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This tensor density is, by definition, a cubic algebraic functional of a tensor density of rank four ${\\cal T}^{ijkl}$, which is antisymmetric in its first two and its last two indices: ${\\cal T}^{ijkl} = - {\\cal T}^{jikl} = - {\\cal T}^{ijlk}$. Thus, ${\\cal K}\\sim {\\cal T}^3$, see Eq.(46). (i) If $\\cal T$ is identified with the electromagnetic response tensor of local and linear media, the Kumme"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1403.3467","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2014-03-14T00:33:07Z","cross_cats_sorted":["hep-th"],"title_canon_sha256":"c63cd98cb78960dfaff12fa54962f3a38694d9d16a00af9eac4680dcc8f5e8c9","abstract_canon_sha256":"7c148f1a9d08890819ace06844049beaa68878b516a581c8c2a76a228c89b2ad"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:44:01.323957Z","signature_b64":"MNCIwwDy+hfMfK9HgIiftkrmvV7iYgreku3Xa27e5n7yOTLCBJZsEE3TP+hVSvsxbP7O4LYbOtNeIJVMKixTCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b2d92e5013dc357e12ba03b271a0ede4e836effd1472eee217a5db8ec14ca207","last_reissued_at":"2026-05-18T01:44:01.323305Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:44:01.323305Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Kummer tensor density in electrodynamics and in gravity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th"],"primary_cat":"gr-qc","authors_text":"Alberto Favaro (Oldenburg), Columbia, Friedrich W. Hehl (Cologne, Missouri), Peter Baekler (Duesseldorf), Yakov Itin (Jerusalem)","submitted_at":"2014-03-14T00:33:07Z","abstract_excerpt":"Guided by results in the premetric electrodynamics of local and linear media, we introduce on 4-dimensional spacetime the new abstract notion of a Kummer tensor density of rank four, ${\\cal K}^{ijkl}$. This tensor density is, by definition, a cubic algebraic functional of a tensor density of rank four ${\\cal T}^{ijkl}$, which is antisymmetric in its first two and its last two indices: ${\\cal T}^{ijkl} = - {\\cal T}^{jikl} = - {\\cal T}^{ijlk}$. Thus, ${\\cal K}\\sim {\\cal T}^3$, see Eq.(46). (i) If $\\cal T$ is identified with the electromagnetic response tensor of local and linear media, the Kumme"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.3467","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1403.3467","created_at":"2026-05-18T01:44:01.323425+00:00"},{"alias_kind":"arxiv_version","alias_value":"1403.3467v2","created_at":"2026-05-18T01:44:01.323425+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.3467","created_at":"2026-05-18T01:44:01.323425+00:00"},{"alias_kind":"pith_short_12","alias_value":"WLMS4UAT3Q2X","created_at":"2026-05-18T12:28:54.890064+00:00"},{"alias_kind":"pith_short_16","alias_value":"WLMS4UAT3Q2X4EV2","created_at":"2026-05-18T12:28:54.890064+00:00"},{"alias_kind":"pith_short_8","alias_value":"WLMS4UAT","created_at":"2026-05-18T12:28:54.890064+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2604.17646","citing_title":"Crystallography, Lorentz violation, and the Standard-Model Extension","ref_index":87,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WLMS4UAT3Q2X4EV2AOZHDIHN4T","json":"https://pith.science/pith/WLMS4UAT3Q2X4EV2AOZHDIHN4T.json","graph_json":"https://pith.science/api/pith-number/WLMS4UAT3Q2X4EV2AOZHDIHN4T/graph.json","events_json":"https://pith.science/api/pith-number/WLMS4UAT3Q2X4EV2AOZHDIHN4T/events.json","paper":"https://pith.science/paper/WLMS4UAT"},"agent_actions":{"view_html":"https://pith.science/pith/WLMS4UAT3Q2X4EV2AOZHDIHN4T","download_json":"https://pith.science/pith/WLMS4UAT3Q2X4EV2AOZHDIHN4T.json","view_paper":"https://pith.science/paper/WLMS4UAT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1403.3467&json=true","fetch_graph":"https://pith.science/api/pith-number/WLMS4UAT3Q2X4EV2AOZHDIHN4T/graph.json","fetch_events":"https://pith.science/api/pith-number/WLMS4UAT3Q2X4EV2AOZHDIHN4T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WLMS4UAT3Q2X4EV2AOZHDIHN4T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WLMS4UAT3Q2X4EV2AOZHDIHN4T/action/storage_attestation","attest_author":"https://pith.science/pith/WLMS4UAT3Q2X4EV2AOZHDIHN4T/action/author_attestation","sign_citation":"https://pith.science/pith/WLMS4UAT3Q2X4EV2AOZHDIHN4T/action/citation_signature","submit_replication":"https://pith.science/pith/WLMS4UAT3Q2X4EV2AOZHDIHN4T/action/replication_record"}},"created_at":"2026-05-18T01:44:01.323425+00:00","updated_at":"2026-05-18T01:44:01.323425+00:00"}