{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:ARKSBT7OQXD3FLIHYBHX7E3OIT","short_pith_number":"pith:ARKSBT7O","canonical_record":{"source":{"id":"2507.12729","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2025-07-17T02:08:14Z","cross_cats_sorted":["cs.CV","cs.NA","math.NA","math.RT"],"title_canon_sha256":"71e85610ffb4ad21b91915106711f62c8083567d7b21a2c2fc8421042721ae61","abstract_canon_sha256":"c4bc835eeed9044b345b850e798d898fd0153c03cf04c3ac0b108bbb945c5255"},"schema_version":"1.0"},"canonical_sha256":"045520cfee85c7b2ad07c04f7f936e44ce7c1593838fb60e5db3415ed1b72b62","source":{"kind":"arxiv","id":"2507.12729","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.12729","created_at":"2026-07-05T11:38:38Z"},{"alias_kind":"arxiv_version","alias_value":"2507.12729v1","created_at":"2026-07-05T11:38:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.12729","created_at":"2026-07-05T11:38:38Z"},{"alias_kind":"pith_short_12","alias_value":"ARKSBT7OQXD3","created_at":"2026-07-05T11:38:38Z"},{"alias_kind":"pith_short_16","alias_value":"ARKSBT7OQXD3FLIH","created_at":"2026-07-05T11:38:38Z"},{"alias_kind":"pith_short_8","alias_value":"ARKSBT7O","created_at":"2026-07-05T11:38:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:ARKSBT7OQXD3FLIHYBHX7E3OIT","target":"record","payload":{"canonical_record":{"source":{"id":"2507.12729","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2025-07-17T02:08:14Z","cross_cats_sorted":["cs.CV","cs.NA","math.NA","math.RT"],"title_canon_sha256":"71e85610ffb4ad21b91915106711f62c8083567d7b21a2c2fc8421042721ae61","abstract_canon_sha256":"c4bc835eeed9044b345b850e798d898fd0153c03cf04c3ac0b108bbb945c5255"},"schema_version":"1.0"},"canonical_sha256":"045520cfee85c7b2ad07c04f7f936e44ce7c1593838fb60e5db3415ed1b72b62","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:38:38.043006Z","signature_b64":"dAFi64K1IN1bNrTGDEuvcXOFV9PhnotkwXt8jSbFywjVBGvpHd2RDOL/yYpltBLgL3yWVF2oB3/sQKLORlZ0BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"045520cfee85c7b2ad07c04f7f936e44ce7c1593838fb60e5db3415ed1b72b62","last_reissued_at":"2026-07-05T11:38:38.042508Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:38:38.042508Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.12729","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":"2026-07-05T11:38:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"t4Uf3e9dtOnzptwas3PppbrcUUkgfdzQZaHoKF0upnHPBc3TernJa5CVv3M451w8znSHOMv95WoZucLylw8wDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T22:21:53.314051Z"},"content_sha256":"40e182bdf57526e96bf773d4c7fa07d6b30fb8c488fa30bb9aaf97acf78a2605","schema_version":"1.0","event_id":"sha256:40e182bdf57526e96bf773d4c7fa07d6b30fb8c488fa30bb9aaf97acf78a2605"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:ARKSBT7OQXD3FLIHYBHX7E3OIT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tensor-Tensor Products, Group Representations, and Semidefinite Programming","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CV","cs.NA","math.NA","math.RT"],"primary_cat":"math.OC","authors_text":"Alex Dunbar, Elizabeth Newman","submitted_at":"2025-07-17T02:08:14Z","abstract_excerpt":"The $\\star_M$-family of tensor-tensor products is a framework which generalizes many properties from linear algebra to third order tensors. Here, we investigate positive semidefiniteness and semidefinite programming under the $\\star_M$-product. Critical to our investigation is a connection between the choice of matrix M in the $\\star_M$-product and the representation theory of an underlying group action. Using this framework, third order tensors equipped with the $\\star_M$-product are a natural setting for the study of invariant semidefinite programs. As applications of the M-SDP framework, we"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.12729","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/2507.12729/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":"2026-07-05T11:38:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VCuqdFkusHuI+7ezn/gnXaIJpqo7bwpuOQI0cIWcmDRdtYsgoxArb2qRRf/o9OuV/44ENxAMCDeh5eilnaJyCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T22:21:53.314658Z"},"content_sha256":"30b1bc2a6e25dde7ad0136a6924a6bdea4b7d457e959dfb915ddbce05d7ca170","schema_version":"1.0","event_id":"sha256:30b1bc2a6e25dde7ad0136a6924a6bdea4b7d457e959dfb915ddbce05d7ca170"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ARKSBT7OQXD3FLIHYBHX7E3OIT/bundle.json","state_url":"https://pith.science/pith/ARKSBT7OQXD3FLIHYBHX7E3OIT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ARKSBT7OQXD3FLIHYBHX7E3OIT/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":"2026-08-18T22:21:53Z","links":{"resolver":"https://pith.science/pith/ARKSBT7OQXD3FLIHYBHX7E3OIT","bundle":"https://pith.science/pith/ARKSBT7OQXD3FLIHYBHX7E3OIT/bundle.json","state":"https://pith.science/pith/ARKSBT7OQXD3FLIHYBHX7E3OIT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ARKSBT7OQXD3FLIHYBHX7E3OIT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:ARKSBT7OQXD3FLIHYBHX7E3OIT","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":"c4bc835eeed9044b345b850e798d898fd0153c03cf04c3ac0b108bbb945c5255","cross_cats_sorted":["cs.CV","cs.NA","math.NA","math.RT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2025-07-17T02:08:14Z","title_canon_sha256":"71e85610ffb4ad21b91915106711f62c8083567d7b21a2c2fc8421042721ae61"},"schema_version":"1.0","source":{"id":"2507.12729","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.12729","created_at":"2026-07-05T11:38:38Z"},{"alias_kind":"arxiv_version","alias_value":"2507.12729v1","created_at":"2026-07-05T11:38:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.12729","created_at":"2026-07-05T11:38:38Z"},{"alias_kind":"pith_short_12","alias_value":"ARKSBT7OQXD3","created_at":"2026-07-05T11:38:38Z"},{"alias_kind":"pith_short_16","alias_value":"ARKSBT7OQXD3FLIH","created_at":"2026-07-05T11:38:38Z"},{"alias_kind":"pith_short_8","alias_value":"ARKSBT7O","created_at":"2026-07-05T11:38:38Z"}],"graph_snapshots":[{"event_id":"sha256:30b1bc2a6e25dde7ad0136a6924a6bdea4b7d457e959dfb915ddbce05d7ca170","target":"graph","created_at":"2026-07-05T11:38:38Z","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/2507.12729/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The $\\star_M$-family of tensor-tensor products is a framework which generalizes many properties from linear algebra to third order tensors. Here, we investigate positive semidefiniteness and semidefinite programming under the $\\star_M$-product. Critical to our investigation is a connection between the choice of matrix M in the $\\star_M$-product and the representation theory of an underlying group action. Using this framework, third order tensors equipped with the $\\star_M$-product are a natural setting for the study of invariant semidefinite programs. As applications of the M-SDP framework, we","authors_text":"Alex Dunbar, Elizabeth Newman","cross_cats":["cs.CV","cs.NA","math.NA","math.RT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2025-07-17T02:08:14Z","title":"Tensor-Tensor Products, Group Representations, and Semidefinite Programming"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.12729","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":"sha256:40e182bdf57526e96bf773d4c7fa07d6b30fb8c488fa30bb9aaf97acf78a2605","target":"record","created_at":"2026-07-05T11:38:38Z","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":"c4bc835eeed9044b345b850e798d898fd0153c03cf04c3ac0b108bbb945c5255","cross_cats_sorted":["cs.CV","cs.NA","math.NA","math.RT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2025-07-17T02:08:14Z","title_canon_sha256":"71e85610ffb4ad21b91915106711f62c8083567d7b21a2c2fc8421042721ae61"},"schema_version":"1.0","source":{"id":"2507.12729","kind":"arxiv","version":1}},"canonical_sha256":"045520cfee85c7b2ad07c04f7f936e44ce7c1593838fb60e5db3415ed1b72b62","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"045520cfee85c7b2ad07c04f7f936e44ce7c1593838fb60e5db3415ed1b72b62","first_computed_at":"2026-07-05T11:38:38.042508Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:38:38.042508Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dAFi64K1IN1bNrTGDEuvcXOFV9PhnotkwXt8jSbFywjVBGvpHd2RDOL/yYpltBLgL3yWVF2oB3/sQKLORlZ0BA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:38:38.043006Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.12729","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:40e182bdf57526e96bf773d4c7fa07d6b30fb8c488fa30bb9aaf97acf78a2605","sha256:30b1bc2a6e25dde7ad0136a6924a6bdea4b7d457e959dfb915ddbce05d7ca170"],"state_sha256":"485f14f3e7e246ae623e34a8f08e25129a7f15cd83a1bcd640c21e800b52f47b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uCszwY1tku+9JE6u53GmV9eDTuWlPPS92PjkVvV33YWcNEW21DxGRUUTG/Z+fp4Zw7TOyvmrKoAkVbLV7sR+AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-18T22:21:53.318635Z","bundle_sha256":"db17e60a646f981f9ab3be0f4b9a5046307a3887a50c2ac7b3a6fbc48538fbcf"}}