hasse_minkowski_extension

⚠️ 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.

Hasse–Minkowski theorem: A quadratic form over ℚ represents 0 nontrivially iff it does over ℝ and over all ℚₚ. This is the prototypical local-global principle. Extensions to higher-degree forms (Brauer-Manin obstruction) and varieties of general type remain active research.

Why this node was retired

The posted statement is

import Mathlib

theorem hasse_minkowski_extension (n : ℕ) (hn : 3 ≤ n)
    (f : Fin n → Fin n → ℤ) (hsym : ∀ i j, f i j = f j i) :
    (∀ p : ℕ, Nat.Prime p →
      ∃ x : Fin n → ZMod p, ∑ i, ∑ j, f i j * x i * x j = 0) →
    (∃ x : Fin n → ℚ, (∃ i, x i ≠ 0) ∧
      ∑ i, ∑ j, (f i j : ℚ) * x i * x j = 0) := by
  sorry

The local hypothesis uses arbitrary vectors mod p and allows zero, omitting nontrivial isotropy over all completions including ℝ.

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

Proposed corrected statement

For a quadratic form Q over ℚ, a nonzero x∈ℚ^n with Q(x)=0 exists iff a nonzero isotropic vector exists over ℝ and over ℚ_p for every prime p. Replace congruences mod p by these completion-valued nonzero witnesses.

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.