{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:OB6QXQEW7RJ3UYOR5MRV5KP6JZ","short_pith_number":"pith:OB6QXQEW","canonical_record":{"source":{"id":"2607.17823","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-20T11:11:55Z","cross_cats_sorted":[],"title_canon_sha256":"68f92077a7114653fdc1188e9cd96252179e9009d99d6543ab4157a0de65f140","abstract_canon_sha256":"863fffa505966f87b7385c9586f149032093a8325b4e26011249a06c60f48c42"},"schema_version":"1.0"},"canonical_sha256":"707d0bc096fc53ba61d1eb235ea9fe4e7febbfa0ea194446b98a1455cd0d722f","source":{"kind":"arxiv","id":"2607.17823","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.17823","created_at":"2026-07-21T02:22:02Z"},{"alias_kind":"arxiv_version","alias_value":"2607.17823v1","created_at":"2026-07-21T02:22:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.17823","created_at":"2026-07-21T02:22:02Z"},{"alias_kind":"pith_short_12","alias_value":"OB6QXQEW7RJ3","created_at":"2026-07-21T02:22:02Z"},{"alias_kind":"pith_short_16","alias_value":"OB6QXQEW7RJ3UYOR","created_at":"2026-07-21T02:22:02Z"},{"alias_kind":"pith_short_8","alias_value":"OB6QXQEW","created_at":"2026-07-21T02:22:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:OB6QXQEW7RJ3UYOR5MRV5KP6JZ","target":"record","payload":{"canonical_record":{"source":{"id":"2607.17823","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-20T11:11:55Z","cross_cats_sorted":[],"title_canon_sha256":"68f92077a7114653fdc1188e9cd96252179e9009d99d6543ab4157a0de65f140","abstract_canon_sha256":"863fffa505966f87b7385c9586f149032093a8325b4e26011249a06c60f48c42"},"schema_version":"1.0"},"canonical_sha256":"707d0bc096fc53ba61d1eb235ea9fe4e7febbfa0ea194446b98a1455cd0d722f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-21T02:22:02.099088Z","signature_b64":"16qavS1naDAcfqLG4ueNCR90YUWxdfnQZFOviPDQIHUE5XmHyWtPAIuhcl64rhWmxMku+8pmeN/I1WJ6stSmBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"707d0bc096fc53ba61d1eb235ea9fe4e7febbfa0ea194446b98a1455cd0d722f","last_reissued_at":"2026-07-21T02:22:02.098277Z","signature_status":"signed_v1","first_computed_at":"2026-07-21T02:22:02.098277Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.17823","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":"2026-07-21T02:22:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"StoWVTbzlwz5CmVhNjkbuaoqG3Jw6CPi+sfaKUVSQg04rGNrFn4MnlHD358Hp4sxxIyKZlNCDRcVJvyxV/AKCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T08:48:49.446953Z"},"content_sha256":"e5ca8d002bfbfe5c9d1263ab4d3afc83667e4edc3aa690c2c5bd1c4117867df4","schema_version":"1.0","event_id":"sha256:e5ca8d002bfbfe5c9d1263ab4d3afc83667e4edc3aa690c2c5bd1c4117867df4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:OB6QXQEW7RJ3UYOR5MRV5KP6JZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Theoretical Foundations of $\\max$@$k$ Reinforcement Learning","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Andrea Celli, Martino Bernasconi, Riccardo Poiani","submitted_at":"2026-07-20T11:11:55Z","abstract_excerpt":"Reinforcement Learning is a cornerstone technique for modern large reasoning models. Usually, for difficult tasks such as code generation and theorem proving, the agent is evaluated by generating $K$ responses rather than sampling a single response, and performance is then measured using a retry-aware metric such as $\\max$@$k$. Despite their practical importance, the theoretical foundations of learning under such criteria remain limited. In this work, we provide a theoretical study of the $\\max$@$k$ learning problem in finite-horizon reinforcement learning. We show that optimizing the $\\max$@$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.17823","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/2607.17823/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":"2026-07-21T02:22:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZFnDJlCNyVpPoaf/6WKX7fmG5TxZzpHFdMAq/Xkz/ITgDQByKwdBo48CkfEz1IJMpr8biC4RzZlOZajYcDCXDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T08:48:49.447851Z"},"content_sha256":"c93306a6caa267b18e687a7cb4b3fc48e4c2ba8fe1b7e74e1a90043b4ed615b9","schema_version":"1.0","event_id":"sha256:c93306a6caa267b18e687a7cb4b3fc48e4c2ba8fe1b7e74e1a90043b4ed615b9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OB6QXQEW7RJ3UYOR5MRV5KP6JZ/bundle.json","state_url":"https://pith.science/pith/OB6QXQEW7RJ3UYOR5MRV5KP6JZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OB6QXQEW7RJ3UYOR5MRV5KP6JZ/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":"2026-08-16T08:48:49Z","links":{"resolver":"https://pith.science/pith/OB6QXQEW7RJ3UYOR5MRV5KP6JZ","bundle":"https://pith.science/pith/OB6QXQEW7RJ3UYOR5MRV5KP6JZ/bundle.json","state":"https://pith.science/pith/OB6QXQEW7RJ3UYOR5MRV5KP6JZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OB6QXQEW7RJ3UYOR5MRV5KP6JZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:OB6QXQEW7RJ3UYOR5MRV5KP6JZ","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":"863fffa505966f87b7385c9586f149032093a8325b4e26011249a06c60f48c42","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-20T11:11:55Z","title_canon_sha256":"68f92077a7114653fdc1188e9cd96252179e9009d99d6543ab4157a0de65f140"},"schema_version":"1.0","source":{"id":"2607.17823","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.17823","created_at":"2026-07-21T02:22:02Z"},{"alias_kind":"arxiv_version","alias_value":"2607.17823v1","created_at":"2026-07-21T02:22:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.17823","created_at":"2026-07-21T02:22:02Z"},{"alias_kind":"pith_short_12","alias_value":"OB6QXQEW7RJ3","created_at":"2026-07-21T02:22:02Z"},{"alias_kind":"pith_short_16","alias_value":"OB6QXQEW7RJ3UYOR","created_at":"2026-07-21T02:22:02Z"},{"alias_kind":"pith_short_8","alias_value":"OB6QXQEW","created_at":"2026-07-21T02:22:02Z"}],"graph_snapshots":[{"event_id":"sha256:c93306a6caa267b18e687a7cb4b3fc48e4c2ba8fe1b7e74e1a90043b4ed615b9","target":"graph","created_at":"2026-07-21T02:22:02Z","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/2607.17823/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Reinforcement Learning is a cornerstone technique for modern large reasoning models. Usually, for difficult tasks such as code generation and theorem proving, the agent is evaluated by generating $K$ responses rather than sampling a single response, and performance is then measured using a retry-aware metric such as $\\max$@$k$. Despite their practical importance, the theoretical foundations of learning under such criteria remain limited. In this work, we provide a theoretical study of the $\\max$@$k$ learning problem in finite-horizon reinforcement learning. We show that optimizing the $\\max$@$","authors_text":"Andrea Celli, Martino Bernasconi, Riccardo Poiani","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-20T11:11:55Z","title":"Theoretical Foundations of $\\max$@$k$ Reinforcement Learning"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.17823","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:e5ca8d002bfbfe5c9d1263ab4d3afc83667e4edc3aa690c2c5bd1c4117867df4","target":"record","created_at":"2026-07-21T02:22:02Z","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":"863fffa505966f87b7385c9586f149032093a8325b4e26011249a06c60f48c42","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-20T11:11:55Z","title_canon_sha256":"68f92077a7114653fdc1188e9cd96252179e9009d99d6543ab4157a0de65f140"},"schema_version":"1.0","source":{"id":"2607.17823","kind":"arxiv","version":1}},"canonical_sha256":"707d0bc096fc53ba61d1eb235ea9fe4e7febbfa0ea194446b98a1455cd0d722f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"707d0bc096fc53ba61d1eb235ea9fe4e7febbfa0ea194446b98a1455cd0d722f","first_computed_at":"2026-07-21T02:22:02.098277Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-21T02:22:02.098277Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"16qavS1naDAcfqLG4ueNCR90YUWxdfnQZFOviPDQIHUE5XmHyWtPAIuhcl64rhWmxMku+8pmeN/I1WJ6stSmBw==","signature_status":"signed_v1","signed_at":"2026-07-21T02:22:02.099088Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.17823","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e5ca8d002bfbfe5c9d1263ab4d3afc83667e4edc3aa690c2c5bd1c4117867df4","sha256:c93306a6caa267b18e687a7cb4b3fc48e4c2ba8fe1b7e74e1a90043b4ed615b9"],"state_sha256":"6c2612094caa74b2217762e84a4bcb99b2b6379b4001bdb81d93aa7fc3cb9ce9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3CdIkGs8VqtjAtAII9XTqMW67KNoRSqn9q/6sruG0qT56LPXj0FthN47bCMidKyW0KyeDRggjKvGW3jcqxGXCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T08:48:49.453166Z","bundle_sha256":"c750380980cb9bed10c95307dc6cf6c21825a33d079f10345a360df55bec16de"}}