{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:QO6DSA5KLWYTREOUROAWVA53AG","short_pith_number":"pith:QO6DSA5K","canonical_record":{"source":{"id":"2606.18458","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-16T20:04:47Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"27a596e81225865ec5c97837269595dd5c68b897a2896edba3b8421f1782f189","abstract_canon_sha256":"dc6d71f2935502c1ea90f6861f64a1d1224f2b697dfe9d7a2d724595f37da9f8"},"schema_version":"1.0"},"canonical_sha256":"83bc3903aa5db13891d48b816a83bb01a9f76e9d1e6d21ae87f8bbda0da1a0aa","source":{"kind":"arxiv","id":"2606.18458","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.18458","created_at":"2026-06-19T16:11:01Z"},{"alias_kind":"arxiv_version","alias_value":"2606.18458v1","created_at":"2026-06-19T16:11:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.18458","created_at":"2026-06-19T16:11:01Z"},{"alias_kind":"pith_short_12","alias_value":"QO6DSA5KLWYT","created_at":"2026-06-19T16:11:01Z"},{"alias_kind":"pith_short_16","alias_value":"QO6DSA5KLWYTREOU","created_at":"2026-06-19T16:11:01Z"},{"alias_kind":"pith_short_8","alias_value":"QO6DSA5K","created_at":"2026-06-19T16:11:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:QO6DSA5KLWYTREOUROAWVA53AG","target":"record","payload":{"canonical_record":{"source":{"id":"2606.18458","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-16T20:04:47Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"27a596e81225865ec5c97837269595dd5c68b897a2896edba3b8421f1782f189","abstract_canon_sha256":"dc6d71f2935502c1ea90f6861f64a1d1224f2b697dfe9d7a2d724595f37da9f8"},"schema_version":"1.0"},"canonical_sha256":"83bc3903aa5db13891d48b816a83bb01a9f76e9d1e6d21ae87f8bbda0da1a0aa","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:11:01.712046Z","signature_b64":"btWZ/lBL3smWHBaj+Di0btv9CL53ZjDnQDD6M2jpd6Fx7Dzu6vGqyd5JA76DXQ1GcAzdizKLEMuHcJDf7kV8Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"83bc3903aa5db13891d48b816a83bb01a9f76e9d1e6d21ae87f8bbda0da1a0aa","last_reissued_at":"2026-06-19T16:11:01.711663Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:11:01.711663Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.18458","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":"2026-06-19T16:11:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LHp1VDcrRN1JFXjXq5RsKev1MusMQNoVryupJbfy5LAOUVzEjXFmG2wGpkrAZVgNErCQ/lB6+Mhpntsa2TlcDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T16:39:36.558056Z"},"content_sha256":"89b4cfe5d94220c175bea39ae4c8f1b4ed1f4c556dc9ce97f4bc5f9dad9f8173","schema_version":"1.0","event_id":"sha256:89b4cfe5d94220c175bea39ae4c8f1b4ed1f4c556dc9ce97f4bc5f9dad9f8173"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:QO6DSA5KLWYTREOUROAWVA53AG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stable size-biasing and the positive scale-mixture order of generalized Gaussian laws","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Domingos S. P. Salazar","submitted_at":"2026-06-16T20:04:47Z","abstract_excerpt":"Let $X_r\\sim N_r(0,1)$ be the centered unit-scale generalized Gaussian random variable with density proportional to $\\exp(-|x|^r/2)$. We prove that, for $p,q>0$, there exists a strictly positive random variable $V$, independent of $X_q$, such that $X_p\\stackrel{d}{=}VX_q$ if and only if $p\\le q$. Moreover, the law of $V$ is unique. For $p<q$, put $a=1/p$, $b=1/q$, and $\\alpha=b/a=p/q$. If $S_\\alpha$ is a positive $\\alpha$-stable random variable with Laplace transform $\\mathbb{E}\\exp(-uS_\\alpha)=\\exp(-u^\\alpha)$, set $W_0=S_\\alpha^{-b}$, let $W$ be the $W_0$-size-biased version of $W_0$, and de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.18458","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/2606.18458/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":"2026-06-19T16:11:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dSoamtJXJdl3E3mgbOV3KDMfeTJzQkJJ8wdZDtTPPM0AypLE7bWq0rcSDHpfiw/vfX/R8ju3XDqzh3YAqkRyAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T16:39:36.558441Z"},"content_sha256":"6529b481ba1918d5ed4376c6b3fbd7cf9478b0be1d78d3ab67a949339ae4de74","schema_version":"1.0","event_id":"sha256:6529b481ba1918d5ed4376c6b3fbd7cf9478b0be1d78d3ab67a949339ae4de74"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QO6DSA5KLWYTREOUROAWVA53AG/bundle.json","state_url":"https://pith.science/pith/QO6DSA5KLWYTREOUROAWVA53AG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QO6DSA5KLWYTREOUROAWVA53AG/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":"2026-06-21T16:39:36Z","links":{"resolver":"https://pith.science/pith/QO6DSA5KLWYTREOUROAWVA53AG","bundle":"https://pith.science/pith/QO6DSA5KLWYTREOUROAWVA53AG/bundle.json","state":"https://pith.science/pith/QO6DSA5KLWYTREOUROAWVA53AG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QO6DSA5KLWYTREOUROAWVA53AG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:QO6DSA5KLWYTREOUROAWVA53AG","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":"dc6d71f2935502c1ea90f6861f64a1d1224f2b697dfe9d7a2d724595f37da9f8","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-16T20:04:47Z","title_canon_sha256":"27a596e81225865ec5c97837269595dd5c68b897a2896edba3b8421f1782f189"},"schema_version":"1.0","source":{"id":"2606.18458","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.18458","created_at":"2026-06-19T16:11:01Z"},{"alias_kind":"arxiv_version","alias_value":"2606.18458v1","created_at":"2026-06-19T16:11:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.18458","created_at":"2026-06-19T16:11:01Z"},{"alias_kind":"pith_short_12","alias_value":"QO6DSA5KLWYT","created_at":"2026-06-19T16:11:01Z"},{"alias_kind":"pith_short_16","alias_value":"QO6DSA5KLWYTREOU","created_at":"2026-06-19T16:11:01Z"},{"alias_kind":"pith_short_8","alias_value":"QO6DSA5K","created_at":"2026-06-19T16:11:01Z"}],"graph_snapshots":[{"event_id":"sha256:6529b481ba1918d5ed4376c6b3fbd7cf9478b0be1d78d3ab67a949339ae4de74","target":"graph","created_at":"2026-06-19T16:11:01Z","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/2606.18458/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $X_r\\sim N_r(0,1)$ be the centered unit-scale generalized Gaussian random variable with density proportional to $\\exp(-|x|^r/2)$. We prove that, for $p,q>0$, there exists a strictly positive random variable $V$, independent of $X_q$, such that $X_p\\stackrel{d}{=}VX_q$ if and only if $p\\le q$. Moreover, the law of $V$ is unique. For $p<q$, put $a=1/p$, $b=1/q$, and $\\alpha=b/a=p/q$. If $S_\\alpha$ is a positive $\\alpha$-stable random variable with Laplace transform $\\mathbb{E}\\exp(-uS_\\alpha)=\\exp(-u^\\alpha)$, set $W_0=S_\\alpha^{-b}$, let $W$ be the $W_0$-size-biased version of $W_0$, and de","authors_text":"Domingos S. P. Salazar","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-16T20:04:47Z","title":"Stable size-biasing and the positive scale-mixture order of generalized Gaussian laws"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.18458","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":"sha256:89b4cfe5d94220c175bea39ae4c8f1b4ed1f4c556dc9ce97f4bc5f9dad9f8173","target":"record","created_at":"2026-06-19T16:11:01Z","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":"dc6d71f2935502c1ea90f6861f64a1d1224f2b697dfe9d7a2d724595f37da9f8","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-16T20:04:47Z","title_canon_sha256":"27a596e81225865ec5c97837269595dd5c68b897a2896edba3b8421f1782f189"},"schema_version":"1.0","source":{"id":"2606.18458","kind":"arxiv","version":1}},"canonical_sha256":"83bc3903aa5db13891d48b816a83bb01a9f76e9d1e6d21ae87f8bbda0da1a0aa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"83bc3903aa5db13891d48b816a83bb01a9f76e9d1e6d21ae87f8bbda0da1a0aa","first_computed_at":"2026-06-19T16:11:01.711663Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-19T16:11:01.711663Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"btWZ/lBL3smWHBaj+Di0btv9CL53ZjDnQDD6M2jpd6Fx7Dzu6vGqyd5JA76DXQ1GcAzdizKLEMuHcJDf7kV8Cw==","signature_status":"signed_v1","signed_at":"2026-06-19T16:11:01.712046Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.18458","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:89b4cfe5d94220c175bea39ae4c8f1b4ed1f4c556dc9ce97f4bc5f9dad9f8173","sha256:6529b481ba1918d5ed4376c6b3fbd7cf9478b0be1d78d3ab67a949339ae4de74"],"state_sha256":"e311932dfc734e58452fa9ead64a713f9773ea5439459131d6a304ae1db74770"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NC3uTFe+/L2omS32a9V4enuJNtBC3yevDaw70BeQPx0xFgjFG8NEc5b3IOneSA4eCyxakrFWfAgB1chZ3lkOCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T16:39:36.560505Z","bundle_sha256":"1054a8554a104d943e42ed93003ccde202c2bae1dab57470c65d3f2d78bcc005"}}