{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:ILX43ZKEHTQOYKWZE6EOULRIHI","short_pith_number":"pith:ILX43ZKE","canonical_record":{"source":{"id":"1608.01656","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-08-04T19:44:34Z","cross_cats_sorted":[],"title_canon_sha256":"98b2ed3041d51d50a3bcb73de08602d3fef0c6a089d5f26aff134dd53f104957","abstract_canon_sha256":"76f2f7273d83d6c7eeed70e702782555f55a8cb0a09ca52b2f304f16ba525e11"},"schema_version":"1.0"},"canonical_sha256":"42efcde5443ce0ec2ad92788ea2e283a0269fb17de9c5e75e428e9b872d8f62c","source":{"kind":"arxiv","id":"1608.01656","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.01656","created_at":"2026-05-18T01:09:51Z"},{"alias_kind":"arxiv_version","alias_value":"1608.01656v1","created_at":"2026-05-18T01:09:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.01656","created_at":"2026-05-18T01:09:51Z"},{"alias_kind":"pith_short_12","alias_value":"ILX43ZKEHTQO","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"ILX43ZKEHTQOYKWZ","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"ILX43ZKE","created_at":"2026-05-18T12:30:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:ILX43ZKEHTQOYKWZE6EOULRIHI","target":"record","payload":{"canonical_record":{"source":{"id":"1608.01656","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-08-04T19:44:34Z","cross_cats_sorted":[],"title_canon_sha256":"98b2ed3041d51d50a3bcb73de08602d3fef0c6a089d5f26aff134dd53f104957","abstract_canon_sha256":"76f2f7273d83d6c7eeed70e702782555f55a8cb0a09ca52b2f304f16ba525e11"},"schema_version":"1.0"},"canonical_sha256":"42efcde5443ce0ec2ad92788ea2e283a0269fb17de9c5e75e428e9b872d8f62c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:51.866822Z","signature_b64":"DkVuv7niOoBbqbncF3e5bw28133CWak7GGDUhmOztW4axuciAd5qZY+y+ONxMCSp/+uGrON089FllDjh1xN/DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"42efcde5443ce0ec2ad92788ea2e283a0269fb17de9c5e75e428e9b872d8f62c","last_reissued_at":"2026-05-18T01:09:51.866230Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:51.866230Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1608.01656","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":"2026-05-18T01:09:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Axe+w+Sv6qEgc/d70AagmX5yLUq+QwnDAVdIlyku+jIBYju2UhshuQr8wTkl37e+gRRiboP39WMIsL8ssUYhDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T20:22:36.262330Z"},"content_sha256":"ca0269511ff4fb19db92d3f42c5018abe7d2c27d28dcb9ab471985963ddda5f7","schema_version":"1.0","event_id":"sha256:ca0269511ff4fb19db92d3f42c5018abe7d2c27d28dcb9ab471985963ddda5f7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:ILX43ZKEHTQOYKWZE6EOULRIHI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Classically Integral Quadratic Forms Excepting at Most Two Values","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andres Mejia, Frederick Saia, Katherine Thompson, Madeleine Barowsky, Nolan Schock, William Damron","submitted_at":"2016-08-04T19:44:34Z","abstract_excerpt":"Let $S \\subseteq \\mathbb{N}$ be finite. Is there a positive definite quadratic form that fails to represent only those elements in $S$? For $S = \\emptyset$, this was solved (for classically integral forms) by the $15$-Theorem of Conway-Schneeberger in the early 1990s and (for all integral forms) by the $290$-Theorem of Bhargava-Hanke in the mid-2000s. In 1938 Halmos attempted to list all weighted sums of four squares that failed to represent $S=\\{m\\}$; of his $88$ candidates, he could provide complete justifications for all but one. In the same spirit, we ask, \"for which $S = \\{m, n\\}$ does th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.01656","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":"2026-05-18T01:09:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bV8b08BiEELka6WLSZrX/H1OjX4NyOTX0N+iq0Rb/vEtB1LSbTgXe/2USpZ7B2NnZPGsTsYczo92PQ3SQNvaAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T20:22:36.262971Z"},"content_sha256":"58fafe41ff9983c571c6f8aa6b9248cec1d02def5d70168d7d5cf0a5a47beb5b","schema_version":"1.0","event_id":"sha256:58fafe41ff9983c571c6f8aa6b9248cec1d02def5d70168d7d5cf0a5a47beb5b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ILX43ZKEHTQOYKWZE6EOULRIHI/bundle.json","state_url":"https://pith.science/pith/ILX43ZKEHTQOYKWZE6EOULRIHI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ILX43ZKEHTQOYKWZE6EOULRIHI/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":"2026-05-23T20:22:36Z","links":{"resolver":"https://pith.science/pith/ILX43ZKEHTQOYKWZE6EOULRIHI","bundle":"https://pith.science/pith/ILX43ZKEHTQOYKWZE6EOULRIHI/bundle.json","state":"https://pith.science/pith/ILX43ZKEHTQOYKWZE6EOULRIHI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ILX43ZKEHTQOYKWZE6EOULRIHI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:ILX43ZKEHTQOYKWZE6EOULRIHI","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":"76f2f7273d83d6c7eeed70e702782555f55a8cb0a09ca52b2f304f16ba525e11","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-08-04T19:44:34Z","title_canon_sha256":"98b2ed3041d51d50a3bcb73de08602d3fef0c6a089d5f26aff134dd53f104957"},"schema_version":"1.0","source":{"id":"1608.01656","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.01656","created_at":"2026-05-18T01:09:51Z"},{"alias_kind":"arxiv_version","alias_value":"1608.01656v1","created_at":"2026-05-18T01:09:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.01656","created_at":"2026-05-18T01:09:51Z"},{"alias_kind":"pith_short_12","alias_value":"ILX43ZKEHTQO","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"ILX43ZKEHTQOYKWZ","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"ILX43ZKE","created_at":"2026-05-18T12:30:22Z"}],"graph_snapshots":[{"event_id":"sha256:58fafe41ff9983c571c6f8aa6b9248cec1d02def5d70168d7d5cf0a5a47beb5b","target":"graph","created_at":"2026-05-18T01:09:51Z","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":"Let $S \\subseteq \\mathbb{N}$ be finite. Is there a positive definite quadratic form that fails to represent only those elements in $S$? For $S = \\emptyset$, this was solved (for classically integral forms) by the $15$-Theorem of Conway-Schneeberger in the early 1990s and (for all integral forms) by the $290$-Theorem of Bhargava-Hanke in the mid-2000s. In 1938 Halmos attempted to list all weighted sums of four squares that failed to represent $S=\\{m\\}$; of his $88$ candidates, he could provide complete justifications for all but one. In the same spirit, we ask, \"for which $S = \\{m, n\\}$ does th","authors_text":"Andres Mejia, Frederick Saia, Katherine Thompson, Madeleine Barowsky, Nolan Schock, William Damron","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-08-04T19:44:34Z","title":"Classically Integral Quadratic Forms Excepting at Most Two Values"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.01656","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:ca0269511ff4fb19db92d3f42c5018abe7d2c27d28dcb9ab471985963ddda5f7","target":"record","created_at":"2026-05-18T01:09:51Z","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":"76f2f7273d83d6c7eeed70e702782555f55a8cb0a09ca52b2f304f16ba525e11","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-08-04T19:44:34Z","title_canon_sha256":"98b2ed3041d51d50a3bcb73de08602d3fef0c6a089d5f26aff134dd53f104957"},"schema_version":"1.0","source":{"id":"1608.01656","kind":"arxiv","version":1}},"canonical_sha256":"42efcde5443ce0ec2ad92788ea2e283a0269fb17de9c5e75e428e9b872d8f62c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"42efcde5443ce0ec2ad92788ea2e283a0269fb17de9c5e75e428e9b872d8f62c","first_computed_at":"2026-05-18T01:09:51.866230Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:09:51.866230Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DkVuv7niOoBbqbncF3e5bw28133CWak7GGDUhmOztW4axuciAd5qZY+y+ONxMCSp/+uGrON089FllDjh1xN/DA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:09:51.866822Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.01656","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ca0269511ff4fb19db92d3f42c5018abe7d2c27d28dcb9ab471985963ddda5f7","sha256:58fafe41ff9983c571c6f8aa6b9248cec1d02def5d70168d7d5cf0a5a47beb5b"],"state_sha256":"f20681016bd4097f96350f6c9bd327201d41d11d113ffe07799e2cbed3e96fe4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wWgMeTUfzE0Qi83Sg14K+BMp7ij3G3jpXlARlcbfHenD0qmmYrzsM/O9FbrjRDIwKfJTRw7Fx8zUjIFG6S2mDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T20:22:36.265691Z","bundle_sha256":"f24a55b57d1d626a88394de74bf079062a984177ecf10b432ec135ef0ac618c0"}}