5c6cf5dbb34564476decb054cbada913843","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eb9f17948c2b4e157e4199d1f703d5c6cf5dbb34564476decb054cbada913843","first_computed_at":"2026-05-20T01:05:25.658175Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T01:05:25.658175Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xUfOa96KV4vkGwgLLoxHeFnHyF2/ocN1WuPfFbYbOW25tZWhyPv5OCwZOWUtpwkvWpyqbG5QOh7froyX11iwDw==","signature_status":"signed_v1","signed_at":"2026-05-20T01:05:25.658894Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.19063","source_kind":"arxiv","source_version":1}}},"equivocations":[{"signer_id":"pith.science","event_type":"integrity_finding","target":"integrity","event_ids":["sha256:2bd8e87b7df587384f2493120c39d62b9906c7f5dcd22a7cb2b830dfb5e9f34a","sha256:4f94158fe8a60c705cda964a1da2a3139499958bcdefa0c4ec4ae93f790b794a"]}],"invalid_events":[],"applied_event_ids":["sha256:46f14456bcc3db95e87ae84b2e159739b1f04373828786a4d3ff147c1273abff","sha256:400cd853b606bbabecef8c4f664236f984b43939d0927e4ae51ebe268736f894"],"state_sha256":"5b95c0a7d2b1f6e65313b71917f3791745efee2bc70453f9a49d2afa2553e30e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0msPKWULKNdDZ7epAtzfMFwdptydF5T2zVtuTyVEPHIK0II76+J7brTHrxAHV9U0Piky7JslusEFJN41K0z6Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-21T03:30:22.274862Z","bundle_sha256":"e01ae10f223f31e6fc26e9292475741b030a969b784209d1e21e7750769b1e11"}}