{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:3DPSCEQ2DMHXMHPUP7BUSGI7MU","short_pith_number":"pith:3DPSCEQ2","canonical_record":{"source":{"id":"2606.02759","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math-ph","submitted_at":"2026-06-01T18:23:38Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"2bedf0c786bd901b505d81b72dd235e5260c8f5710017a875dcf7cf792452082","abstract_canon_sha256":"77f04fd9f7d74e13a06a28729cfdbaa638c13e48a2913c79686f281c6cf2fafd"},"schema_version":"1.0"},"canonical_sha256":"d8df21121a1b0f761df47fc349191f6515164551816d4976f8bb59bbcf240fc5","source":{"kind":"arxiv","id":"2606.02759","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.02759","created_at":"2026-06-03T01:05:21Z"},{"alias_kind":"arxiv_version","alias_value":"2606.02759v1","created_at":"2026-06-03T01:05:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.02759","created_at":"2026-06-03T01:05:21Z"},{"alias_kind":"pith_short_12","alias_value":"3DPSCEQ2DMHX","created_at":"2026-06-03T01:05:21Z"},{"alias_kind":"pith_short_16","alias_value":"3DPSCEQ2DMHXMHPU","created_at":"2026-06-03T01:05:21Z"},{"alias_kind":"pith_short_8","alias_value":"3DPSCEQ2","created_at":"2026-06-03T01:05:21Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:3DPSCEQ2DMHXMHPUP7BUSGI7MU","target":"record","payload":{"canonical_record":{"source":{"id":"2606.02759","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math-ph","submitted_at":"2026-06-01T18:23:38Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"2bedf0c786bd901b505d81b72dd235e5260c8f5710017a875dcf7cf792452082","abstract_canon_sha256":"77f04fd9f7d74e13a06a28729cfdbaa638c13e48a2913c79686f281c6cf2fafd"},"schema_version":"1.0"},"canonical_sha256":"d8df21121a1b0f761df47fc349191f6515164551816d4976f8bb59bbcf240fc5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T01:05:21.098355Z","signature_b64":"ugU9Guq62gahNjxo7R6KMQIelNehf5ZhvcZyjtwzLHX/W/2drCMp8Cieg+sS5CAPS7Bq20DSEFwmuynDNcvOAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d8df21121a1b0f761df47fc349191f6515164551816d4976f8bb59bbcf240fc5","last_reissued_at":"2026-06-03T01:05:21.097877Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T01:05:21.097877Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.02759","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":"2026-06-03T01:05:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"11fDqSXNC44IikEbEViuvlH7I05FRA7q02yiosEzdp783Us8rjg73RK5RSLbI8i2gB6PW9yZiZwtXFHk9K3YDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T17:52:14.045387Z"},"content_sha256":"8c09332ec9c10a2d4b07cc6eb5551f1b721ef7522a6b7479ec00369cd77d4e7d","schema_version":"1.0","event_id":"sha256:8c09332ec9c10a2d4b07cc6eb5551f1b721ef7522a6b7479ec00369cd77d4e7d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:3DPSCEQ2DMHXMHPUP7BUSGI7MU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bounds on the Bogoliubov--Hartree--Fock Energy of the Pauli--Fierz Hamiltonian","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Matthias Herdzik, Volker Bach","submitted_at":"2026-06-01T18:23:38Z","abstract_excerpt":"A variational analysis of the Bogoliubov--Hartree--Fock (BHF) energy of the translation-invariant, spinless Pauli--Fierz Hamiltonian with massless dispersion relation built up on work of the first author, Breteaux, and Tzaneteas (2013) and of the first author and Hach (2022) is presented. The main results are lower and upper bounds on the BHF energy for fixed total momentum expressed through simpler variational problems defined on the space of positive Hilbert--Schmidt operators and a new variational formulation of the upper bound for zero total momentum. Specifically, we introduce a change of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02759","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/2606.02759/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":"2026-06-03T01:05:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iokO+VZ+9QVJ6ZWStisa+PE6C1BeJvXtRwAwa86FLG6XSVsGyioCVrNEcXt70LoHpuEpSUy4HahpdV6ESUH7CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T17:52:14.045775Z"},"content_sha256":"342dd69320961dc0bcd9fb7454ea45f5f810ab44d52d4325bf57e441977ac170","schema_version":"1.0","event_id":"sha256:342dd69320961dc0bcd9fb7454ea45f5f810ab44d52d4325bf57e441977ac170"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3DPSCEQ2DMHXMHPUP7BUSGI7MU/bundle.json","state_url":"https://pith.science/pith/3DPSCEQ2DMHXMHPUP7BUSGI7MU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3DPSCEQ2DMHXMHPUP7BUSGI7MU/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":"2026-06-21T17:52:14Z","links":{"resolver":"https://pith.science/pith/3DPSCEQ2DMHXMHPUP7BUSGI7MU","bundle":"https://pith.science/pith/3DPSCEQ2DMHXMHPUP7BUSGI7MU/bundle.json","state":"https://pith.science/pith/3DPSCEQ2DMHXMHPUP7BUSGI7MU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3DPSCEQ2DMHXMHPUP7BUSGI7MU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:3DPSCEQ2DMHXMHPUP7BUSGI7MU","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":"77f04fd9f7d74e13a06a28729cfdbaa638c13e48a2913c79686f281c6cf2fafd","cross_cats_sorted":["math.MP"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math-ph","submitted_at":"2026-06-01T18:23:38Z","title_canon_sha256":"2bedf0c786bd901b505d81b72dd235e5260c8f5710017a875dcf7cf792452082"},"schema_version":"1.0","source":{"id":"2606.02759","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.02759","created_at":"2026-06-03T01:05:21Z"},{"alias_kind":"arxiv_version","alias_value":"2606.02759v1","created_at":"2026-06-03T01:05:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.02759","created_at":"2026-06-03T01:05:21Z"},{"alias_kind":"pith_short_12","alias_value":"3DPSCEQ2DMHX","created_at":"2026-06-03T01:05:21Z"},{"alias_kind":"pith_short_16","alias_value":"3DPSCEQ2DMHXMHPU","created_at":"2026-06-03T01:05:21Z"},{"alias_kind":"pith_short_8","alias_value":"3DPSCEQ2","created_at":"2026-06-03T01:05:21Z"}],"graph_snapshots":[{"event_id":"sha256:342dd69320961dc0bcd9fb7454ea45f5f810ab44d52d4325bf57e441977ac170","target":"graph","created_at":"2026-06-03T01:05:21Z","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/2606.02759/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A variational analysis of the Bogoliubov--Hartree--Fock (BHF) energy of the translation-invariant, spinless Pauli--Fierz Hamiltonian with massless dispersion relation built up on work of the first author, Breteaux, and Tzaneteas (2013) and of the first author and Hach (2022) is presented. The main results are lower and upper bounds on the BHF energy for fixed total momentum expressed through simpler variational problems defined on the space of positive Hilbert--Schmidt operators and a new variational formulation of the upper bound for zero total momentum. Specifically, we introduce a change of","authors_text":"Matthias Herdzik, Volker Bach","cross_cats":["math.MP"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math-ph","submitted_at":"2026-06-01T18:23:38Z","title":"Bounds on the Bogoliubov--Hartree--Fock Energy of the Pauli--Fierz Hamiltonian"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02759","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":"sha256:8c09332ec9c10a2d4b07cc6eb5551f1b721ef7522a6b7479ec00369cd77d4e7d","target":"record","created_at":"2026-06-03T01:05:21Z","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":"77f04fd9f7d74e13a06a28729cfdbaa638c13e48a2913c79686f281c6cf2fafd","cross_cats_sorted":["math.MP"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math-ph","submitted_at":"2026-06-01T18:23:38Z","title_canon_sha256":"2bedf0c786bd901b505d81b72dd235e5260c8f5710017a875dcf7cf792452082"},"schema_version":"1.0","source":{"id":"2606.02759","kind":"arxiv","version":1}},"canonical_sha256":"d8df21121a1b0f761df47fc349191f6515164551816d4976f8bb59bbcf240fc5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d8df21121a1b0f761df47fc349191f6515164551816d4976f8bb59bbcf240fc5","first_computed_at":"2026-06-03T01:05:21.097877Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-03T01:05:21.097877Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ugU9Guq62gahNjxo7R6KMQIelNehf5ZhvcZyjtwzLHX/W/2drCMp8Cieg+sS5CAPS7Bq20DSEFwmuynDNcvOAA==","signature_status":"signed_v1","signed_at":"2026-06-03T01:05:21.098355Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.02759","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8c09332ec9c10a2d4b07cc6eb5551f1b721ef7522a6b7479ec00369cd77d4e7d","sha256:342dd69320961dc0bcd9fb7454ea45f5f810ab44d52d4325bf57e441977ac170"],"state_sha256":"5439828351739e1470fd021a9939b1acced9b4f82663c745b6d192a75dd33fe3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ak8USeOV8CeROVTutI2iZGd8qcBR4pt8aQ5ygZe6T22zRk5IiKcAhAazLY7m8jjoSP0cgxG4glUoAftKPtgKCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T17:52:14.047979Z","bundle_sha256":"57586f6e8cd9978f03be49a5e4ecbf539f3e3bcbe3f6b783f63c7d3ece5e25b9"}}