Exponentials.lean`

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Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 2 contain sorry, 0 declared as axioms

Exponentials conjectures and theorems

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

theorem Exponentials.four_exponentials_conjecture Four exponentials conjecture Let $x_0, x_1$ and $y_0, y_1$ be $\mathbb Q$-linearly independent pairs of complex numbers, then some $e^{x_i y_j}$ is transcendental sorry line 38 source
Source documentation
Four exponentials conjecture Let $x_0, x_1$ and $y_0, y_1$ be $\mathbb Q$-linearly independent pairs of complex numbers, then some $e^{x_i y_j}$ is transcendental.

depends on sorry what this means
Statement uses: Fin, OfNat.ofNat, Nat, Complex, LinearIndependent, Rat, Exists, Transcendental, Complex.exp, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
52 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem four_exponentials_conjecture (x : Fin 2 → ℂ) (y : Fin 2 → ℂ)
    (h1 : LinearIndependent ℚ x) (h2 : LinearIndependent ℚ y) :
    ∃ i j : Fin 2, Transcendental ℚ (exp (x i * y j)) := by
…
```
theorem Exponentials.two_pow_three_pow_transcendental The four exponential conjecture would imply that for any irrational number $t$, at least one of the numbers $2^t$ and $3^t$ is transcendental sorry line 48 source
Source documentation
The four exponential conjecture would imply that for any irrational number $t$, at least one of the numbers $2^t$ and $3^t$ is transcendental.

depends on sorry what this means
Statement uses: Real, Irrational, Or, Transcendental, Rat, HPow.hPow, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem two_pow_three_pow_transcendental (t : ℝ) (h : Irrational t) :
    Transcendental ℚ (2 ^ t : ℝ) ∨ Transcendental ℚ (3 ^ t : ℝ) := by
…
```