{"ID":2870562,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.16239","arxiv_id":"2509.16239","title":"Gödel Mirror: A Formal System For Contradiction-Driven Recursion","abstract":"We introduce the Gödel Mirror, a formal system defined in Lean 4 that treats contradiction as a control signal for recursive structural evolution. Inspired by Gödelian self-reference, our system's operational semantics encode symbolic paradoxes as deterministic transitions. Unlike systems designed to guarantee normalization, the Gödel Mirror is a minimal and verifiable architecture that leverages a controlled, non-terminating loop as a productive feature. Our Lean 4 mechanization proves that self-referential paradoxes are deterministically encapsulated and resolved into new structures without leading to logical explosion, yielding a paraconsistent inference loop: Paradox -\u003e Encapsulate -\u003e Reenter -\u003e Node We argue that this calculus opens a new class of symbolic systems in which contradiction is metabolized into structure, providing a formal basis for agents capable of resolving internal inconsistencies.","short_abstract":"We introduce the Gödel Mirror, a formal system defined in Lean 4 that treats contradiction as a control signal for recursive structural evolution. Inspired by Gödelian self-reference, our system's operational semantics encode symbolic paradoxes as deterministic transitions. Unlike systems designed to guarantee normaliz...","url_abs":"https://arxiv.org/abs/2509.16239","url_pdf":"https://arxiv.org/pdf/2509.16239v1","authors":"[\"Jhet Chan\"]","published":"2025-09-16T17:41:16Z","proceeding":"cs.LO","tasks":"[\"cs.LO\",\"cs.PL\"]","methods":"[]","has_code":false}