waring_problem_matrices

⚠️ 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.

Waring's problem for matrices: Every n×n matrix over ℝ is a sum of k-th powers of matrices. The minimum number of summands (the matrix Waring number) depends on n and k. Related to polynomial maps and power sum representations.

Why this node was retired

The posted statement is

import Mathlib

theorem waring_problem_matrices (n k : ℕ) (hn : 1 ≤ n) (hk : 2 ≤ k) :
    ∃ s : ℕ, ∀ M : Matrix (Fin n) (Fin n) ℝ,
      ∃ (As : Fin s → Matrix (Fin n) (Fin n) ℝ),
        M = ∑ i, (As i) ^ k := by
  sorry

The target includes 1×1 real matrices with even powers and arbitrary signs.

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

Proposed corrected statement

A safe substantive repair is the odd-power version: for n≥1 and odd k≥3, seek s such that every real n×n matrix is a sum of s kth powers. If retaining even k, specify the intended matrix-dimension and field hypotheses from the matrix-Waring source; this audit has not validated a general n≥2 repair.

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.