Prove2Me

Asymptotic power bound for the complete band-prime arithmetic budget

Proved
ThreeSum.BandPrimeCosts.timeBudget_powerBound

by wurtle · Oct 14, 2026 · Mathlib 0df444a (Lean v4.33.1)

3sumalgorithmsfine-grained-complexitythree-sum-1962word-ram

Fix a natural exponent kk. Let Tk(n)T_k(n) be the explicit band-prime time budget: prime-pool construction plus the product of the prime-chain depth allowance, candidate-band count, squared pool-size allowance, and per-triple arithmetic budget, with its factor 1515. Let E=128H2(1/256)E=128H_2(1/256), where H2H_2 is binary entropy, and use the fixed parameters ε=1/5\varepsilon=1/5 and σ=−1/20\sigma=-1/20.

For every real η>0\eta>0, all sufficiently large natural nn satisfy

Tk(n)≤n 2+ε(4/5−(2/5)E−2σ)+η.T_k(n)\le n^{\,2+\varepsilon(4/5-(2/5)E-2\sigma)+\eta}.

The threshold may depend on kk and η\eta.

This bounds all executed arithmetic in the band-prime stage at its stated exponent, while allowing arbitrarily small asymptotic slack.

Preamble
import Aesop
import Definitions.Def_APSPSource_RemainingDefinitions
import Definitions.Def_APSPSource_ThreeSumApsp_Lang_Logic
import Definitions.Def_APSPSource_ThreeSumApsp_Lang_Regions
import Definitions.Def_ThreeSum1962Source_Bundle01
import Definitions.Def_ThreeSum1962Source_Bundle05
import Definitions.Def_ThreeSum1962Source_Bundle07
import Definitions.Def_ThreeSum1962Source_Bundle09
import Definitions.Def_ThreeSum1962Source_Bundle10
import Definitions.Def_TrulySubcubicAPSP_SourceSpecification
import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.BigOperators.Group.Finset.Powerset
import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Field.ZMod
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Int.Defs
import Mathlib.Algebra.GroupWithZero.Nat
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Algebra.Order.BigOperators.GroupWithZero.List
import Mathlib.Algebra.Order.Chebyshev
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Floor.Div
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Ring.Nat
import Mathlib.Analysis.Complex.ExponentialBounds
import Mathlib.Analysis.Convex.Deriv
import Mathlib.Analysis.SpecialFunctions.BinaryEntropy
import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Combinatorics.Colex
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Fin.Embedding
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Max
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Finset.Range
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sets
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Int.Notation
import Mathlib.Data.Int.SuccPred
import Mathlib.Data.List.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.GetD
import Mathlib.Data.List.MinMax
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Pi
import Mathlib.Data.List.ProdSigma
import Mathlib.Data.List.TakeDrop
import Mathlib.Data.Nat.Cast.Order.Ring
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Choose.Bounds
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Notation
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Nat.Size
import Mathlib.Data.Nat.Sqrt
import Mathlib.Data.Nat.SuccPred
import Mathlib.Data.Real.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Index
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.Matrix.Defs
import Mathlib.LinearAlgebra.Matrix.Dual
import Mathlib.LinearAlgebra.Matrix.Notation
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Logic.Function.Basic
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.NumberTheory.Bertrand
import Mathlib.NumberTheory.Chebyshev
import Mathlib.NumberTheory.PrimeCounting
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.Probability.Independence.Basic
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.Abel
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.DeriveFintype
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.Push
import Mathlib.Tactic.Ring
import Mathlib.Tactic.SplitIfs
import Mathlib.Tactic.Tauto
import Theorems.Thm_ThreeSum_ArithmeticStageBounds_perTriple_powerBound
import Theorems.Thm_ThreeSum_BandPrimeCosts_pool_time_le
import Theorems.Thm_ThreeSum_HierarchyScales_bandCount_log_rate
import Theorems.Thm_ThreeSum_LightPrimePool_prefix_increasing
import Theorems.Thm_ThreeSum_MassScales_candidateCount_log_rate
import Theorems.Thm_ThreeSum_Scales_natSqrt_ratio_tendsto_zero
import Theorems.Thm_ThreeSum_Scales_primeDepth_log_ratio_tendsto_zero
import Theorems.Thm_ThreeSum_StageBounds_PowerBound_add_same


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.ArithmeticStageBounds
end ThreeSum.ArithmeticStageBounds
namespace ThreeSum.BandScales
end ThreeSum.BandScales
namespace ThreeSum.CostLedger
end ThreeSum.CostLedger
namespace ThreeSum.HierarchyScales
end ThreeSum.HierarchyScales
namespace ThreeSum.MassScales
end ThreeSum.MassScales
namespace ThreeSum.ModuloCosts
end ThreeSum.ModuloCosts
namespace ThreeSum.Parameters
end ThreeSum.Parameters
namespace ThreeSum.Scales
end ThreeSum.Scales
namespace ThreeSum.StageBounds
end ThreeSum.StageBounds
namespace ThreeSum.Switch
end ThreeSum.Switch
namespace ThreeSum.TrimScales
end ThreeSum.TrimScales
namespace ThreeSum.UniformStageBounds
end ThreeSum.UniformStageBounds
Formal statement
theorem ThreeSum.BandPrimeCosts.timeBudget_powerBound : ∀ (k : Nat),
  ThreeSum.StageBounds.PowerBound (ThreeSum.BandPrimeCosts.timeBudget k)
    (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd)
      (@OfNat.ofNat.{0} Real (nat_lit 2)
        (@instOfNatAtLeastTwo.{0} Real (nat_lit 2) Real.instNatCast
          (@Nat.instAtLeastTwoHAddOfNat (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))
            (@Nat.instNeZeroSucc (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))
      (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) ThreeSum.Parameters.epsilon
        (@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub)
          (@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub)
            (@HDiv.hDiv.{0, 0, 0} Real Real Real
              (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))
              (@OfNat.ofNat.{0} Real (nat_lit 4)
                (@instOfNatAtLeastTwo.{0} Real (nat_lit 4) Real.instNatCast
                  (@Nat.instAtLeastTwoHAddOfNat (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))
                    (@Nat.instNeZeroSucc (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))
              (@OfNat.ofNat.{0} Real (nat_lit 5)
                (@instOfNatAtLeastTwo.{0} Real (nat_lit 5) Real.instNatCast
                  (@Nat.instAtLeastTwoHAddOfNat (@OfNat.ofNat.{0} Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))
                    (@Nat.instNeZeroSucc (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))))))
            (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)
              (@HDiv.hDiv.{0, 0, 0} Real Real Real
                (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))
                (@OfNat.ofNat.{0} Real (nat_lit 2)
                  (@instOfNatAtLeastTwo.{0} Real (nat_lit 2) Real.instNatCast
                    (@Nat.instAtLeastTwoHAddOfNat (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))
                      (@Nat.instNeZeroSucc (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))
                (@OfNat.ofNat.{0} Real (nat_lit 5)
                  (@instOfNatAtLeastTwo.{0} Real (nat_lit 5) Real.instNatCast
                    (@Nat.instAtLeastTwoHAddOfNat (@OfNat.ofNat.{0} Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))
                      (@Nat.instNeZeroSucc (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))))))
              ThreeSum.Parameters.E))
          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)
            (@OfNat.ofNat.{0} Real (nat_lit 2)
              (@instOfNatAtLeastTwo.{0} Real (nat_lit 2) Real.instNatCast
                (@Nat.instAtLeastTwoHAddOfNat (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))
                  (@Nat.instNeZeroSucc (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))
            ThreeSum.Parameters.bandShift)))) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/BandPrimeCostBounds.lean#L54-L69

View graph