lean_workbook_plus_30108

⚠️ 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.

Prove that the function $T_4 : \mathbb{Z}_{107} \to \mathbb{Z}_{107}$ defined by $T_4(x)=x^4$ for every $x\in\mathbb{Z}_{107}$ is 2-to-1.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_30108 (T : ZMod 107 → ZMod 107) (hT: T = fun x ↦ x^4) :
  ∀ y : ZMod 107, ∃ x : ZMod 107, T x = y ∨ T x = y   :=  by sorry

Duplicating a disjunct expresses surjectivity, not two-to-one. The accepted note also overstates the source claim at zero: its image has54 elements, but the zero fiber is a singleton.

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

Proposed corrected statement

On the nonzero residues modulo107, state that x↦x⁴ has exactly two preimages for each value in its image, and no preimages elsewhere; zero has exactly one preimage on the full field.

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.