{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:M6RWMEXSHB2ACVMUHQRAWUZQOM","short_pith_number":"pith:M6RWMEXS","canonical_record":{"source":{"id":"2510.03382","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2025-10-03T15:52:14Z","cross_cats_sorted":["math-ph","math.MP","math.PR"],"title_canon_sha256":"655d18f8204514639ec085112d547fe2440c89f4b1ee7eeff67f42aae9a2fb17","abstract_canon_sha256":"1e62d94852568a00a048d935dffdba6c7b8cd2b06d4714bdc073b1090eec1647"},"schema_version":"1.0"},"canonical_sha256":"67a36612f238740155943c220b5330731c9ea3eb2fefd43a5e4aaf315acac395","source":{"kind":"arxiv","id":"2510.03382","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2510.03382","created_at":"2026-05-20T00:00:25Z"},{"alias_kind":"arxiv_version","alias_value":"2510.03382v5","created_at":"2026-05-20T00:00:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2510.03382","created_at":"2026-05-20T00:00:25Z"},{"alias_kind":"pith_short_12","alias_value":"M6RWMEXSHB2A","created_at":"2026-05-20T00:00:25Z"},{"alias_kind":"pith_short_16","alias_value":"M6RWMEXSHB2ACVMU","created_at":"2026-05-20T00:00:25Z"},{"alias_kind":"pith_short_8","alias_value":"M6RWMEXS","created_at":"2026-05-20T00:00:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:M6RWMEXSHB2ACVMUHQRAWUZQOM","target":"record","payload":{"canonical_record":{"source":{"id":"2510.03382","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2025-10-03T15:52:14Z","cross_cats_sorted":["math-ph","math.MP","math.PR"],"title_canon_sha256":"655d18f8204514639ec085112d547fe2440c89f4b1ee7eeff67f42aae9a2fb17","abstract_canon_sha256":"1e62d94852568a00a048d935dffdba6c7b8cd2b06d4714bdc073b1090eec1647"},"schema_version":"1.0"},"canonical_sha256":"67a36612f238740155943c220b5330731c9ea3eb2fefd43a5e4aaf315acac395","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:00:25.269795Z","signature_b64":"sycAieLgewZh1wwhjhXiVrABTeJghE60RFTNip2aDmD+rt0IkYV9LPr+BjJ09BRGPS3c15KpY2R+OVujXBH2Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"67a36612f238740155943c220b5330731c9ea3eb2fefd43a5e4aaf315acac395","last_reissued_at":"2026-05-20T00:00:25.268741Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:00:25.268741Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2510.03382","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-20T00:00:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XYDRINJT/ihYRrmlIYyZ5VRqMNhqwLSWuTq2HtfANWpvhpozuETE0RJqvVXZgJySRSfJKB12mFJyYd3gdQUeCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T03:17:14.100681Z"},"content_sha256":"d194420e1cb672f196409b27ae4ab28962c423b9eb19059aca4a08a102775f73","schema_version":"1.0","event_id":"sha256:d194420e1cb672f196409b27ae4ab28962c423b9eb19059aca4a08a102775f73"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:M6RWMEXSHB2ACVMUHQRAWUZQOM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spectral results for free random variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"If the epsilon-derivative of the log-potential admits a real analytic extension through zero, then lambda lies outside the spectrum of a.","cross_cats":["math-ph","math.MP","math.PR"],"primary_cat":"math.OA","authors_text":"Brian C. Hall, Ching-Wei Ho","submitted_at":"2025-10-03T15:52:14Z","abstract_excerpt":"Let $(\\mathcal{A},\\mathrm{tr})$ be a von Neumann algebra with a faithful, normal trace $\\mathrm{tr}:\\mathcal{A}\\rightarrow\\mathbb{C}.$ For each $a\\in\\mathcal{A},$ define \\[ S(\\lambda,\\varepsilon)=\\mathrm{tr}[\\log((a-\\lambda)^{\\ast}(a-\\lambda )+\\varepsilon)],\\quad\\lambda\\in\\mathbb{C},~\\varepsilon>0, \\] so that the limit as $\\varepsilon\\rightarrow0^{+}$ of $S$ is the log potential of the Brown measure of $a.$ Suppose that for a fixed $\\lambda\\in\\mathbb{C},$ the function \\[ \\varepsilon\\mapsto\\frac{\\partial S}{\\partial\\varepsilon}(\\lambda ,\\varepsilon)=\\mathrm{tr}[((a-\\lambda)^{\\ast}(a-\\lambda)+\\v"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Suppose that for a fixed λ∈ℂ, the function ε↦∂S/∂ε(λ,ε)=tr[((a−λ)∗(a−λ)+ε)−1] admits a real analytic extension to a neighborhood of 0 in ℝ. Then λ is outside the spectrum of a.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The implication from real-analytic extendability of the ε-derivative at zero to λ lying in the resolvent set relies on the specific functional-analytic properties of the trace and the definition of the Brown measure via the limit of S as ε→0+ (abstract, first paragraph).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A criterion linking real-analytic extendability of the ε-derivative of the log-potential to λ being outside the spectrum is established and used to show spectrum equals Brown-measure support for circular, elliptic, and free multiplicative Brownian motion elements.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"If the epsilon-derivative of the log-potential admits a real analytic extension through zero, then lambda lies outside the spectrum of a.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3c3dc3adcbbf12a45a8e8e815ceb16034ec9dfe37c3f88e3b709173199ad152d"},"source":{"id":"2510.03382","kind":"arxiv","version":5},"verdict":{"id":"5b3d44c3-eb67-4b2a-91b1-31956ca9c97e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T10:57:16.369272Z","strongest_claim":"Suppose that for a fixed λ∈ℂ, the function ε↦∂S/∂ε(λ,ε)=tr[((a−λ)∗(a−λ)+ε)−1] admits a real analytic extension to a neighborhood of 0 in ℝ. Then λ is outside the spectrum of a.","one_line_summary":"A criterion linking real-analytic extendability of the ε-derivative of the log-potential to λ being outside the spectrum is established and used to show spectrum equals Brown-measure support for circular, elliptic, and free multiplicative Brownian motion elements.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The implication from real-analytic extendability of the ε-derivative at zero to λ lying in the resolvent set relies on the specific functional-analytic properties of the trace and the definition of the Brown measure via the limit of S as ε→0+ (abstract, first paragraph).","pith_extraction_headline":"If the epsilon-derivative of the log-potential admits a real analytic extension through zero, then lambda lies outside the spectrum of a."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2510.03382/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":2,"snapshot_sha256":"49c4873d961dd61b8e20e068194f052cf1192992a5ad6d21199e68d7bcd7c8af"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":"5b3d44c3-eb67-4b2a-91b1-31956ca9c97e"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-20T00:00:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8Os2W9cE7AEZd4RcP3uaqyfRyKqpUWFJmEgHasjBagITJTzlQ4CvtoubudPJ3CLGz/BN3ADlZZbJfC5Sx7itAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T03:17:14.101720Z"},"content_sha256":"233da145bdb6c5f359348fae75e67b95babe24862b9b66f2fa2d7b5f4661a7ef","schema_version":"1.0","event_id":"sha256:233da145bdb6c5f359348fae75e67b95babe24862b9b66f2fa2d7b5f4661a7ef"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/M6RWMEXSHB2ACVMUHQRAWUZQOM/bundle.json","state_url":"https://pith.science/pith/M6RWMEXSHB2ACVMUHQRAWUZQOM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/M6RWMEXSHB2ACVMUHQRAWUZQOM/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":"2026-05-27T03:17:14Z","links":{"resolver":"https://pith.science/pith/M6RWMEXSHB2ACVMUHQRAWUZQOM","bundle":"https://pith.science/pith/M6RWMEXSHB2ACVMUHQRAWUZQOM/bundle.json","state":"https://pith.science/pith/M6RWMEXSHB2ACVMUHQRAWUZQOM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/M6RWMEXSHB2ACVMUHQRAWUZQOM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:M6RWMEXSHB2ACVMUHQRAWUZQOM","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":"1e62d94852568a00a048d935dffdba6c7b8cd2b06d4714bdc073b1090eec1647","cross_cats_sorted":["math-ph","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2025-10-03T15:52:14Z","title_canon_sha256":"655d18f8204514639ec085112d547fe2440c89f4b1ee7eeff67f42aae9a2fb17"},"schema_version":"1.0","source":{"id":"2510.03382","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2510.03382","created_at":"2026-05-20T00:00:25Z"},{"alias_kind":"arxiv_version","alias_value":"2510.03382v5","created_at":"2026-05-20T00:00:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2510.03382","created_at":"2026-05-20T00:00:25Z"},{"alias_kind":"pith_short_12","alias_value":"M6RWMEXSHB2A","created_at":"2026-05-20T00:00:25Z"},{"alias_kind":"pith_short_16","alias_value":"M6RWMEXSHB2ACVMU","created_at":"2026-05-20T00:00:25Z"},{"alias_kind":"pith_short_8","alias_value":"M6RWMEXS","created_at":"2026-05-20T00:00:25Z"}],"graph_snapshots":[{"event_id":"sha256:233da145bdb6c5f359348fae75e67b95babe24862b9b66f2fa2d7b5f4661a7ef","target":"graph","created_at":"2026-05-20T00:00:25Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"Suppose that for a fixed λ∈ℂ, the function ε↦∂S/∂ε(λ,ε)=tr[((a−λ)∗(a−λ)+ε)−1] admits a real analytic extension to a neighborhood of 0 in ℝ. Then λ is outside the spectrum of a."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The implication from real-analytic extendability of the ε-derivative at zero to λ lying in the resolvent set relies on the specific functional-analytic properties of the trace and the definition of the Brown measure via the limit of S as ε→0+ (abstract, first paragraph)."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"A criterion linking real-analytic extendability of the ε-derivative of the log-potential to λ being outside the spectrum is established and used to show spectrum equals Brown-measure support for circular, elliptic, and free multiplicative Brownian motion elements."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"If the epsilon-derivative of the log-potential admits a real analytic extension through zero, then lambda lies outside the spectrum of a."}],"snapshot_sha256":"3c3dc3adcbbf12a45a8e8e815ceb16034ec9dfe37c3f88e3b709173199ad152d"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"49c4873d961dd61b8e20e068194f052cf1192992a5ad6d21199e68d7bcd7c8af"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2510.03382/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $(\\mathcal{A},\\mathrm{tr})$ be a von Neumann algebra with a faithful, normal trace $\\mathrm{tr}:\\mathcal{A}\\rightarrow\\mathbb{C}.$ For each $a\\in\\mathcal{A},$ define \\[ S(\\lambda,\\varepsilon)=\\mathrm{tr}[\\log((a-\\lambda)^{\\ast}(a-\\lambda )+\\varepsilon)],\\quad\\lambda\\in\\mathbb{C},~\\varepsilon>0, \\] so that the limit as $\\varepsilon\\rightarrow0^{+}$ of $S$ is the log potential of the Brown measure of $a.$ Suppose that for a fixed $\\lambda\\in\\mathbb{C},$ the function \\[ \\varepsilon\\mapsto\\frac{\\partial S}{\\partial\\varepsilon}(\\lambda ,\\varepsilon)=\\mathrm{tr}[((a-\\lambda)^{\\ast}(a-\\lambda)+\\v","authors_text":"Brian C. Hall, Ching-Wei Ho","cross_cats":["math-ph","math.MP","math.PR"],"headline":"If the epsilon-derivative of the log-potential admits a real analytic extension through zero, then lambda lies outside the spectrum of a.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2025-10-03T15:52:14Z","title":"Spectral results for free random variables"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2510.03382","kind":"arxiv","version":5},"verdict":{"created_at":"2026-05-18T10:57:16.369272Z","id":"5b3d44c3-eb67-4b2a-91b1-31956ca9c97e","model_set":{"reader":"grok-4.3"},"one_line_summary":"A criterion linking real-analytic extendability of the ε-derivative of the log-potential to λ being outside the spectrum is established and used to show spectrum equals Brown-measure support for circular, elliptic, and free multiplicative Brownian motion elements.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"If the epsilon-derivative of the log-potential admits a real analytic extension through zero, then lambda lies outside the spectrum of a.","strongest_claim":"Suppose that for a fixed λ∈ℂ, the function ε↦∂S/∂ε(λ,ε)=tr[((a−λ)∗(a−λ)+ε)−1] admits a real analytic extension to a neighborhood of 0 in ℝ. Then λ is outside the spectrum of a.","weakest_assumption":"The implication from real-analytic extendability of the ε-derivative at zero to λ lying in the resolvent set relies on the specific functional-analytic properties of the trace and the definition of the Brown measure via the limit of S as ε→0+ (abstract, first paragraph)."}},"verdict_id":"5b3d44c3-eb67-4b2a-91b1-31956ca9c97e"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d194420e1cb672f196409b27ae4ab28962c423b9eb19059aca4a08a102775f73","target":"record","created_at":"2026-05-20T00:00:25Z","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":"1e62d94852568a00a048d935dffdba6c7b8cd2b06d4714bdc073b1090eec1647","cross_cats_sorted":["math-ph","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2025-10-03T15:52:14Z","title_canon_sha256":"655d18f8204514639ec085112d547fe2440c89f4b1ee7eeff67f42aae9a2fb17"},"schema_version":"1.0","source":{"id":"2510.03382","kind":"arxiv","version":5}},"canonical_sha256":"67a36612f238740155943c220b5330731c9ea3eb2fefd43a5e4aaf315acac395","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"67a36612f238740155943c220b5330731c9ea3eb2fefd43a5e4aaf315acac395","first_computed_at":"2026-05-20T00:00:25.268741Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:00:25.268741Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sycAieLgewZh1wwhjhXiVrABTeJghE60RFTNip2aDmD+rt0IkYV9LPr+BjJ09BRGPS3c15KpY2R+OVujXBH2Aw==","signature_status":"signed_v1","signed_at":"2026-05-20T00:00:25.269795Z","signed_message":"canonical_sha256_bytes"},"source_id":"2510.03382","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d194420e1cb672f196409b27ae4ab28962c423b9eb19059aca4a08a102775f73","sha256:233da145bdb6c5f359348fae75e67b95babe24862b9b66f2fa2d7b5f4661a7ef"],"state_sha256":"08106b49d6ead270c00cd9b2fe00e3a90c09a24da50bab9b78365972b638a15f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jK2tI5KSBYOe2dGDmchgBGPRJK/6/1/4OsUESS8m5f0XVd4dX7L0hT7+to+NYQ0KSX99NODfqtyuGNUiloXGDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T03:17:14.106116Z","bundle_sha256":"73f6d00a85e7eca7bf8ecedc322ca8c6f75ded77ec6b969cd79396ddac8844a0"}}