Gödel Mirror: A Formal System For Contradiction-Driven Recursion
Jhet Chan
cs.LO
Sep 16, 2025 · v1
TL;DR
Defines and mechanizes the Gödel Mirror, a contradiction-driven recursion calculus, in Lean 4 with machine-checked metatheory.
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 -> Encapsulate -> Reenter -> 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.
Problem
Standard formal systems either guarantee termination (ruling out productive loops) or lack mechanisms to handle contradiction constructively. There is no framework that treats self-referential paradox as a deterministic control signal for structural evolution.
Approach
The Godel Mirror is a formal system defined in Lean 4 whose operational semantics encode symbolic paradoxes as deterministic transitions. The system uses a controlled non-terminating loop: Paradox -> Encapsulate -> Reenter -> Node. The Lean 4 mechanization proves that self-referential paradoxes are deterministically encapsulated and resolved into new structures without logical explosion.
Results
The Lean 4 formalization demonstrates that the system yields a paraconsistent inference loop where contradictions are metabolized into structure. The authors prove deterministic encapsulation of paradoxes and absence of logical explosion within the defined calculus.
