{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:22DX7ERO4WRCQJDVLMYWQFZ5BS","short_pith_number":"pith:22DX7ERO","canonical_record":{"source":{"id":"1409.5772","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2014-09-19T19:23:21Z","cross_cats_sorted":[],"title_canon_sha256":"35c0982def2ed1772722136a615bf843b0d08c7cd54262a769bebcb2f3da29f5","abstract_canon_sha256":"a05fdd3719339e5616ec681551929058aa035099ccd47b70e3929f70833a600f"},"schema_version":"1.0"},"canonical_sha256":"d6877f922ee5a22824755b3168173d0c9cefc141325db77194f6efa54dd25b7d","source":{"kind":"arxiv","id":"1409.5772","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.5772","created_at":"2026-05-17T23:42:14Z"},{"alias_kind":"arxiv_version","alias_value":"1409.5772v1","created_at":"2026-05-17T23:42:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.5772","created_at":"2026-05-17T23:42:14Z"},{"alias_kind":"pith_short_12","alias_value":"22DX7ERO4WRC","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"22DX7ERO4WRCQJDV","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"22DX7ERO","created_at":"2026-05-18T12:28:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:22DX7ERO4WRCQJDVLMYWQFZ5BS","target":"record","payload":{"canonical_record":{"source":{"id":"1409.5772","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2014-09-19T19:23:21Z","cross_cats_sorted":[],"title_canon_sha256":"35c0982def2ed1772722136a615bf843b0d08c7cd54262a769bebcb2f3da29f5","abstract_canon_sha256":"a05fdd3719339e5616ec681551929058aa035099ccd47b70e3929f70833a600f"},"schema_version":"1.0"},"canonical_sha256":"d6877f922ee5a22824755b3168173d0c9cefc141325db77194f6efa54dd25b7d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:14.578363Z","signature_b64":"UVEg7yYAAs48XXKN/UdIi/HcsuB9WPsV1V/DAJlwtKSw7HFHQAFUKvJP6aaqeRR7pZJ59GqigICqiSAnAFu2Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d6877f922ee5a22824755b3168173d0c9cefc141325db77194f6efa54dd25b7d","last_reissued_at":"2026-05-17T23:42:14.577553Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:14.577553Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1409.5772","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":"2026-05-17T23:42:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S9GcR8Iv17HrGHn3EamrC4bCAVDjvP0BWwAmjqMnT5avN+UunxR0DKZZawaTYppwaGW4iiiZJl4YTJY7SuB3Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T18:16:52.773200Z"},"content_sha256":"b3b120f316ede13e124b0fab29e093af3b0e50b2f6cd78f9257114a5c9a9d698","schema_version":"1.0","event_id":"sha256:b3b120f316ede13e124b0fab29e093af3b0e50b2f6cd78f9257114a5c9a9d698"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:22DX7ERO4WRCQJDVLMYWQFZ5BS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Swiss Cheese Theorem for Linear Operators with Two Invariant Subspaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Audrey Moore, Markus Schmidmeier","submitted_at":"2014-09-19T19:23:21Z","abstract_excerpt":"We study systems $(V,T,U_1,U_2)$ consisting of a finite dimensional vector space $V$, a nilpotent $k$-linear operator $T:V\\to V$ and two $T$-invariant subspaces $U_1\\subset U_2\\subset V$. Let $\\mathcal S(n)$ be the category of such systems where the operator $T$ acts with nilpotency index at most $n$. We determine the dimension types $(\\dim U_1, \\dim U_2/U_1, \\dim V/U_2)$ of indecomposable systems in $\\mathcal S(n)$ for $n\\leq 4$. It turns out that in the case where $n=4$ there are infinitely many such triples $(x,y,z)$, they all lie in the cylinder given by $|x-y|,|y-z|,|z-x|\\leq 4$. But not "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.5772","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":""},"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":"2026-05-17T23:42:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cbLSSOUiq+IdBYTGNDw5WpCBqiuGup9HCg+Kk9wSb01Ytp+0lLKr7NxyV7IdB+Z7kUbzgTh1+IIHlWyLoBIaDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T18:16:52.773883Z"},"content_sha256":"f054cbde2167c6d452c3433b9a7a999ca8c356f9ccb6fb617b8fd4b47b0cf9ad","schema_version":"1.0","event_id":"sha256:f054cbde2167c6d452c3433b9a7a999ca8c356f9ccb6fb617b8fd4b47b0cf9ad"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/22DX7ERO4WRCQJDVLMYWQFZ5BS/bundle.json","state_url":"https://pith.science/pith/22DX7ERO4WRCQJDVLMYWQFZ5BS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/22DX7ERO4WRCQJDVLMYWQFZ5BS/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":"2026-06-07T18:16:52Z","links":{"resolver":"https://pith.science/pith/22DX7ERO4WRCQJDVLMYWQFZ5BS","bundle":"https://pith.science/pith/22DX7ERO4WRCQJDVLMYWQFZ5BS/bundle.json","state":"https://pith.science/pith/22DX7ERO4WRCQJDVLMYWQFZ5BS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/22DX7ERO4WRCQJDVLMYWQFZ5BS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:22DX7ERO4WRCQJDVLMYWQFZ5BS","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":"a05fdd3719339e5616ec681551929058aa035099ccd47b70e3929f70833a600f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2014-09-19T19:23:21Z","title_canon_sha256":"35c0982def2ed1772722136a615bf843b0d08c7cd54262a769bebcb2f3da29f5"},"schema_version":"1.0","source":{"id":"1409.5772","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.5772","created_at":"2026-05-17T23:42:14Z"},{"alias_kind":"arxiv_version","alias_value":"1409.5772v1","created_at":"2026-05-17T23:42:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.5772","created_at":"2026-05-17T23:42:14Z"},{"alias_kind":"pith_short_12","alias_value":"22DX7ERO4WRC","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"22DX7ERO4WRCQJDV","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"22DX7ERO","created_at":"2026-05-18T12:28:09Z"}],"graph_snapshots":[{"event_id":"sha256:f054cbde2167c6d452c3433b9a7a999ca8c356f9ccb6fb617b8fd4b47b0cf9ad","target":"graph","created_at":"2026-05-17T23:42:14Z","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 study systems $(V,T,U_1,U_2)$ consisting of a finite dimensional vector space $V$, a nilpotent $k$-linear operator $T:V\\to V$ and two $T$-invariant subspaces $U_1\\subset U_2\\subset V$. Let $\\mathcal S(n)$ be the category of such systems where the operator $T$ acts with nilpotency index at most $n$. We determine the dimension types $(\\dim U_1, \\dim U_2/U_1, \\dim V/U_2)$ of indecomposable systems in $\\mathcal S(n)$ for $n\\leq 4$. It turns out that in the case where $n=4$ there are infinitely many such triples $(x,y,z)$, they all lie in the cylinder given by $|x-y|,|y-z|,|z-x|\\leq 4$. But not ","authors_text":"Audrey Moore, Markus Schmidmeier","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2014-09-19T19:23:21Z","title":"The Swiss Cheese Theorem for Linear Operators with Two Invariant Subspaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.5772","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:b3b120f316ede13e124b0fab29e093af3b0e50b2f6cd78f9257114a5c9a9d698","target":"record","created_at":"2026-05-17T23:42:14Z","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":"a05fdd3719339e5616ec681551929058aa035099ccd47b70e3929f70833a600f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2014-09-19T19:23:21Z","title_canon_sha256":"35c0982def2ed1772722136a615bf843b0d08c7cd54262a769bebcb2f3da29f5"},"schema_version":"1.0","source":{"id":"1409.5772","kind":"arxiv","version":1}},"canonical_sha256":"d6877f922ee5a22824755b3168173d0c9cefc141325db77194f6efa54dd25b7d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d6877f922ee5a22824755b3168173d0c9cefc141325db77194f6efa54dd25b7d","first_computed_at":"2026-05-17T23:42:14.577553Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:42:14.577553Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UVEg7yYAAs48XXKN/UdIi/HcsuB9WPsV1V/DAJlwtKSw7HFHQAFUKvJP6aaqeRR7pZJ59GqigICqiSAnAFu2Aw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:42:14.578363Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.5772","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b3b120f316ede13e124b0fab29e093af3b0e50b2f6cd78f9257114a5c9a9d698","sha256:f054cbde2167c6d452c3433b9a7a999ca8c356f9ccb6fb617b8fd4b47b0cf9ad"],"state_sha256":"47363bb698245b8e3c820ada7f111813482ab1514faa268946a93212cb52a954"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jAscWkvtD05T9yPmsJbdGqCXfd2RCclyJTfI3y5oVFYfz6FfWHY3233GFDh7PA+toFcsqaQ3yOD4t/dO92n/Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T18:16:52.777952Z","bundle_sha256":"9b1c6c88ddcde819a9b672b2d6e946955ffca9f71a36a618c418ac2aa8588c50"}}