lean_workbook_plus_81097

⚠️ 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.

Assuming $\ (x-\frac{1}{3})(y-\frac{1}{3}) \geq 0$ $\ \Leftrightarrow xy+\frac{1}{9}\geq \frac{1}{3}\left ( x+y \right )\Leftrightarrow 2xy\geq \frac{2}{3}\left ( \frac{2}{3}-z \right )\left ( \star \right )$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_81097 : ∀ x y z : ℝ, (x - 1 / 3) * (y - 1 / 3) ≥ 0 → 2 * x * y ≥ 2 / 3 * ((2 / 3 - z) * (x + y))   :=  by sorry

A relation involving z from the source reduction was omitted, and an extra factor(x+y) was inserted on the RHS.

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

Proposed corrected statement

Recover x+y+z=1 if that is the intended normalization, then derive2xy≥(2/3)(2/3−z) from the stated product condition; do not multiply again by x+y.

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.