selberg_eigenvalue_conjecture

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Proved status carries no information about that problem. Do not import this node or use it as a dependency.

Selberg's eigenvalue conjecture (1965): For any congruence subgroup Γ of SL(2,ℤ), the non-zero eigenvalues of the Laplace–Beltrami operator on L²(Γ\ℍ) are all ≥ 1/4. Known for Γ = SL(2,ℤ) itself; best known bound is 975/4096 (Kim–Sarnak 2003) instead of 1/4 for general congruence subgroups.

Why this node was retired

The posted statement is

import Mathlib

theorem selberg_eigenvalue_conjecture :
    ∀ (Gamma : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ))
      [Gamma.Normal] (hfin : Gamma.index ≠ 0),
    ∀ (lambda : ℝ),
      (∃ (f : (Matrix.SpecialLinearGroup (Fin 2) ℤ) → ℂ),
        (∀ g gamma, f (gamma * g) = f g) ∧
        f ≠ 0 ∧
        (∀ g : Matrix.SpecialLinearGroup (Fin 2) ℤ,
          (lambda : ℂ) * f g = f g)) →
      (1 : ℝ) / 4 ≤ lambda := by
  sorry

The alleged eigenfunction equation is λf=f, not a Laplace eigenvalue equation; nonzero f forces λ=1.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

Define the hyperbolic Laplacian on the appropriate congruence quotient, take a nonzero square-integrable Laplace eigenfunction with positive eigenvalue λ, and ask λ≥1/4. Include the quotient, domain and cuspidal/spectral conditions.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.