Prove2Me

Power bound for active work under the uniform false-mass budget

Proved
ThreeSum.UniformStageBounds.budget_active_powerBound

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

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

For a fixed natural magnitude exponent kk, let Fk(n)=⌊falseNumerator⁡(k,n)/radix⁡(n)⌋F_k(n)=\lfloor\operatorname{falseNumerator}(k,n)/\operatorname{radix}(n)\rfloor. Define active work as Ak(n)=Fk(n)R(n)G(n)Q(n)A_k(n)=F_k(n)R(n)G(n)Q(n), where RR is the representation bound, GG the local balanced-partition count, and QQ the sum of query-layer counts. Set ε=1/5\varepsilon=1/5, δ=−1/20\delta=-1/20, b=191/1000b=191/1000, ρ=277/1000\rho=277/1000, and κ=512H2(1/512)−256H2(1/256)−1\kappa=512H_2(1/512)-256H_2(1/256)-1, with binary entropy H2H_2. Then

PowerBound⁡(Ak, 2−ε(1/2−δ−κ−ρ(1+b−2δ))).\operatorname{PowerBound}(A_k,\,2-\varepsilon(1/2-\delta-\kappa-\rho(1+b-2\delta))).

PowerBound means that for every η>0\eta>0, eventually Ak(n)≤na+ηA_k(n)\le n^{a+\eta} for the displayed exponent aa.

This bounds the active component of the finite work ledger uniformly over the size-only false-mass budget.

Preamble
import Aesop
import Definitions.Def_ThreeSum1962Source_Bundle01
import Definitions.Def_ThreeSum1962Source_Bundle05
import Definitions.Def_ThreeSum1962Source_Bundle07
import Definitions.Def_ThreeSum1962Source_Bundle08
import Definitions.Def_ThreeSum1962Source_Bundle09
import Definitions.Def_ThreeSum1962Source_Bundle10
import Definitions.Def_ThreeSum1962Source_Bundle11
import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Field.ZMod
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Chebyshev
import Mathlib.Algebra.Order.Floor.Div
import Mathlib.Algebra.Order.Ring.Pow
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.Pow.Asymptotics
import Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Max
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.List.MinMax
import Mathlib.Data.List.Pi
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Sqrt
import Mathlib.Data.Real.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Index
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.Matrix.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.NumberTheory.Bertrand
import Mathlib.NumberTheory.Chebyshev
import Mathlib.Order.Monotone.Basic
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.FinCases
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 Theorems.Thm_ThreeSum_CostLedger_queryTerms_log_bound
import Theorems.Thm_ThreeSum_HierarchyScales_hierarchyEnvelope_log_rate
import Theorems.Thm_ThreeSum_MassScales_falseNumerator_log_rate
import Theorems.Thm_ThreeSum_TrimScales_representationBound_log_rate


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
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.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
Formal statement
theorem ThreeSum.UniformStageBounds.budget_active_powerBound : ∀ (k : Nat),
  ThreeSum.StageBounds.PowerBound (ThreeSum.StageBounds.activeCount (ThreeSum.UniformStageBounds.falseBudget k))
    (@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub)
      (@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)
            (@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 1) (@One.toOfNat1.{0} Real Real.instOne))
                (@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)
            ThreeSum.Parameters.kappa)
          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) ThreeSum.Parameters.entropyRatio
            (@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub)
              (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd)
                (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) ThreeSum.Parameters.boxSaving)
              (@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/UniformStageBounds.lean#L62-L78

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