Is D1Tuple — ordinary tail incompatible
ProvedD1Bound5.IsD1Tuple.ordinary_tail_incompatiblediophantine-tuplesnumber-theory
The ordinary tail is genuinely infinite. A single checked overlap at orders 5 and 6 propagates because the actual quadratic lower rate exceeds the height rate; geometric lower endpoints are unbounded.
Preamble
import Lean.Elab.Tactic.Omega import Mathlib.Algebra.BigOperators.Fin import Mathlib.Algebra.BigOperators.Group.Finset.Sigma import Mathlib.Algebra.BigOperators.NatAntidiagonal import Mathlib.Algebra.CharZero.Infinite import Mathlib.Algebra.Order.Archimedean.Real.Basic import Mathlib.Algebra.Polynomial.BigOperators import Mathlib.Algebra.Polynomial.Degree.Operations import Mathlib.Algebra.Polynomial.Div import Mathlib.Algebra.Polynomial.PartialFractions import Mathlib.Algebra.Polynomial.Reverse import Mathlib.Algebra.Polynomial.RingDivision import Mathlib.Algebra.Polynomial.Roots import Mathlib.Algebra.QuadraticAlgebra.Basic import Mathlib.Algebra.Ring.Basic import Mathlib.Algebra.Ring.Equiv import Mathlib.Analysis.Analytic.Binomial import Mathlib.Analysis.Complex.Basic import Mathlib.Analysis.Complex.Norm import Mathlib.Analysis.Normed.Ring.Basic import Mathlib.Analysis.Normed.Ring.InfiniteSum import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral import Mathlib.Analysis.SpecialFunctions.Pow.Real import Mathlib.Analysis.SpecificLimits.Normed import Mathlib.Combinatorics.Enumerative.Catalan.Basic import Mathlib.Data.Finset.Card import Mathlib.Data.Finset.Insert import Mathlib.Data.Int.GCD import Mathlib.Data.Sym.Basic import Mathlib.Data.Sym.Card import Mathlib.LinearAlgebra.Dimension.Free import Mathlib.LinearAlgebra.Lagrange import Mathlib.LinearAlgebra.Matrix.NonsingularInverse import Mathlib.LinearAlgebra.Matrix.Polynomial import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex import Mathlib.NumberTheory.NumberField.Norm import Mathlib.RingTheory.Binomial import Mathlib.RingTheory.MvPolynomial.Symmetric.Defs import Mathlib.RingTheory.PowerSeries.Binomial import Mathlib.RingTheory.PowerSeries.Trunc import Mathlib.Tactic.FieldSimp import Mathlib.Tactic.FinCases import Mathlib.Tactic.GCongr import Mathlib.Tactic.Linarith import Mathlib.Tactic.LinearCombination import Mathlib.Tactic.NormNum import Mathlib.Tactic.Positivity import Mathlib.Tactic.Ring import Mathlib.Topology.Algebra.InfiniteSum.Order import Definitions.Def_D1Bound5_IsD1Tuple import Definitions.Def_D1Bound5_OrdinaryClosedIntervals_Definitions
Formal statement
section
open scoped NumberField
open Module
namespace D1Bound5
theorem _root_.D1Bound5.IsD1Tuple.ordinary_tail_incompatible
{K : Type*} [Field K] [NumberField K] [NumberField.IsTotallyComplex K]
(hK : finrank ℚ K = 2) (phi : K →+* ℂ)
(t : Fin 4 → 𝓞 K) (ht : IsD1Tuple t)
(hAB : ‖phi (t 0 : K)‖ ≤ ‖phi (t 1 : K)‖)
(hAC : 1000 ≤ ‖phi (t 0 : K)‖*‖phi (t 2 : K)‖)
(hC : 4*‖phi (t 1 : K)‖ < ‖phi (t 2 : K)‖)
(hCD : ‖phi (t 2 : K)‖ ≤ ‖phi (t 3 : K)‖)
(hrate : 42*‖phi (t 1 : K)‖^3*‖phi (t 2 : K)‖ <
2*‖phi (t 0 : K)‖*‖phi (t 2 : K)‖^2/(5*‖phi (t 1 : K)‖))
(hoverlap : ordinaryWindowLower ‖phi (t 1 : K)‖ ‖phi (t 2 : K)‖ 6 ≤
ordinaryWindowUpper ‖phi (t 0 : K)‖ ‖phi (t 1 : K)‖ ‖phi (t 2 : K)‖ 5)
(hstart : ordinaryWindowLower ‖phi (t 1 : K)‖ ‖phi (t 2 : K)‖ 5 ≤
‖phi (t 3 : K)‖) : False := by sorry
end D1Bound5
end
Source
D1Bound5/OrdinaryClosedIntervals.lean, lines 135-205, SHA-256 2a5f04eb6285acfb92a512be32794797bc3bc72ea1071063ed81f784bc551a9b. Project: https://github.com/tomislav-math/d1-bound5-lean