{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:OCSSEQAKE4O75YDNNCMAXWPH3D","short_pith_number":"pith:OCSSEQAK","canonical_record":{"source":{"id":"1612.08565","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-12-27T10:38:04Z","cross_cats_sorted":["math-ph","math.AP","math.MP"],"title_canon_sha256":"8e96c3b34831ad36be802a4d39e9c1b303d51e2703e6f127bf98fc0b77321dbe","abstract_canon_sha256":"b0bdf45114f98515d9978dc084a7babdcb9b79471138e69b086163091aa77c4d"},"schema_version":"1.0"},"canonical_sha256":"70a522400a271dfee06d68980bd9e7d8d5e61d53b645ffcd169b7b4c661fcf5c","source":{"kind":"arxiv","id":"1612.08565","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.08565","created_at":"2026-05-18T00:51:29Z"},{"alias_kind":"arxiv_version","alias_value":"1612.08565v2","created_at":"2026-05-18T00:51:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.08565","created_at":"2026-05-18T00:51:29Z"},{"alias_kind":"pith_short_12","alias_value":"OCSSEQAKE4O7","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"OCSSEQAKE4O75YDN","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"OCSSEQAK","created_at":"2026-05-18T12:30:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:OCSSEQAKE4O75YDNNCMAXWPH3D","target":"record","payload":{"canonical_record":{"source":{"id":"1612.08565","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-12-27T10:38:04Z","cross_cats_sorted":["math-ph","math.AP","math.MP"],"title_canon_sha256":"8e96c3b34831ad36be802a4d39e9c1b303d51e2703e6f127bf98fc0b77321dbe","abstract_canon_sha256":"b0bdf45114f98515d9978dc084a7babdcb9b79471138e69b086163091aa77c4d"},"schema_version":"1.0"},"canonical_sha256":"70a522400a271dfee06d68980bd9e7d8d5e61d53b645ffcd169b7b4c661fcf5c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:29.937575Z","signature_b64":"+eMXdZorphicN/5o98+oCBi4CExb53a/eKngFJ+avPjhLkg8nMICSaPVEAKwkBSg1uxGohxVsnx8MYTF62e1DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"70a522400a271dfee06d68980bd9e7d8d5e61d53b645ffcd169b7b4c661fcf5c","last_reissued_at":"2026-05-18T00:51:29.937040Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:29.937040Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.08565","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:51:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jyqy4LzpzCoBzg+kl2x2WOw6xnTGOX8AbjsTiopOGQ3r1YlOrAHRmrbd/v7lvt1Fn6BFw5Aq9fsbYxZPw/90Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T18:11:23.050173Z"},"content_sha256":"a70ccca7be26e2f5718073ad73afeaa242904677269b0530bbc14a673db9e096","schema_version":"1.0","event_id":"sha256:a70ccca7be26e2f5718073ad73afeaa242904677269b0530bbc14a673db9e096"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:OCSSEQAKE4O75YDNNCMAXWPH3D","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Spectral Gap Estimate and Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AP","math.MP"],"primary_cat":"math.SP","authors_text":"Bogdan Georgiev, Mayukh Mukherjee, Stefan Steinerberger","submitted_at":"2016-12-27T10:38:04Z","abstract_excerpt":"We consider the Schr\\\"odinger operator $$-\\frac{d^2}{d x^2} + V \\qquad \\mbox{on an interval}~~[a,b]~\\mbox{with Dirichlet boundary conditions},$$ where $V$ is bounded from below and prove a lower bound on the first eigenvalue $\\lambda_1$ in terms of sublevel estimates: if $ w_V(y) = |I_y|,\\text{ where } I_y := \\left\\{ x \\in [a,b]: V(x) \\leq y \\right\\},$ then $$ \\lambda_1 \\geq \\frac{1}{250} \\min_{y > \\min V}{\\left(\\frac{1}{w_V(y)^2} + y\\right)}.$$ The result is sharp up to a universal constant if $\\left\\{ x \\in [a,b]: V(x) \\leq y \\right\\}$ is an interval for the value of $y$ solving the minimiza"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.08565","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:51:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1u3erCEaQg6P0ehr+ZxVz3n54udfxGgXnLv0FO4ENdaY2s4pB5kpNJHv49Q6IHdpx+eKFjW2875q0tw0psFkBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T18:11:23.050920Z"},"content_sha256":"d5196858512ec9b6d36366d6eb66afeba5205d2ce4a0968c118825455b247222","schema_version":"1.0","event_id":"sha256:d5196858512ec9b6d36366d6eb66afeba5205d2ce4a0968c118825455b247222"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OCSSEQAKE4O75YDNNCMAXWPH3D/bundle.json","state_url":"https://pith.science/pith/OCSSEQAKE4O75YDNNCMAXWPH3D/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OCSSEQAKE4O75YDNNCMAXWPH3D/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":"2026-06-09T18:11:23Z","links":{"resolver":"https://pith.science/pith/OCSSEQAKE4O75YDNNCMAXWPH3D","bundle":"https://pith.science/pith/OCSSEQAKE4O75YDNNCMAXWPH3D/bundle.json","state":"https://pith.science/pith/OCSSEQAKE4O75YDNNCMAXWPH3D/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OCSSEQAKE4O75YDNNCMAXWPH3D/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:OCSSEQAKE4O75YDNNCMAXWPH3D","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":"b0bdf45114f98515d9978dc084a7babdcb9b79471138e69b086163091aa77c4d","cross_cats_sorted":["math-ph","math.AP","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-12-27T10:38:04Z","title_canon_sha256":"8e96c3b34831ad36be802a4d39e9c1b303d51e2703e6f127bf98fc0b77321dbe"},"schema_version":"1.0","source":{"id":"1612.08565","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.08565","created_at":"2026-05-18T00:51:29Z"},{"alias_kind":"arxiv_version","alias_value":"1612.08565v2","created_at":"2026-05-18T00:51:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.08565","created_at":"2026-05-18T00:51:29Z"},{"alias_kind":"pith_short_12","alias_value":"OCSSEQAKE4O7","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"OCSSEQAKE4O75YDN","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"OCSSEQAK","created_at":"2026-05-18T12:30:36Z"}],"graph_snapshots":[{"event_id":"sha256:d5196858512ec9b6d36366d6eb66afeba5205d2ce4a0968c118825455b247222","target":"graph","created_at":"2026-05-18T00:51:29Z","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"},"paper":{"abstract_excerpt":"We consider the Schr\\\"odinger operator $$-\\frac{d^2}{d x^2} + V \\qquad \\mbox{on an interval}~~[a,b]~\\mbox{with Dirichlet boundary conditions},$$ where $V$ is bounded from below and prove a lower bound on the first eigenvalue $\\lambda_1$ in terms of sublevel estimates: if $ w_V(y) = |I_y|,\\text{ where } I_y := \\left\\{ x \\in [a,b]: V(x) \\leq y \\right\\},$ then $$ \\lambda_1 \\geq \\frac{1}{250} \\min_{y > \\min V}{\\left(\\frac{1}{w_V(y)^2} + y\\right)}.$$ The result is sharp up to a universal constant if $\\left\\{ x \\in [a,b]: V(x) \\leq y \\right\\}$ is an interval for the value of $y$ solving the minimiza","authors_text":"Bogdan Georgiev, Mayukh Mukherjee, Stefan Steinerberger","cross_cats":["math-ph","math.AP","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-12-27T10:38:04Z","title":"A Spectral Gap Estimate and Applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.08565","kind":"arxiv","version":2},"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:a70ccca7be26e2f5718073ad73afeaa242904677269b0530bbc14a673db9e096","target":"record","created_at":"2026-05-18T00:51:29Z","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":"b0bdf45114f98515d9978dc084a7babdcb9b79471138e69b086163091aa77c4d","cross_cats_sorted":["math-ph","math.AP","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-12-27T10:38:04Z","title_canon_sha256":"8e96c3b34831ad36be802a4d39e9c1b303d51e2703e6f127bf98fc0b77321dbe"},"schema_version":"1.0","source":{"id":"1612.08565","kind":"arxiv","version":2}},"canonical_sha256":"70a522400a271dfee06d68980bd9e7d8d5e61d53b645ffcd169b7b4c661fcf5c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"70a522400a271dfee06d68980bd9e7d8d5e61d53b645ffcd169b7b4c661fcf5c","first_computed_at":"2026-05-18T00:51:29.937040Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:51:29.937040Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+eMXdZorphicN/5o98+oCBi4CExb53a/eKngFJ+avPjhLkg8nMICSaPVEAKwkBSg1uxGohxVsnx8MYTF62e1DA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:51:29.937575Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.08565","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a70ccca7be26e2f5718073ad73afeaa242904677269b0530bbc14a673db9e096","sha256:d5196858512ec9b6d36366d6eb66afeba5205d2ce4a0968c118825455b247222"],"state_sha256":"2b0a2f3aba32b2fc8dd90e24c0801987a35d29a86847b8730eb2b136c84078c8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y8W8AQLCM/051V+q6yphSieVdQMKsC76t2Pf8BoapsiGtVmzk05LYWMB6llj4ArkOhxgIjn3VlBYSbM4nXM8AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T18:11:23.055180Z","bundle_sha256":"5650cd2a8d7e70d795126ecee1ce7e6ae1ba723cee10553d0f64a6698cd1941a"}}