Is D1Tuple — retained start of frontier

Proved

D1Bound5.IsD1Tuple.retained_start_of_frontier

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

diophantine-tuplesnumber-theory

The retained start estimates for an actual ordered sextuple in the general H32 frontier regime. The four-anchor form is proved nonzero using its actual zero window and the repeated five-entry power gap, then its full-ring integral lower bound supplies the final seventh-power gap.

Preamble

import Lean.Util.CollectAxioms
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.QuadraticAlgebra.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Analysis.Analytic.Binomial
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.Normed.Ring.Basic
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Insert
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.Sym.Basic
import Mathlib.Data.Sym.Card
import Mathlib.Data.ZMod.Basic
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.LinearAlgebra.Lagrange
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.RingTheory.Binomial
import Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
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.Positivity
import Mathlib.Tactic.Ring
import Definitions.Def_D1Bound5_IsD1Tuple

Formal statement

section
open Finset
open scoped NumberField
open Module

namespace D1Bound5







theorem _root_.D1Bound5.IsD1Tuple.retained_start_of_frontier
    {K : Type*} [Field K] [NumberField K] [NumberField.IsTotallyComplex K]
    (hK : finrank ℚ K = 2) (phi : K →+* ℂ)
    (a : Fin 6 → 𝓞 K) (ha : IsD1Tuple a)
    (hmon : Monotone (fun i => ‖phi (a i : K)‖))
    (hB : 3 ≤ ‖phi (a 1 : K)‖)
    (hfrontier : 2*‖phi (a 0 : K)‖*‖phi (a 1 : K)‖*‖phi (a 2 : K)‖ < ‖phi (a 3 : K)‖)
    (hAT32 : 32 ≤ ‖phi (a 0 : K)‖*‖phi (a 3 : K)‖) :
    let A := ‖phi (a 0 : K)‖
    let B := ‖phi (a 1 : K)‖
    let S := ‖phi (a 2 : K)‖
    let T := ‖phi (a 3 : K)‖
    let C := ‖phi (a 4 : K)‖
    let D := ‖phi (a 5 : K)‖
    A^2*T^5/S^2 < C ∧ A^2*C^5/S^2 < D ∧ A^2*C^7/(S^2*T^4) < D ∧
      1000 < C ∧ 200*S < C ∧ 4*T < C ∧ 16*B*S*T < C ∧ 1000*C < D := by sorry

end D1Bound5
end

Source

D1Bound5/CoherentRetainedFrontierStart.lean, lines 39-167, SHA-256 572ee94f938aba9d899196d56e8ccc5e5237ee23027c97c709c1a088bcb4c1af. Project: https://github.com/tomislav-math/d1-bound5-lean

View graph