{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:NXPF7FXWOZU3PD35ZUEAFLXKZU","short_pith_number":"pith:NXPF7FXW","canonical_record":{"source":{"id":"0906.0278","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-06-01T13:18:07Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"38014a693cd9b5580d872415e4554a56c16768b00dc231b3cabafc6df079b29d","abstract_canon_sha256":"1a602a1078a2fb91667a0a13563c590b4d11ec89b46b9780ab6b90b9604c04c9"},"schema_version":"1.0"},"canonical_sha256":"6dde5f96f67669b78f7dcd0802aeeacd2fc56772d5512465034d10101679644c","source":{"kind":"arxiv","id":"0906.0278","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.0278","created_at":"2026-05-18T02:13:30Z"},{"alias_kind":"arxiv_version","alias_value":"0906.0278v1","created_at":"2026-05-18T02:13:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.0278","created_at":"2026-05-18T02:13:30Z"},{"alias_kind":"pith_short_12","alias_value":"NXPF7FXWOZU3","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"NXPF7FXWOZU3PD35","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"NXPF7FXW","created_at":"2026-05-18T12:26:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:NXPF7FXWOZU3PD35ZUEAFLXKZU","target":"record","payload":{"canonical_record":{"source":{"id":"0906.0278","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-06-01T13:18:07Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"38014a693cd9b5580d872415e4554a56c16768b00dc231b3cabafc6df079b29d","abstract_canon_sha256":"1a602a1078a2fb91667a0a13563c590b4d11ec89b46b9780ab6b90b9604c04c9"},"schema_version":"1.0"},"canonical_sha256":"6dde5f96f67669b78f7dcd0802aeeacd2fc56772d5512465034d10101679644c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:13:30.119305Z","signature_b64":"9a+8GB1ANGidFrrM80QxkTIbBcWa/mdW9doZ5M+JmOdZSdQEW/TnrLmIH+q0pZL8r9qjtSw7tkhz/POW9DqeCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6dde5f96f67669b78f7dcd0802aeeacd2fc56772d5512465034d10101679644c","last_reissued_at":"2026-05-18T02:13:30.118699Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:13:30.118699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0906.0278","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":"2026-05-18T02:13:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JG7OVheHVqSDp3Y23Dat8W7NL3vtaPeCYqFbTyZXlgcV5lZb4PqUmossIlbuLNzyQHq5VMsPZ6KKAYWvNpsVBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T07:34:36.510363Z"},"content_sha256":"6ae2dc16d71ea5e78695fe551317d7c9af99cf379416b40a9c23d980eaefb583","schema_version":"1.0","event_id":"sha256:6ae2dc16d71ea5e78695fe551317d7c9af99cf379416b40a9c23d980eaefb583"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:NXPF7FXWOZU3PD35ZUEAFLXKZU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Algebraic Shape Invariant Potentials as the Generalized Deformed Oscillator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Wang-Chang Su","submitted_at":"2009-06-01T13:18:07Z","abstract_excerpt":"Within the framework of supersymmetric quantum mechanics, we study the simplified version of potential algebra of shape invariance condition in k steps, where k is an arbitrary positive integer. The associated potential algebra is found to be equivalent to the generalized deformed oscillator algebra that has a built-in Z_k-grading structure. The algebraic realization of shape invariance condition in k steps is therefore formulated by the method of Z_k-graded deformed oscillator. Based on this formulation, we explicitly construct the general algebraic properties for shape invariant potentials i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.0278","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":"2026-05-18T02:13:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nidwjkS569orx8exXrS5UqQ6X6NlNJktQ1YqknDN4mW0+DU82dyKk8y++d5IKg8gSDT/dg5JoXgo3EBHj67zCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T07:34:36.510718Z"},"content_sha256":"f967a3df28a1f8d195c1b2c1bd7d88c28f192c4496a772704e5d46837b4f12f0","schema_version":"1.0","event_id":"sha256:f967a3df28a1f8d195c1b2c1bd7d88c28f192c4496a772704e5d46837b4f12f0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NXPF7FXWOZU3PD35ZUEAFLXKZU/bundle.json","state_url":"https://pith.science/pith/NXPF7FXWOZU3PD35ZUEAFLXKZU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NXPF7FXWOZU3PD35ZUEAFLXKZU/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":"2026-05-28T07:34:36Z","links":{"resolver":"https://pith.science/pith/NXPF7FXWOZU3PD35ZUEAFLXKZU","bundle":"https://pith.science/pith/NXPF7FXWOZU3PD35ZUEAFLXKZU/bundle.json","state":"https://pith.science/pith/NXPF7FXWOZU3PD35ZUEAFLXKZU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NXPF7FXWOZU3PD35ZUEAFLXKZU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:NXPF7FXWOZU3PD35ZUEAFLXKZU","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":"1a602a1078a2fb91667a0a13563c590b4d11ec89b46b9780ab6b90b9604c04c9","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-06-01T13:18:07Z","title_canon_sha256":"38014a693cd9b5580d872415e4554a56c16768b00dc231b3cabafc6df079b29d"},"schema_version":"1.0","source":{"id":"0906.0278","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.0278","created_at":"2026-05-18T02:13:30Z"},{"alias_kind":"arxiv_version","alias_value":"0906.0278v1","created_at":"2026-05-18T02:13:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.0278","created_at":"2026-05-18T02:13:30Z"},{"alias_kind":"pith_short_12","alias_value":"NXPF7FXWOZU3","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"NXPF7FXWOZU3PD35","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"NXPF7FXW","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:f967a3df28a1f8d195c1b2c1bd7d88c28f192c4496a772704e5d46837b4f12f0","target":"graph","created_at":"2026-05-18T02:13:30Z","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":"Within the framework of supersymmetric quantum mechanics, we study the simplified version of potential algebra of shape invariance condition in k steps, where k is an arbitrary positive integer. The associated potential algebra is found to be equivalent to the generalized deformed oscillator algebra that has a built-in Z_k-grading structure. The algebraic realization of shape invariance condition in k steps is therefore formulated by the method of Z_k-graded deformed oscillator. Based on this formulation, we explicitly construct the general algebraic properties for shape invariant potentials i","authors_text":"Wang-Chang Su","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-06-01T13:18:07Z","title":"Algebraic Shape Invariant Potentials as the Generalized Deformed Oscillator"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.0278","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":"sha256:6ae2dc16d71ea5e78695fe551317d7c9af99cf379416b40a9c23d980eaefb583","target":"record","created_at":"2026-05-18T02:13:30Z","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":"1a602a1078a2fb91667a0a13563c590b4d11ec89b46b9780ab6b90b9604c04c9","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-06-01T13:18:07Z","title_canon_sha256":"38014a693cd9b5580d872415e4554a56c16768b00dc231b3cabafc6df079b29d"},"schema_version":"1.0","source":{"id":"0906.0278","kind":"arxiv","version":1}},"canonical_sha256":"6dde5f96f67669b78f7dcd0802aeeacd2fc56772d5512465034d10101679644c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6dde5f96f67669b78f7dcd0802aeeacd2fc56772d5512465034d10101679644c","first_computed_at":"2026-05-18T02:13:30.118699Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:13:30.118699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9a+8GB1ANGidFrrM80QxkTIbBcWa/mdW9doZ5M+JmOdZSdQEW/TnrLmIH+q0pZL8r9qjtSw7tkhz/POW9DqeCw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:13:30.119305Z","signed_message":"canonical_sha256_bytes"},"source_id":"0906.0278","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6ae2dc16d71ea5e78695fe551317d7c9af99cf379416b40a9c23d980eaefb583","sha256:f967a3df28a1f8d195c1b2c1bd7d88c28f192c4496a772704e5d46837b4f12f0"],"state_sha256":"22e6d0760753e8cf7aeb7b8015d4fba288c0910642830ac396919e97159aca9b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SKXhFRbC63k7SJ12aVTRM7wna1P09Vk4zXAfT+GSWi5yVOiYkGlOzzd+i1urXzPvThEQMCQ1VkMbcp4Io90cCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T07:34:36.512799Z","bundle_sha256":"69b7d318ca706ac84b19701cfaec72dd090c28a2a84067711d764cdf6d32ef22"}}