{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:QJMGWKYJ7D5HZZH7NAMR4EH53S","short_pith_number":"pith:QJMGWKYJ","canonical_record":{"source":{"id":"2607.20210","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2026-07-22T14:30:28Z","cross_cats_sorted":[],"title_canon_sha256":"ac288d3bdc958d4c70f3420a7c09d8ebad5786f1b5947e21b57d43d9ff45c48a","abstract_canon_sha256":"b552f5835a92f83a9aef585b6304b471a9cdf97ab559e1cd16f575262ad16e1c"},"schema_version":"1.0"},"canonical_sha256":"82586b2b09f8fa7ce4ff68191e10fddc90c8d3d929699f33f827fb3234eb6fb6","source":{"kind":"arxiv","id":"2607.20210","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.20210","created_at":"2026-07-23T01:25:07Z"},{"alias_kind":"arxiv_version","alias_value":"2607.20210v1","created_at":"2026-07-23T01:25:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.20210","created_at":"2026-07-23T01:25:07Z"},{"alias_kind":"pith_short_12","alias_value":"QJMGWKYJ7D5H","created_at":"2026-07-23T01:25:07Z"},{"alias_kind":"pith_short_16","alias_value":"QJMGWKYJ7D5HZZH7","created_at":"2026-07-23T01:25:07Z"},{"alias_kind":"pith_short_8","alias_value":"QJMGWKYJ","created_at":"2026-07-23T01:25:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:QJMGWKYJ7D5HZZH7NAMR4EH53S","target":"record","payload":{"canonical_record":{"source":{"id":"2607.20210","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2026-07-22T14:30:28Z","cross_cats_sorted":[],"title_canon_sha256":"ac288d3bdc958d4c70f3420a7c09d8ebad5786f1b5947e21b57d43d9ff45c48a","abstract_canon_sha256":"b552f5835a92f83a9aef585b6304b471a9cdf97ab559e1cd16f575262ad16e1c"},"schema_version":"1.0"},"canonical_sha256":"82586b2b09f8fa7ce4ff68191e10fddc90c8d3d929699f33f827fb3234eb6fb6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-23T01:25:07.911224Z","signature_b64":"hxOMs4I3iWAEqv90tfA6ETHeJ0FDBBahiLC1enwCirveHROls0/D+cT2j6q19AXpxWfh53fKHWxuJCiFCCRyCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"82586b2b09f8fa7ce4ff68191e10fddc90c8d3d929699f33f827fb3234eb6fb6","last_reissued_at":"2026-07-23T01:25:07.910378Z","signature_status":"signed_v1","first_computed_at":"2026-07-23T01:25:07.910378Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.20210","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":"2026-07-23T01:25:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"h4So6xUUtSZpNNuanPBZ8jE+R3U9zfHLLXfMO7dxRkcvGBpvHM/I3nhtDJ2t3AalpjRP2D8yZbPEaJxqNpTUAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T18:59:44.758358Z"},"content_sha256":"dc25cfa3ec605d5d763b47c7643668e093b5f681a1fe0a5b972cc6c9c28eb194","schema_version":"1.0","event_id":"sha256:dc25cfa3ec605d5d763b47c7643668e093b5f681a1fe0a5b972cc6c9c28eb194"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:QJMGWKYJ7D5HZZH7NAMR4EH53S","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Graded Keller maps and the Jacobian Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"T. Shaska","submitted_at":"2026-07-22T14:30:28Z","abstract_excerpt":"Rational points on weighted projective spaces are sparse: a point of $\\mathbb{P}^n(\\mathbb{Q})$ lifts along the Veronese morphism only when a Kummer condition holds at every prime (arXiv:2509.02319). We ask what this viewpoint says about the Jacobian Conjecture. The map recently announced as a counterexample is equivariant for the grading $\\mathrm{wt}(x,y,z)=(1,-1,-2)$, and we show that the sign pattern of such a grading decides everything. If the weights are all positive, the setting of weighted projective spaces, an equivariant Keller map is always an automorphism, so no counterexample can b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.20210","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/2607.20210/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":"2026-07-23T01:25:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gkKmpjkcYG+1YaOQrHH0IXCdisTKRa/+3DAhN8RhEl1gNRBdElHVBXV2nQUX0rz4eWvGX73ffpbv1jZ6Z2NxCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T18:59:44.759185Z"},"content_sha256":"7b12850b5d1309ec24395e4fe44be21c6ef96f6026ab0bc5873085bb0e1bd85b","schema_version":"1.0","event_id":"sha256:7b12850b5d1309ec24395e4fe44be21c6ef96f6026ab0bc5873085bb0e1bd85b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QJMGWKYJ7D5HZZH7NAMR4EH53S/bundle.json","state_url":"https://pith.science/pith/QJMGWKYJ7D5HZZH7NAMR4EH53S/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QJMGWKYJ7D5HZZH7NAMR4EH53S/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":"2026-08-15T18:59:44Z","links":{"resolver":"https://pith.science/pith/QJMGWKYJ7D5HZZH7NAMR4EH53S","bundle":"https://pith.science/pith/QJMGWKYJ7D5HZZH7NAMR4EH53S/bundle.json","state":"https://pith.science/pith/QJMGWKYJ7D5HZZH7NAMR4EH53S/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QJMGWKYJ7D5HZZH7NAMR4EH53S/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:QJMGWKYJ7D5HZZH7NAMR4EH53S","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":"b552f5835a92f83a9aef585b6304b471a9cdf97ab559e1cd16f575262ad16e1c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2026-07-22T14:30:28Z","title_canon_sha256":"ac288d3bdc958d4c70f3420a7c09d8ebad5786f1b5947e21b57d43d9ff45c48a"},"schema_version":"1.0","source":{"id":"2607.20210","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.20210","created_at":"2026-07-23T01:25:07Z"},{"alias_kind":"arxiv_version","alias_value":"2607.20210v1","created_at":"2026-07-23T01:25:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.20210","created_at":"2026-07-23T01:25:07Z"},{"alias_kind":"pith_short_12","alias_value":"QJMGWKYJ7D5H","created_at":"2026-07-23T01:25:07Z"},{"alias_kind":"pith_short_16","alias_value":"QJMGWKYJ7D5HZZH7","created_at":"2026-07-23T01:25:07Z"},{"alias_kind":"pith_short_8","alias_value":"QJMGWKYJ","created_at":"2026-07-23T01:25:07Z"}],"graph_snapshots":[{"event_id":"sha256:7b12850b5d1309ec24395e4fe44be21c6ef96f6026ab0bc5873085bb0e1bd85b","target":"graph","created_at":"2026-07-23T01:25:07Z","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/2607.20210/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Rational points on weighted projective spaces are sparse: a point of $\\mathbb{P}^n(\\mathbb{Q})$ lifts along the Veronese morphism only when a Kummer condition holds at every prime (arXiv:2509.02319). We ask what this viewpoint says about the Jacobian Conjecture. The map recently announced as a counterexample is equivariant for the grading $\\mathrm{wt}(x,y,z)=(1,-1,-2)$, and we show that the sign pattern of such a grading decides everything. If the weights are all positive, the setting of weighted projective spaces, an equivariant Keller map is always an automorphism, so no counterexample can b","authors_text":"T. Shaska","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2026-07-22T14:30:28Z","title":"Graded Keller maps and the Jacobian Conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.20210","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:dc25cfa3ec605d5d763b47c7643668e093b5f681a1fe0a5b972cc6c9c28eb194","target":"record","created_at":"2026-07-23T01:25:07Z","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":"b552f5835a92f83a9aef585b6304b471a9cdf97ab559e1cd16f575262ad16e1c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2026-07-22T14:30:28Z","title_canon_sha256":"ac288d3bdc958d4c70f3420a7c09d8ebad5786f1b5947e21b57d43d9ff45c48a"},"schema_version":"1.0","source":{"id":"2607.20210","kind":"arxiv","version":1}},"canonical_sha256":"82586b2b09f8fa7ce4ff68191e10fddc90c8d3d929699f33f827fb3234eb6fb6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"82586b2b09f8fa7ce4ff68191e10fddc90c8d3d929699f33f827fb3234eb6fb6","first_computed_at":"2026-07-23T01:25:07.910378Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-23T01:25:07.910378Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hxOMs4I3iWAEqv90tfA6ETHeJ0FDBBahiLC1enwCirveHROls0/D+cT2j6q19AXpxWfh53fKHWxuJCiFCCRyCg==","signature_status":"signed_v1","signed_at":"2026-07-23T01:25:07.911224Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.20210","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dc25cfa3ec605d5d763b47c7643668e093b5f681a1fe0a5b972cc6c9c28eb194","sha256:7b12850b5d1309ec24395e4fe44be21c6ef96f6026ab0bc5873085bb0e1bd85b"],"state_sha256":"b673919f3b116bdd5b34f313a0dbdc3299665b6496ff4f0714b82af724153c4d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"By+AvU6In+7uln8/CLRCPWQZDbqtOnJ/omntLhuZPOp9Jpk+uuYMgH8FHPX2sPNS/p6t6uC6ngJYPbJTQhw/DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-15T18:59:44.765189Z","bundle_sha256":"3ced0d48f06a44b010c78bc4e15a5ea0f8bca9e58ff9e2f4637f86da6ee61116"}}