{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:RTR2IYAP4H7IARRUV4IC4KVVKU","short_pith_number":"pith:RTR2IYAP","canonical_record":{"source":{"id":"2606.05475","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.FA","submitted_at":"2026-06-03T21:53:52Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"0562dd21b8785887ba504ee3400729ecd14430ff04bf5b4a28cb3c2679bc7782","abstract_canon_sha256":"ea8211b2386273196790a25fc42f2b19085f98793970e933deab971e70442ab9"},"schema_version":"1.0"},"canonical_sha256":"8ce3a4600fe1fe804634af102e2ab5552cb1af7767e1439f9e3664fb334bd238","source":{"kind":"arxiv","id":"2606.05475","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.05475","created_at":"2026-06-05T01:14:52Z"},{"alias_kind":"arxiv_version","alias_value":"2606.05475v1","created_at":"2026-06-05T01:14:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.05475","created_at":"2026-06-05T01:14:52Z"},{"alias_kind":"pith_short_12","alias_value":"RTR2IYAP4H7I","created_at":"2026-06-05T01:14:52Z"},{"alias_kind":"pith_short_16","alias_value":"RTR2IYAP4H7IARRU","created_at":"2026-06-05T01:14:52Z"},{"alias_kind":"pith_short_8","alias_value":"RTR2IYAP","created_at":"2026-06-05T01:14:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:RTR2IYAP4H7IARRUV4IC4KVVKU","target":"record","payload":{"canonical_record":{"source":{"id":"2606.05475","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.FA","submitted_at":"2026-06-03T21:53:52Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"0562dd21b8785887ba504ee3400729ecd14430ff04bf5b4a28cb3c2679bc7782","abstract_canon_sha256":"ea8211b2386273196790a25fc42f2b19085f98793970e933deab971e70442ab9"},"schema_version":"1.0"},"canonical_sha256":"8ce3a4600fe1fe804634af102e2ab5552cb1af7767e1439f9e3664fb334bd238","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-05T01:14:52.365719Z","signature_b64":"tzmt87KEaCKHnvwM3Z215Xa+8b89txCmb1XoSyI/y3fWSQqeu0OwW2LFhnEngddCeKdyNiUXVZUg0AgBwtzsDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ce3a4600fe1fe804634af102e2ab5552cb1af7767e1439f9e3664fb334bd238","last_reissued_at":"2026-06-05T01:14:52.364905Z","signature_status":"signed_v1","first_computed_at":"2026-06-05T01:14:52.364905Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.05475","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":"2026-06-05T01:14:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9fYV/gnyISxLwnfgBaaGBZmFBUwSYZpkt0P5nDHeFd3af37WJuwmUNqjgM/RLOL/eHP26it/MWmLTqTotF+RBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T06:48:14.679985Z"},"content_sha256":"855955dc4072ce4fd8e044ef6816d96e7d16f5ed95f63a7419d268b94556514f","schema_version":"1.0","event_id":"sha256:855955dc4072ce4fd8e044ef6816d96e7d16f5ed95f63a7419d268b94556514f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:RTR2IYAP4H7IARRUV4IC4KVVKU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Reverse inequalities for super-Riesz transforms on graphs with a slow diffusion","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.FA","authors_text":"Joseph Feneuil","submitted_at":"2026-06-03T21:53:52Z","abstract_excerpt":"In the $D$-dimensional Vicsek graph, we prove that the Riesz-like inequality $ \\|\\nabla f\\|_p \\leq C \\|\\Delta^\\gamma f\\|_p $ holds for every $p\\in(1,\\infty)$ and every $ 0<\\gamma<\\gamma^*(p):=\\frac{1}{D+1}+\\frac{D-1}{D+1}\\,\\frac{1}{p}, $ while it fails whenever $p\\in(1,\\infty)$ and $\\gamma^*(p)<\\gamma<1$. Thus, the validity of the inequality remains open only at the critical exponent $\\gamma=\\gamma^*(p)$. This provides the first example of an $L^p$-bounded ``super-Riesz transform'', namely an operator of the form $\\nabla \\Delta^{-\\gamma}$ with $\\gamma$ strictly larger than the Euclidean thresh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.05475","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/2606.05475/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":"2026-06-05T01:14:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IeNvStj8qmcbm0Nil4vcpvpIO7sSAXxsnzOzSs2mVc9KllqH+sf27QlTohvQhfRTcYQmAybz+okzezGS40H0Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T06:48:14.680311Z"},"content_sha256":"91c2cd2cc22e18181c5aa55208571169376ece57f15754f0b35bda22926e1d23","schema_version":"1.0","event_id":"sha256:91c2cd2cc22e18181c5aa55208571169376ece57f15754f0b35bda22926e1d23"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RTR2IYAP4H7IARRUV4IC4KVVKU/bundle.json","state_url":"https://pith.science/pith/RTR2IYAP4H7IARRUV4IC4KVVKU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RTR2IYAP4H7IARRUV4IC4KVVKU/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":"2026-08-14T06:48:14Z","links":{"resolver":"https://pith.science/pith/RTR2IYAP4H7IARRUV4IC4KVVKU","bundle":"https://pith.science/pith/RTR2IYAP4H7IARRUV4IC4KVVKU/bundle.json","state":"https://pith.science/pith/RTR2IYAP4H7IARRUV4IC4KVVKU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RTR2IYAP4H7IARRUV4IC4KVVKU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:RTR2IYAP4H7IARRUV4IC4KVVKU","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":"ea8211b2386273196790a25fc42f2b19085f98793970e933deab971e70442ab9","cross_cats_sorted":["math.PR"],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.FA","submitted_at":"2026-06-03T21:53:52Z","title_canon_sha256":"0562dd21b8785887ba504ee3400729ecd14430ff04bf5b4a28cb3c2679bc7782"},"schema_version":"1.0","source":{"id":"2606.05475","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.05475","created_at":"2026-06-05T01:14:52Z"},{"alias_kind":"arxiv_version","alias_value":"2606.05475v1","created_at":"2026-06-05T01:14:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.05475","created_at":"2026-06-05T01:14:52Z"},{"alias_kind":"pith_short_12","alias_value":"RTR2IYAP4H7I","created_at":"2026-06-05T01:14:52Z"},{"alias_kind":"pith_short_16","alias_value":"RTR2IYAP4H7IARRU","created_at":"2026-06-05T01:14:52Z"},{"alias_kind":"pith_short_8","alias_value":"RTR2IYAP","created_at":"2026-06-05T01:14:52Z"}],"graph_snapshots":[{"event_id":"sha256:91c2cd2cc22e18181c5aa55208571169376ece57f15754f0b35bda22926e1d23","target":"graph","created_at":"2026-06-05T01:14:52Z","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/2606.05475/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In the $D$-dimensional Vicsek graph, we prove that the Riesz-like inequality $ \\|\\nabla f\\|_p \\leq C \\|\\Delta^\\gamma f\\|_p $ holds for every $p\\in(1,\\infty)$ and every $ 0<\\gamma<\\gamma^*(p):=\\frac{1}{D+1}+\\frac{D-1}{D+1}\\,\\frac{1}{p}, $ while it fails whenever $p\\in(1,\\infty)$ and $\\gamma^*(p)<\\gamma<1$. Thus, the validity of the inequality remains open only at the critical exponent $\\gamma=\\gamma^*(p)$. This provides the first example of an $L^p$-bounded ``super-Riesz transform'', namely an operator of the form $\\nabla \\Delta^{-\\gamma}$ with $\\gamma$ strictly larger than the Euclidean thresh","authors_text":"Joseph Feneuil","cross_cats":["math.PR"],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.FA","submitted_at":"2026-06-03T21:53:52Z","title":"Reverse inequalities for super-Riesz transforms on graphs with a slow diffusion"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.05475","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":"sha256:855955dc4072ce4fd8e044ef6816d96e7d16f5ed95f63a7419d268b94556514f","target":"record","created_at":"2026-06-05T01:14:52Z","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":"ea8211b2386273196790a25fc42f2b19085f98793970e933deab971e70442ab9","cross_cats_sorted":["math.PR"],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.FA","submitted_at":"2026-06-03T21:53:52Z","title_canon_sha256":"0562dd21b8785887ba504ee3400729ecd14430ff04bf5b4a28cb3c2679bc7782"},"schema_version":"1.0","source":{"id":"2606.05475","kind":"arxiv","version":1}},"canonical_sha256":"8ce3a4600fe1fe804634af102e2ab5552cb1af7767e1439f9e3664fb334bd238","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8ce3a4600fe1fe804634af102e2ab5552cb1af7767e1439f9e3664fb334bd238","first_computed_at":"2026-06-05T01:14:52.364905Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-05T01:14:52.364905Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tzmt87KEaCKHnvwM3Z215Xa+8b89txCmb1XoSyI/y3fWSQqeu0OwW2LFhnEngddCeKdyNiUXVZUg0AgBwtzsDA==","signature_status":"signed_v1","signed_at":"2026-06-05T01:14:52.365719Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.05475","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:855955dc4072ce4fd8e044ef6816d96e7d16f5ed95f63a7419d268b94556514f","sha256:91c2cd2cc22e18181c5aa55208571169376ece57f15754f0b35bda22926e1d23"],"state_sha256":"e33ef9b9dc8f231896ecd67132e341eada4384f19c3c2947b15f6d7fe289ae46"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vaK3qmih3gCCwz5yvAh4J+ItpwCPQlkTgXb3Adkomr34xKIhe3aO2DVYrv/7qi4YKZBm/yVWWX7peN3YXOHOAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T06:48:14.683223Z","bundle_sha256":"6e2f888c3fcb3423f8431605b776d0812c83bf3d0ea392aa6a11e4afd69568ce"}}