ProofAtlas

Lean evidence record

Tao’s Almost-Bounded Collatz Orbits: Lean evidence

This technical record binds the exact theorem statement to its commit-pinned Lean source, checker results, assumptions, and publication-review status.

Read theorem page
Lean buildpassed
Unfinished proof stepsNone
Publication reviewsAccepted

Line counts exclude blank lines; comments and documentation count. The total is the commit-pinned first-party Lean import closure; Mathlib and other third-party dependencies are excluded.

Exact formal proposition

Hypotheses and conclusion

This is the meaningful proposition proved by the checked wrapper declaration. It is extracted from the same commit-pinned Lean source.

def TaoAlmostBoundedColMinStatement : Prop :=
  ∀ f : ℕ → ℝ,
    GrowsToInfinity f →
      HasLogDensity {N : ℕ | 0 < N ∧ (collatzOrbitMin N : ℝ) < f N} 1
Definitions used in this proposition

GrowsToInfinity

def GrowsToInfinity (f : ℕ → ℝ) : Prop :=
  Tendsto f atTop atTop

HasLogDensity

def HasLogDensity (s : Set ℕ) (d : ℝ) : Prop :=
  Tendsto (fun N : ℕ => logCountingRatio s N) atTop (nhds d)

collatzOrbitMin

noncomputable def collatzOrbitMin (N : ℕ) : ℕ :=
  sInf (collatzOrbitValues N)
Lean wrapper declaration

The checked endpoint names the expanded proposition above. Both come from the same commit-pinned source.

theorem taoAlmostBoundedColMin_checked : TaoAlmostBoundedColMinStatement
Open main Lean file

Technical evidence record

Source identity, checker results, and assumptions

Main Lean declaration
taoAlmostBoundedColMin_checked
Source commit
d53c8de00056

Mechanical evidence

How Lean checked the proof

These fields support the exact Lean declaration, not a broader informal claim.

Artifact ID
artifact.known-tao-almost-bounded-orbits.colmin.v001
Declarations covered by recorded evidence
Erdos1135.Tao.taoAlmostBoundedColMin_checked
Erdos1135.Tao.taoAlmostBounded_checked
Lean build
passed
Recorded build time
4.5 s one machine-dependent evidence run, not a benchmark
Evidence collected
· clean-source provenance recorded
Unfinished proof check
passed
Lean toolchain
leanprover/lean4:v4.30.0-rc2
Recorded source commit
d53c8de00056fb05999e44589f2399d7efa026e4
Source SHA-256
sha256:ca16df30d1c105812c238360a63fa27d3ef1933631713aa4f6cced71b9f9267d
Statement alignment
accepted

Lean foundations

Standard foundations used by the proof

Lean reports the logical foundations below through Mathlib. They are standard proof-system foundations, not conjectural mathematical assumptions about this theorem. The recorded closure stays within the approved classical_mathlib_standard profile, with no unexpected axiom or unfinished-proof placeholder.

  • Classical.choice
  • Quot.sound
  • propext

Files and machine-readable evidence

Reproduce or inspect the recorded check

Use the complete first-party source bundle for reconstruction, or inspect the exact main file and checker evidence separately. Mathlib and other third-party dependencies are identified but not rebundled.

Checked source + agent continuation package · 397 Lean files · 1 evidence record · Download ZIPBrowse counted source onlineDownload main Lean fileDownload checker evidence (JSON)Download Apache-2.0 license

Review bindings

Publication reviews accepted

All four required publication-review gates are accepted for the retained presentation of this exact theorem. The review bindings are recorded separately from the Lean build and do not broaden the formal statement.

Read the publication-review details