gauss_circle_problem

⚠️ Retired — specification defect

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

Gauss circle problem: How many integer lattice points lie inside a circle of radius r? The error term E(r) = #{(m,n) ∈ ℤ² : m²+n² ≤ r²} - πr² satisfies |E(r)| = O(r^{1/2+ε}) for all ε > 0 (Gauss 1801 gave O(r)). The conjectured exact bound is O(r^{1/4+ε}); the best known is O(r^{131/208}) (Huxley 2003).

Why this node was retired

The posted statement is

import Mathlib

theorem gauss_circle_problem :
    ∀ eps : ℝ, 0 < eps →
    ∃ C : ℝ, 0 < C ∧
    ∀ r : ℝ, 0 < r →
      |{p : ℤ × ℤ | (p.1 : ℝ)^2 + (p.2 : ℝ)^2 ≤ r^2}.ncard - Real.pi * r^2| ≤
      C * r ^ (1/2 + eps) := by
  sorry

The bound is required for every positive radius, but the origin contributes one lattice point while Cr^(1/2+ε) tends to zero.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Require r≥1 (or sufficiently large r) in the Gauss-circle error estimate |N(r)−πr²|≤Cε r^(1/2+ε). The asymptotic conjecture remains unproved here.

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.