Is D1Tuple — frontier fifteen growth

Proved

D1Bound5.IsD1Tuple.frontier_fifteen_growth

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

diophantine-tuplesnumber-theory

The sharpened fourth-entry gap used for the S>=15 one-small branch. It holds for every actual ordered quadruple with A<3, B>=3 and S>=15, without a first-pair table or a determinant certificate.

Preamble

import Lean.Util.CollectAxioms
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.QuadraticAlgebra.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Analysis.CStarAlgebra.Matrix
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Complex.Norm
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Int.GCD
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.Sqrt
import Mathlib.Data.Real.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Permutation
import Mathlib.LinearAlgebra.Matrix.SchurComplement
import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.IntervalCases
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Definitions.Def_D1Bound5_IsD1Tuple

Formal statement

section
open scoped NumberField Matrix.Norms.L2Operator BigOperators
open Matrix Module

namespace D1Bound5


theorem _root_.D1Bound5.IsD1Tuple.frontier_fifteen_growth
    {K : Type*} [Field K] [NumberField K] [NumberField.IsTotallyComplex K]
    (hK : finrank ℚ K = 2) (phi : K →+* ℂ)
    (a : Fin 4 → 𝓞 K) (ha : IsD1Tuple a)
    (hmon : Monotone (fun i => ‖phi (a i : K)‖))
    (hA3 : ‖phi (a 0 : K)‖ < 3) (hB : 3 ≤ ‖phi (a 1 : K)‖)
    (hC : 15 ≤ ‖phi (a 2 : K)‖) :
    (59/20 : ℝ)*‖phi (a 0 : K)‖*‖phi (a 1 : K)‖*‖phi (a 2 : K)‖ <
      ‖phi (a 3 : K)‖ := by sorry

end D1Bound5
end

Source

D1Bound5/FrontierFifteenTuple.lean, lines 8-94, SHA-256 4a92712c822542b20d5e7b8390c19074244d70629fca5b37a5c2dc68ee97ae70. Project: https://github.com/tomislav-math/d1-bound5-lean

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