{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:TIB4MVMREAY4M3TO5NP4DCAPBN","short_pith_number":"pith:TIB4MVMR","canonical_record":{"source":{"id":"2605.29121","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-05-27T21:29:30Z","cross_cats_sorted":["cs.AI","cs.LG"],"title_canon_sha256":"fbaa843b49c1f72b0afd65a1eedcd3cfeef09f7492a4c7652249f612992004f2","abstract_canon_sha256":"88e057ea0474d4816d62602ff7c3019238e83a5d891bbde3c8435ba1ea22caf1"},"schema_version":"1.0"},"canonical_sha256":"9a03c655912031c66e6eeb5fc1880f0b52f92dbca6351d8ae267331036dea2a6","source":{"kind":"arxiv","id":"2605.29121","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.29121","created_at":"2026-05-29T01:05:19Z"},{"alias_kind":"arxiv_version","alias_value":"2605.29121v1","created_at":"2026-05-29T01:05:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.29121","created_at":"2026-05-29T01:05:19Z"},{"alias_kind":"pith_short_12","alias_value":"TIB4MVMREAY4","created_at":"2026-05-29T01:05:19Z"},{"alias_kind":"pith_short_16","alias_value":"TIB4MVMREAY4M3TO","created_at":"2026-05-29T01:05:19Z"},{"alias_kind":"pith_short_8","alias_value":"TIB4MVMR","created_at":"2026-05-29T01:05:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:TIB4MVMREAY4M3TO5NP4DCAPBN","target":"record","payload":{"canonical_record":{"source":{"id":"2605.29121","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-05-27T21:29:30Z","cross_cats_sorted":["cs.AI","cs.LG"],"title_canon_sha256":"fbaa843b49c1f72b0afd65a1eedcd3cfeef09f7492a4c7652249f612992004f2","abstract_canon_sha256":"88e057ea0474d4816d62602ff7c3019238e83a5d891bbde3c8435ba1ea22caf1"},"schema_version":"1.0"},"canonical_sha256":"9a03c655912031c66e6eeb5fc1880f0b52f92dbca6351d8ae267331036dea2a6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-29T01:05:19.318697Z","signature_b64":"aKejXH513/XttewubhLi/MrDKtc+V0WWpUa5RuOQ4z6gQLnb1G9Gg+HSW/C3+v6tPL5iFZsJaOHeQBS0B/vUDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9a03c655912031c66e6eeb5fc1880f0b52f92dbca6351d8ae267331036dea2a6","last_reissued_at":"2026-05-29T01:05:19.317913Z","signature_status":"signed_v1","first_computed_at":"2026-05-29T01:05:19.317913Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.29121","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":"2026-05-29T01:05:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PqZrtfRZJ1QMf34us7VUPWx32Vq329L4OizIOCnY6NSHi4lBwM1noEa6y2l7kdlBdozQjOelzDOuD4EGJps9BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T07:11:22.069917Z"},"content_sha256":"1ad4a0c9933a5be8ccda76c39195929677fca066c91ee9bcd86918a5c81acc36","schema_version":"1.0","event_id":"sha256:1ad4a0c9933a5be8ccda76c39195929677fca066c91ee9bcd86918a5c81acc36"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:TIB4MVMREAY4M3TO5NP4DCAPBN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Minimal Bifurcation Model of Load Imbalance in a Softmax Mixture-of-Experts Router","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.AI","cs.LG"],"primary_cat":"math.DS","authors_text":"Innopolis, O. M. Kiselev (Innopolis University, Russia)","submitted_at":"2026-05-27T21:29:30Z","abstract_excerpt":"We propose a minimal dynamical model of adaptive softmax routing for a two-expert Mixture-of-Experts (MoE) layer. The model is obtained as a mean-field limit of a discrete reinforcement rule: the selected expert receives a small score increment, while all scores undergo regularizing decay. In the symmetric case the limiting system has a supercritical pitchfork bifurcation: for weak feedback there is a unique stable balanced state, whereas above a critical feedback strength two stable asymmetric states appear. When an external asymmetry is added, the pitchfork unfolds into a pair of fold bifurc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.29121","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/2605.29121/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":"2026-05-29T01:05:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QQi8KXNFlBu1dCTfnhhjbvmC2AXAyzVKjS9RMVzXtaHXlCEOuvJIOQZBcYq+tgA+b4SZULS9I4xJqAIaQbauCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T07:11:22.070607Z"},"content_sha256":"e5f2392bfd9cfc0f0677e34f0bd305039f9b504ef8bed378c28b8b19ef0cbeb7","schema_version":"1.0","event_id":"sha256:e5f2392bfd9cfc0f0677e34f0bd305039f9b504ef8bed378c28b8b19ef0cbeb7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TIB4MVMREAY4M3TO5NP4DCAPBN/bundle.json","state_url":"https://pith.science/pith/TIB4MVMREAY4M3TO5NP4DCAPBN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TIB4MVMREAY4M3TO5NP4DCAPBN/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":"2026-06-11T07:11:22Z","links":{"resolver":"https://pith.science/pith/TIB4MVMREAY4M3TO5NP4DCAPBN","bundle":"https://pith.science/pith/TIB4MVMREAY4M3TO5NP4DCAPBN/bundle.json","state":"https://pith.science/pith/TIB4MVMREAY4M3TO5NP4DCAPBN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TIB4MVMREAY4M3TO5NP4DCAPBN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:TIB4MVMREAY4M3TO5NP4DCAPBN","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":"88e057ea0474d4816d62602ff7c3019238e83a5d891bbde3c8435ba1ea22caf1","cross_cats_sorted":["cs.AI","cs.LG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-05-27T21:29:30Z","title_canon_sha256":"fbaa843b49c1f72b0afd65a1eedcd3cfeef09f7492a4c7652249f612992004f2"},"schema_version":"1.0","source":{"id":"2605.29121","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.29121","created_at":"2026-05-29T01:05:19Z"},{"alias_kind":"arxiv_version","alias_value":"2605.29121v1","created_at":"2026-05-29T01:05:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.29121","created_at":"2026-05-29T01:05:19Z"},{"alias_kind":"pith_short_12","alias_value":"TIB4MVMREAY4","created_at":"2026-05-29T01:05:19Z"},{"alias_kind":"pith_short_16","alias_value":"TIB4MVMREAY4M3TO","created_at":"2026-05-29T01:05:19Z"},{"alias_kind":"pith_short_8","alias_value":"TIB4MVMR","created_at":"2026-05-29T01:05:19Z"}],"graph_snapshots":[{"event_id":"sha256:e5f2392bfd9cfc0f0677e34f0bd305039f9b504ef8bed378c28b8b19ef0cbeb7","target":"graph","created_at":"2026-05-29T01:05:19Z","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/2605.29121/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We propose a minimal dynamical model of adaptive softmax routing for a two-expert Mixture-of-Experts (MoE) layer. The model is obtained as a mean-field limit of a discrete reinforcement rule: the selected expert receives a small score increment, while all scores undergo regularizing decay. In the symmetric case the limiting system has a supercritical pitchfork bifurcation: for weak feedback there is a unique stable balanced state, whereas above a critical feedback strength two stable asymmetric states appear. When an external asymmetry is added, the pitchfork unfolds into a pair of fold bifurc","authors_text":"Innopolis, O. M. Kiselev (Innopolis University, Russia)","cross_cats":["cs.AI","cs.LG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-05-27T21:29:30Z","title":"A Minimal Bifurcation Model of Load Imbalance in a Softmax Mixture-of-Experts Router"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.29121","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":"sha256:1ad4a0c9933a5be8ccda76c39195929677fca066c91ee9bcd86918a5c81acc36","target":"record","created_at":"2026-05-29T01:05:19Z","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":"88e057ea0474d4816d62602ff7c3019238e83a5d891bbde3c8435ba1ea22caf1","cross_cats_sorted":["cs.AI","cs.LG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-05-27T21:29:30Z","title_canon_sha256":"fbaa843b49c1f72b0afd65a1eedcd3cfeef09f7492a4c7652249f612992004f2"},"schema_version":"1.0","source":{"id":"2605.29121","kind":"arxiv","version":1}},"canonical_sha256":"9a03c655912031c66e6eeb5fc1880f0b52f92dbca6351d8ae267331036dea2a6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9a03c655912031c66e6eeb5fc1880f0b52f92dbca6351d8ae267331036dea2a6","first_computed_at":"2026-05-29T01:05:19.317913Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-29T01:05:19.317913Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aKejXH513/XttewubhLi/MrDKtc+V0WWpUa5RuOQ4z6gQLnb1G9Gg+HSW/C3+v6tPL5iFZsJaOHeQBS0B/vUDA==","signature_status":"signed_v1","signed_at":"2026-05-29T01:05:19.318697Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.29121","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1ad4a0c9933a5be8ccda76c39195929677fca066c91ee9bcd86918a5c81acc36","sha256:e5f2392bfd9cfc0f0677e34f0bd305039f9b504ef8bed378c28b8b19ef0cbeb7"],"state_sha256":"f429ee1e06712d285d94c8760136abd02ac276ec0b66669c04e280feb6082f30"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8+p9902/6x3onsV3EJpyH5udTyYKXB2d5ULdGk882eh4VWBtk8YSYvdiBogpvySurxriKZLDw2g/zMfaB2oMAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T07:11:22.076202Z","bundle_sha256":"6c826410056a80ba52d5acef087916e08ba3e10b7781e3a5f9639b6ffd4aae48"}}