{"paper":{"title":"The Topological Dual of a Dataset: A Logic-to-Topology Encoding for AlphaGeometry-Style Data","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A logic-to-topology encoder transforms datasets into topological duals to reveal structural invariants preserved in neural latent spaces under input changes.","cross_cats":["cs.LO"],"primary_cat":"cs.AI","authors_text":"Anthony Bordg","submitted_at":"2026-04-20T10:18:08Z","abstract_excerpt":"AlphaGeometry represents a milestone in neuro-symbolic reasoning, yet its architecture faces a log-linear scaling bottleneck within its symbolic deduction engine that limits its efficiency as problem complexity increases. Recent technical reports suggest that current domain-specific languages may be isomorphic as input representations to natural language, interchanging them acts as a performance-invariant transformation, implying that current neural guidance relies on superficial encodings rather than structural understanding. This paper addresses this representation bottleneck by proposing a "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"By leveraging the Logic of Observation, we utilize the duality between provability in observable theories and topologies to propose a logic-to-topology encoder for the input space. We introduce the concept of the 'topological dual of a dataset', a transformation that bridges formal logic, topology, and neural processing.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the duality between provability in observable theories and topologies can be turned into a practical encoder that actually reveals structural invariants in a neural model's latent space under input transformations, rather than remaining a high-level analogy.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The topological dual of a dataset is introduced as a transformation that encodes logical structures into topological ones to expose invariants in neural latent spaces for AlphaGeometry-style reasoning.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A logic-to-topology encoder transforms datasets into topological duals to reveal structural invariants preserved in neural latent spaces under input changes.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"249e553ee1f7da463c1374e15a28c048b0e3dbf660b841c289d947efd5c982bc"},"source":{"id":"2604.18050","kind":"arxiv","version":2},"verdict":{"id":"51330bfa-7779-4c8e-b100-e870d3be8639","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T04:55:09.001658Z","strongest_claim":"By leveraging the Logic of Observation, we utilize the duality between provability in observable theories and topologies to propose a logic-to-topology encoder for the input space. We introduce the concept of the 'topological dual of a dataset', a transformation that bridges formal logic, topology, and neural processing.","one_line_summary":"The topological dual of a dataset is introduced as a transformation that encodes logical structures into topological ones to expose invariants in neural latent spaces for AlphaGeometry-style reasoning.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the duality between provability in observable theories and topologies can be turned into a practical encoder that actually reveals structural invariants in a neural model's latent space under input transformations, rather than remaining a high-level analogy.","pith_extraction_headline":"A logic-to-topology encoder transforms datasets into topological duals to reveal structural invariants preserved in neural latent spaces under input changes."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.18050/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-20T04:30:49.516529Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"0176650613012b612562fbab020f780e3dc12c3583b990a839610ed87ffcf8b2"},"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"}