optimal_packing_circles

⚠️ 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.

Circle packing in a circle: Pack n non-overlapping unit circles inside a larger circle of minimum radius. Exact optimal configurations known for n ≤ 11; general n is open.

Why this node was retired

The posted statement is

import Mathlib

theorem optimal_packing_circles (n : ℕ) (hn : 2 ≤ n) :
    ∃ (centers : Fin n → ℝ × ℝ) (r : ℝ), 0 < r ∧
      (∀ i j : Fin n, i ≠ j → dist (centers i) (centers j) ≥ 2 * r) ∧
      ∀ i : Fin n, (centers i).1 ^ 2 + (centers i).2 ^ 2 ≤ 1 ∧
      r = (1 / (1 + Real.sqrt 2)) := by
  sorry

The statement uses the sup metric on ℝ×ℝ and an asserted fixed radius; at ten centers a 3×3 grid pigeonhole contradicts the required separation.

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

Proposed corrected statement

Use Euclidean ℝ² and define the optimal radius as a supremum over configurations satisfying Euclidean separation and containment of the entire disks in the container. Do not assert the posted all-n closed radius formula without a separate correct theorem.

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.