{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:4L3JUU4TLUEOGRRNS5AF5BYBEZ","short_pith_number":"pith:4L3JUU4T","canonical_record":{"source":{"id":"2607.00237","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-30T22:29:05Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"7780b221be0a318712f813e79052abe37bb8460284ab3644ed8d1e74755e7914","abstract_canon_sha256":"e7ca235418fc84224fef0cd9bf7f863142247025926a93a56290e4ceafd7897f"},"schema_version":"1.0"},"canonical_sha256":"e2f69a53935d08e3462d97405e8701264d8a470d78eed8fc74b2406f9b26ecdd","source":{"kind":"arxiv","id":"2607.00237","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.00237","created_at":"2026-07-02T00:18:40Z"},{"alias_kind":"arxiv_version","alias_value":"2607.00237v1","created_at":"2026-07-02T00:18:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.00237","created_at":"2026-07-02T00:18:40Z"},{"alias_kind":"pith_short_12","alias_value":"4L3JUU4TLUEO","created_at":"2026-07-02T00:18:40Z"},{"alias_kind":"pith_short_16","alias_value":"4L3JUU4TLUEOGRRN","created_at":"2026-07-02T00:18:40Z"},{"alias_kind":"pith_short_8","alias_value":"4L3JUU4T","created_at":"2026-07-02T00:18:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:4L3JUU4TLUEOGRRNS5AF5BYBEZ","target":"record","payload":{"canonical_record":{"source":{"id":"2607.00237","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-30T22:29:05Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"7780b221be0a318712f813e79052abe37bb8460284ab3644ed8d1e74755e7914","abstract_canon_sha256":"e7ca235418fc84224fef0cd9bf7f863142247025926a93a56290e4ceafd7897f"},"schema_version":"1.0"},"canonical_sha256":"e2f69a53935d08e3462d97405e8701264d8a470d78eed8fc74b2406f9b26ecdd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-02T00:18:40.506466Z","signature_b64":"JOoECkQdNq2h8ac4+HErdR0jDhoJv2InU2ZglGYwRAMLvaTL030ophFHeSm9a1GNjwTI/C5mS+OXTHHmDiBVCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e2f69a53935d08e3462d97405e8701264d8a470d78eed8fc74b2406f9b26ecdd","last_reissued_at":"2026-07-02T00:18:40.505949Z","signature_status":"signed_v1","first_computed_at":"2026-07-02T00:18:40.505949Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.00237","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":"2026-07-02T00:18:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MfKg22MvBMyP8wf/25p+CdvwdZ4x4n2pYLRpGjRBqvEicgEOhcdSRuef6W458T65zbqqCVV3HurRmjmeHcAPBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T08:36:45.545723Z"},"content_sha256":"c4e22eceb2c50cd552f23331d04c51ccc737cd2aea8a32b4f915ffdb18d50a5b","schema_version":"1.0","event_id":"sha256:c4e22eceb2c50cd552f23331d04c51ccc737cd2aea8a32b4f915ffdb18d50a5b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:4L3JUU4TLUEOGRRNS5AF5BYBEZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tropical Geometry as a Restricted Architecture for Physics-Informed Neural Networks: Applications in Nonlinear Fluid-Structure Examples","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.AP","authors_text":"Alonso Andapia-Viveros, Carla Valencia-Negrete, Cristhian Garay-Lopez, Marco Favela-Rodriguez","submitted_at":"2026-06-30T22:29:05Z","abstract_excerpt":"Nonlinear algebraic (polynomial) differential equations that govern fluid-structure interactions, such as those modeling vortex-induced vibrations, and shock waves, often lack analytical solutions, creating significant challenges to efficient prediction and control. While Physics-Informed Neural Networks (PINNs) offer a mesh-free numerical alternative, they frequently suffer from convergence stagnation when optimizing over chaotic landscapes or stiff singularities. This paper introduces a hybrid methodology that integrates tropical differential algebraic geometry with deep learning. Using trop"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.00237","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/2607.00237/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":"2026-07-02T00:18:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3EfACHf6fSvTgSSH3Y6V8Y6m7UXPBsHxOftsfnaimI4qTG/wkniAa0I8nScSam2WTNVqrJwOyvU2FGTBG34uAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T08:36:45.546118Z"},"content_sha256":"76eb032ea908a5dd9f8a8a8af05e7e7d9465d542d5bc973efe2a587405508f62","schema_version":"1.0","event_id":"sha256:76eb032ea908a5dd9f8a8a8af05e7e7d9465d542d5bc973efe2a587405508f62"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4L3JUU4TLUEOGRRNS5AF5BYBEZ/bundle.json","state_url":"https://pith.science/pith/4L3JUU4TLUEOGRRNS5AF5BYBEZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4L3JUU4TLUEOGRRNS5AF5BYBEZ/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":"2026-07-03T08:36:45Z","links":{"resolver":"https://pith.science/pith/4L3JUU4TLUEOGRRNS5AF5BYBEZ","bundle":"https://pith.science/pith/4L3JUU4TLUEOGRRNS5AF5BYBEZ/bundle.json","state":"https://pith.science/pith/4L3JUU4TLUEOGRRNS5AF5BYBEZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4L3JUU4TLUEOGRRNS5AF5BYBEZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:4L3JUU4TLUEOGRRNS5AF5BYBEZ","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":"e7ca235418fc84224fef0cd9bf7f863142247025926a93a56290e4ceafd7897f","cross_cats_sorted":["math.AC"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-30T22:29:05Z","title_canon_sha256":"7780b221be0a318712f813e79052abe37bb8460284ab3644ed8d1e74755e7914"},"schema_version":"1.0","source":{"id":"2607.00237","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.00237","created_at":"2026-07-02T00:18:40Z"},{"alias_kind":"arxiv_version","alias_value":"2607.00237v1","created_at":"2026-07-02T00:18:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.00237","created_at":"2026-07-02T00:18:40Z"},{"alias_kind":"pith_short_12","alias_value":"4L3JUU4TLUEO","created_at":"2026-07-02T00:18:40Z"},{"alias_kind":"pith_short_16","alias_value":"4L3JUU4TLUEOGRRN","created_at":"2026-07-02T00:18:40Z"},{"alias_kind":"pith_short_8","alias_value":"4L3JUU4T","created_at":"2026-07-02T00:18:40Z"}],"graph_snapshots":[{"event_id":"sha256:76eb032ea908a5dd9f8a8a8af05e7e7d9465d542d5bc973efe2a587405508f62","target":"graph","created_at":"2026-07-02T00:18:40Z","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/2607.00237/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Nonlinear algebraic (polynomial) differential equations that govern fluid-structure interactions, such as those modeling vortex-induced vibrations, and shock waves, often lack analytical solutions, creating significant challenges to efficient prediction and control. While Physics-Informed Neural Networks (PINNs) offer a mesh-free numerical alternative, they frequently suffer from convergence stagnation when optimizing over chaotic landscapes or stiff singularities. This paper introduces a hybrid methodology that integrates tropical differential algebraic geometry with deep learning. Using trop","authors_text":"Alonso Andapia-Viveros, Carla Valencia-Negrete, Cristhian Garay-Lopez, Marco Favela-Rodriguez","cross_cats":["math.AC"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-30T22:29:05Z","title":"Tropical Geometry as a Restricted Architecture for Physics-Informed Neural Networks: Applications in Nonlinear Fluid-Structure Examples"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.00237","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:c4e22eceb2c50cd552f23331d04c51ccc737cd2aea8a32b4f915ffdb18d50a5b","target":"record","created_at":"2026-07-02T00:18:40Z","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":"e7ca235418fc84224fef0cd9bf7f863142247025926a93a56290e4ceafd7897f","cross_cats_sorted":["math.AC"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-30T22:29:05Z","title_canon_sha256":"7780b221be0a318712f813e79052abe37bb8460284ab3644ed8d1e74755e7914"},"schema_version":"1.0","source":{"id":"2607.00237","kind":"arxiv","version":1}},"canonical_sha256":"e2f69a53935d08e3462d97405e8701264d8a470d78eed8fc74b2406f9b26ecdd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e2f69a53935d08e3462d97405e8701264d8a470d78eed8fc74b2406f9b26ecdd","first_computed_at":"2026-07-02T00:18:40.505949Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-02T00:18:40.505949Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JOoECkQdNq2h8ac4+HErdR0jDhoJv2InU2ZglGYwRAMLvaTL030ophFHeSm9a1GNjwTI/C5mS+OXTHHmDiBVCQ==","signature_status":"signed_v1","signed_at":"2026-07-02T00:18:40.506466Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.00237","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c4e22eceb2c50cd552f23331d04c51ccc737cd2aea8a32b4f915ffdb18d50a5b","sha256:76eb032ea908a5dd9f8a8a8af05e7e7d9465d542d5bc973efe2a587405508f62"],"state_sha256":"d4415ac3442ad3803b785f3d00fbcf0748c361cf8ea2774aa4cac0855b31e827"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Pmw9TQKdjkfNbdaVNnKXPuVo0+REAI/TQiQH0FYnSUdg9ZzMjkUzicWDIjSPiKBhBvCKpAPYETG5NxEzOWuPAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T08:36:45.548174Z","bundle_sha256":"238da1b7076aacd3e6dd48575be909cea331cae052b942108d17fcffa48ce4e4"}}