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February 2, 20260 citationsOpen Access

1=0 Lean 4 Formalization of the Fundamental Arithmetic Contradiction

Key Points

To demonstrate the collapse of the associative property in mixed number arithmetic using Lean 4.
Developed a Lean 4 formalization to analyze mixed number arithmetic.
Evaluated the expression 1(1/3)6 under both left and right association.
Used Lean 4's type checker to verify computational steps.
Found that 1(1/3)6 evaluates to different values under left (8) and right (22/3) association.
Derived a contradiction as both evaluations should equal each other, leading to 8 = 22/3.
Concluded with the false statement 1 = 0 as a result of the contradiction.

Abstract

We present a complete Lean 4 formalization demonstrating the collapse of the associative property in mixed number arithmetic. The expression 1(1/3)6 evaluates to 8 under left-association and 22/3 under right-association. Since associativity requires both values to be equal, we derive 8 = 22/3, leading to ⊥ (False) and consequently 1 = 0. All computational steps are verified by Lean 4's type checker.

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Kaoru Aguilera Katayama (Thu,) studied this question.

synapsesocial.com/papers/6980fd9dc1c9540dea80f602 https://doi.org/https://doi.org/10.5281/zenodo.18417972

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