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A Lean formalization of Roy's strong six exponentials theorem

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by Shawn Krisman

Version 2 / Oct 18, 2026 / CC BY 4.0

Abstract

We formalize in Lean 4 with Mathlib Roy's strong six exponentials theorem: if x1,x2x_1,x_2 are linearly independent over Q‾\overline{\mathbb{Q}} and y1,y2,y3y_1,y_2,y_3 are linearly independent over Q‾\overline{\mathbb{Q}}, then one of the six products xiyjx_iy_j lies outside the Q‾\overline{\mathbb{Q}}-vector space spanned by 11 and the logarithms of algebraic numbers. The proof closure uses only Lean's three standard axioms. Along the way the development proves Waldschmidt's transcendence theorem on Gad0×Gmd1\mathbb{G}_a^{d_0}\times\mathbb{G}_m^{d_1} in the form used by Roy, and Philippon's multiplicity estimate with derivatives on Gad0×Gmd1\mathbb{G}_a^{d_0}\times\mathbb{G}_m^{d_1} over every algebraically closed field of characteristic zero, following Roy's account in Waldschmidt's book. We reproduce the public Lean statements, summarize the mathematical route, and record how the artifact is verified. The code is available at https://github.com/telaviv/strong-six-exponentials.

Provenance statement

This was done through a single session with claude code.

Tools used

Anthropic
Claude OpusVersion 5.5

References

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  1. The mathlib Community. The Lean Mathematical Library. Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP 2020), pp. 367–381. 2020
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Version history

  1. v1 · 9 pagesSubmitted by Shawn KrismanInitial depositSupersededOct 18, 2026
  2. v2 · 9 pagesSubmitted by Shawn KrismanSubmitted as a new versionCurrentNov 06, 2026