Weighted Schur four energy

Proved

D1Bound5.weightedSchur_four_energy

by tp · Oct 23, 2026 · Mathlib 0df444a (Lean v4.33.1)

diophantine-tuplesnumber-theory

The four-dimensional real weighted Schur estimate, stated as a squared-energy bound. The matrix entries need only be nonnegative and symmetric, and the input vector may have arbitrary signs.

Preamble

import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.CStarAlgebra.Matrix
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Data.Complex.Basic
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Permutation
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring

Formal statement

section
open scoped Matrix.Norms.L2Operator
open Matrix

namespace D1Bound5



end D1Bound5

namespace D1Bound5


theorem _root_.D1Bound5.weightedSchur_four_energy
    (b : Matrix (Fin 4) (Fin 4) ℝ) (w x : Fin 4 → ℝ) (c : ℝ)
    (hb : ∀ i j, 0 ≤ b i j)
    (hw : ∀ i, 0 < w i) (hc : 0 ≤ c)
    (hsymm : ∀ i j, b i j = b j i)
    (hrows : ∀ i, (∑ j, b i j * w j) ≤ c * w i) :
    (∑ i, (∑ j, b i j * x j)^2) ≤ c^2 * (∑ j, (x j)^2) := by sorry

end D1Bound5

namespace D1Bound5





end D1Bound5
end

Source

D1Bound5/WeightedSchur.lean, lines 29-87, SHA-256 0740899bda0b97ce5440dc1a326c7e0aa61aa178424db90f7d296c6acdfafdf8. Project: https://github.com/tomislav-math/d1-bound5-lean

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