Datasets:

FineEnvs
/

openai-math

Context: this statement belongs to family 293 of the release, A counterexample to the hyperinvariant-subspace problem (Operator algebras). The family as a whole: Constructs a nonzero norm-quasinilpotent operator on every infinite-dimensional separable complex Hilbert space with no nonzero proper closed subspace invariant under every commuting operator. The construction also gives operators with no nontrivial invariant projection in the hyperfinite type $\mathrm{II}_1$ factor.

The challenge is BackwardIntertwiners, also at /opt/openai-math/challenges/BackwardIntertwiners.lean:

import Mathlib

namespace OAI

namespace Hyperinvariant233
open Filter
open scoped Topology
universe u
variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

def commutant (T : H →L[ℂ] H) : Subalgebra ℂ (H →L[ℂ] H) :=
  Subalgebra.centralizer ℂ ({T} : Set (H →L[ℂ] H))

def TransitiveCommutant (T : H →L[ℂ] H) : Prop :=
  ∀ K : Submodule ℂ H, IsClosed (K : Set H) →
    (∀ A : H →L[ℂ] H, A * T = T * A → ∀ x ∈ K, A x ∈ K) →
    K = ⊥ ∨ K = ⊤

@[instance_reducible]
def strongOperatorTopology : TopologicalSpace (H →L[ℂ] H) :=
  TopologicalSpace.induced (fun A : H →L[ℂ] H => (fun x : H => A x)) inferInstance

def FullClaim : Prop :=
  ∃ T : H →L[ℂ] H, T ≠ 0 ∧
    Tendsto (fun n : ℕ => ‖T ^ n‖ ^ (1 / (n : ℝ))) atTop (𝓝 0) ∧
    TransitiveCommutant T ∧ commutant T ≠ ⊤ ∧
    @IsClosed (H →L[ℂ] H) strongOperatorTopology (commutant T : Set (H →L[ℂ] H))

end Hyperinvariant233

namespace BackwardIntertwiners
open Filter Topology MeasureTheory Set
noncomputable section
open scoped ENNReal

theorem direct_algebra_corollary (H : Type*) [NormedAddCommGroup H] [InnerProductSpace ℂ H]
    [CompleteSpace H] [TopologicalSpace.SeparableSpace H] (hInf : ¬FiniteDimensional ℂ H) :
    Hyperinvariant233.FullClaim (H:=H) := by
  sorry

end
end BackwardIntertwiners

end OAI

What to submit

Write /workspace/Submission.lean. Start from a copy of the challenge:

cp /opt/openai-math/challenges/BackwardIntertwiners.lean /workspace/Submission.lean

then replace every sorry with a proof. The file is graded on three things:

Same statements. The theorem OAI.BackwardIntertwiners.direct_algebra_corollary must keep exactly the statement shown above: same names, namespaces, binders and types. Every definition the statements use must stay exactly as written. Change nothing except the proofs.
Standard axioms only. Proofs may use only propext, Quot.sound and Classical.choice. sorry, admit, new axioms and native_decide (it introduces an axiom of its own) are rejected.
Keep the challenge's declarations as they are. Put new lemmas and instances after the definitions the statements use, or in a separate Submission/*.lean module. A declaration added before them can change how they elaborate, and then they no longer match the challenge.
Kernel-checked. The proofs are re-checked by the Lean kernel, not just the elaborator.

Long proofs can be split into modules under /workspace/Submission/ (module names Submission.Foo, Submission.Foo.Bar) imported from Submission.lean. Only .lean files at those two paths are graded.

Environment

Lean v4.34.1 and Mathlib at commit d13f23b are installed and prebuilt; /workspace is a Lake project.
The sandbox has 4 CPUs and 8 GB of memory; LEAN_NUM_THREADS=3 keeps lake build to three parallel jobs. Check your work with cd /workspace && lake build Submission. Add #print axioms <name> to see which axioms a proof uses.
There is no internet access. OpenAI's own proofs are not installed.

Grading

When you finish, Submission.lean and Submission/**.lean are copied to a fresh machine and checked with Comparator, the Lean FRO's proof checker. The reward is 1 if Comparator accepts the proof and 0 otherwise. A partial proof scores 0.

openai/math challenge BackwardIntertwiners (family 293)

What to submit

Environment

Grading

openai/math challenge `BackwardIntertwiners` (family 293)