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We prove the inequalities A [g N^2 (N-1)^2]^{1/3} < E < B [g N^2 (N-1)^2]^{1/3}, where A = 2.33810741 and B = [81/(2 pi)]^{1/3} = 2.3447779. The average of these bounds determines E with an error less than 0.15% for all N > 1."},"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":"math-ph/0311032","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2003-11-20T17:03:18Z","cross_cats_sorted":["hep-ph","hep-th","math.MP"],"title_canon_sha256":"b20310459e1c9a337670d1b445534c6d033ec99af425fbba13bf4480ec8eedcb","abstract_canon_sha256":"2cc600c9138ad664aee6e6cdae97442b6b8fc1be48b103e2cfaefaa773f1b159"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:38:33.640898Z","signature_b64":"Gh1ByRTG+HgGSvslE5jmEUJpJo94PMjwddU7tfX34EJ657B12l3hCYkW7MDiee6s3DU9Fa/Tod1HNe/KpqGoAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2fc86ca1a20da6a9240dfd376f5cefe26ee06f3fbde70fb124093f5086b98285","last_reissued_at":"2026-05-18T01:38:33.640473Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:38:33.640473Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The energy of a system of relativistic massless bosons bound by oscillator pair potentials","license":"","headline":"","cross_cats":["hep-ph","hep-th","math.MP"],"primary_cat":"math-ph","authors_text":"Franz F. Schoeberl, Richard L. Hall, Wolfgang Lucha","submitted_at":"2003-11-20T17:03:18Z","abstract_excerpt":"We study the lowest energy E of a semirelativistic system of N identical massless bosons with Hamiltonian H= sum{i=1 to N} sqrt(p_i^2)+ sum{j>i=1 to N} g |r_i - r_j|^2, g > 0. We prove the inequalities A [g N^2 (N-1)^2]^{1/3} < E < B [g N^2 (N-1)^2]^{1/3}, where A = 2.33810741 and B = [81/(2 pi)]^{1/3} = 2.3447779. 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