Prove2Me

A nonzero tensor factor cannot defeat the rank exponent

Proved
OAI.MatrixMultiplication.AuxiliarySeparation.tensor_catalytic_obstruction

by marwahaha · Oct 7, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

matrix-multiplicationoai-matmul-modulartensor-rank

For a field FF, let RF(n)R_F(n) be the exact rank of the square matrix multiplication coefficient tensor of size nn, and let νF=inf⁡n≥2log⁡RF(n)/log⁡n\nu_F=\inf_{n\ge2}\log R_F(n)/\log n. In the semiring of finite tensors over FF modulo mutual restriction, let D,sD,s be tensor classes with s≠0s\ne0. Let d,k∈Nd,k\in\mathbb N with d>0d>0, and let [Td][T_d] be the matrix multiplication class of size dd. If

D+[Td]s≤D+ks,D+[T_d]s\le D+ks,

then dνF≤kd^{\nu_F}\le k. Here addition is direct sum, multiplication is tensor product, and the order is restriction. The result holds over any field and does not require rank additivity.

Preamble
import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.AddTorsor.Basic
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Ring.Nat
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
import Mathlib.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
import Mathlib.Algebra.Polynomial.Eval.Defs
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Ring.GeomSum
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
import Mathlib.Analysis.Convex.Caratheodory
import Mathlib.Analysis.Convex.Cone.Dual
import Mathlib.Analysis.Convex.GaugeRescale
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.PartitionOfUnity
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Affine.AddTorsorBases
import Mathlib.Analysis.SpecialFunctions.BinaryEntropy
import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecialFunctions.Stirling
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Matrix.Mul
import Mathlib.Data.Nat.Choose.Multinomial
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Nat.Log
import Mathlib.Data.Rat.BigOperators
import Mathlib.Data.Rat.Floor
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Real.Basic
import Mathlib.Data.Set.Finite.Basic
import Mathlib.FieldTheory.IntermediateField.Adjoin.Basic
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.Geometry.Convex.Cone.Pointed
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.MonoidLocalization.GrothendieckGroup
import Mathlib.GroupTheory.Perm.DomMulAct
import Mathlib.LinearAlgebra.Basis.Defs
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.LinearAlgebra.Finsupp.LinearCombination
import Mathlib.LinearAlgebra.Lagrange
import Mathlib.LinearAlgebra.Matrix.Rank
import Mathlib.LinearAlgebra.PiTensorProduct
import Mathlib.LinearAlgebra.PiTensorProduct.Basis
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.LinearAlgebra.TensorProduct.Pi
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Logic.Equiv.Prod
import Mathlib.Logic.Equiv.Sum
import Mathlib.Order.Antisymmetrization
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.RingTheory.Polynomial.DegreeLT
import Mathlib.RingTheory.RootsOfUnity.AlgebraicallyClosed
import Mathlib.Tactic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.FunProp
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.Topology.Algebra.Module.FiniteDimension
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Order.MonotoneConvergence
import Definitions.Def_OAI429_GenericCharacterCore
import Definitions.Def_OAI429_GenericRank
import Definitions.Def_OAI429_GenericSemiring_Part1
import Definitions.Def_OAI429_GenericSemiring_Part2
import Definitions.Def_mme_omega

set_option autoImplicit false
set_option linter.all false
open scoped Classical
Formal statement
theorem OAI.MatrixMultiplication.AuxiliarySeparation.tensor_catalytic_obstruction.{u_1} :
    ∀ {𝕜 : Type u_1} [inst : Field 𝕜]
  {D s : @OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst} {d k : Nat},
  @LT.lt Nat instLTNat (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) d →
    @Ne (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst) s
        (@OfNat.ofNat (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst) (nat_lit 0)
          (@Zero.toOfNat0 (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
            (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.instZeroTensorClass 𝕜 inst))) →
      @LE.le (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
          (@Preorder.toLE (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
            (@PartialOrder.toPreorder (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
              (@instPartialOrderAntisymmetrization
                (OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.FiniteTensor 𝕜)
                (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.FiniteTensor.instPreorder 𝕜 inst))))
          (@HAdd.hAdd (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
            (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
            (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
            (@instHAdd (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
              (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.instAddTensorClass 𝕜 inst))
            D
            (@HMul.hMul (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
              (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
              (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
              (@instHMul (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
                (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.instMulTensorClass 𝕜 inst))
              (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.matrixClass 𝕜 inst d) s))
          (@HAdd.hAdd (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
            (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
            (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
            (@instHAdd (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
              (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.instAddTensorClass 𝕜 inst))
            D
            (@HMul.hMul (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
              (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
              (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
              (@instHMul (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
                (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.instMulTensorClass 𝕜 inst))
              (@Nat.cast (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
                (@AddMonoidWithOne.toNatCast
                  (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
                  (@AddCommMonoidWithOne.toAddMonoidWithOne
                    (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
                    (@NonAssocSemiring.toAddCommMonoidWithOne
                      (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
                      (@Semiring.toNonAssocSemiring
                        (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
                        (@CommSemiring.toSemiring
                          (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass 𝕜 inst)
                          (@OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.instCommSemiringTensorClass 𝕜
                            inst))))))
                k)
              s)) →
        @LE.le Real Real.instLE
          (@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) (@Nat.cast Real Real.instNatCast d)
            (@OAI.MatrixMultiplication.AuxiliarySeparation.exactRankExponent 𝕜 inst))
          (@Nat.cast Real Real.instNatCast k)
:= by sorry
Source
https://prove2.me/submissions/facf5b05-d302-42c1-a107-8ff40a0c2134; OAI.MatrixMultiplication.AuxiliarySeparation.tensor_catalytic_obstruction, lines 8228-8269

View graph