Logarithmic growth rate of the false-incidence numerator

Proved

ThreeSum.MassScales.falseNumerator_log_rate

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

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

Fix $k\in\mathbb N$ . Let $P_n$ be the prescribed prime-window scale, $q_n$ the number of candidate primes in $(P_n,4P_n]$ , $g_n$ the band size, and $b_n$ the band count. Let $d_n$ be the prescribed prime-chain depth, $m_n=\lfloor\log_2 n\rfloor$ , and define the false-incidence numerator

F_k(n)=q_n(3b_n^2g_n^3)\,[16(k+2)(m_n+1)]^{d_n}\,(32P_n4^{d_n}).

With $E=128H_2(1/256)$ and band shift $\sigma=-1/20$ , its exact logarithmic rate is

\lim_{n\to\infty}\frac{\log F_k(n)}{\log n}=2+\frac{E+\sigma}{5}.

Here the limit is along natural input sizes, and $H_2$ denotes binary entropy.

This isolates the polynomial growth of the false-incidence numerator from its subpolynomial prime and score losses.

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 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_BandScales_bandSize_log_rate
import Theorems.Thm_ThreeSum_HierarchyScales_bandCount_log_rate
import Theorems.Thm_ThreeSum_MassScales_candidateCount_log_rate
import Theorems.Thm_ThreeSum_Scales_primeLoss_log_ratio_tendsto_zero
import Theorems.Thm_ThreeSum_Scales_scoreLoss_log_ratio_tendsto_zero


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.Parameters
end ThreeSum.Parameters
namespace ThreeSum.Scales
end ThreeSum.Scales
namespace ThreeSum.Switch
end ThreeSum.Switch
namespace ThreeSum.TrimScales
end ThreeSum.TrimScales

Formal statement

theorem ThreeSum.MassScales.falseNumerator_log_rate : ∀ (k : Nat),
  @Filter.Tendsto.{0, 0} Nat Real
    (fun (n : Nat) =>
      @HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))
        (Real.log (@Nat.cast.{0} Real Real.instNatCast (ThreeSum.MassScales.falseNumerator k n)))
        (Real.log (@Nat.cast.{0} Real Real.instNatCast n)))
    (@Filter.atTop.{0} Nat Nat.instPreorder)
    (@nhds.{0} Real
      (@UniformSpace.toTopologicalSpace.{0} Real (@PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))
      (@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)))))))
        (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))
          (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd) ThreeSum.Parameters.E
            ThreeSum.Parameters.bandShift)
          (@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)))))))))) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/MassScales.lean#L161-L174

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