Lean statement

Exact zero case for the companion totient counts

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Theorem selected for verification: OAI.TotientAsymptotic.companion_zero_case

Lean statement file: TotientCompanionZero.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 024 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem companion_zero_case (k : ℕ) (_hk : 0 < k)    (hno : ¬ ∃ d : ℕ, IsTotient d ∧ k*d < ell d) :    (∀ x : ℝ, 0 < x → N k x = 0) ∧    (∀ s ∈ Set.Ico (0 : ℝ) 1, A (fk k) s = 0) := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

definition OAI.TotientAsymptotic.IsTotient line 13

def IsTotient (v : ℕ) : Prop := ∃ n : ℕ, 0 < n ∧ n.totient = v

definition OAI.TotientAsymptotic.ell line 15

def ell (v : ℕ) : ℕ := by  classical  exact if h : IsTotient v then Nat.find h else 0

definition OAI.TotientAsymptotic.a line 19

def a (j : ℕ) : ℝ := (j + 1 : ℝ) * Real.log (j + 1) - j * Real.log j - 1

definition OAI.TotientAsymptotic.RenewalRootSet line 21

def RenewalRootSet : Set ℝ :=  {z | 0 < z ∧ z < 1 ∧ (∑' j : ℕ, a (j+1) * z^(j+1)) = 1}

definition OAI.TotientAsymptotic.rho line 24
```
def rho : ℝ := sInf RenewalRootSet
```
definition OAI.TotientAsymptotic.lam line 26
```
def lam : ℝ := Real.log (1 / rho)
```

definition OAI.TotientAsymptotic.gamma line 28

def gamma : ℝ := (∑' j : ℕ, (j+1 : ℝ) * a (j+1) * rho^(j+1))⁻¹

definition OAI.TotientAsymptotic.P line 30

def P (H : ℕ) : ℕ := ⌊Real.log (Real.log (H : ℝ))⌋₊

definition OAI.TotientAsymptotic.alpha line 31

def alpha (s : ℝ) : ℝ := lam * Real.exp (lam * s)

definition OAI.TotientAsymptotic.largestPrimeFactor line 33

def largestPrimeFactor (n : ℕ) : ℕ := max 1 (n.primeFactors.sup id)

structure OAI.TotientAsymptotic.TailDatum line 35

structure TailDatum (H : ℕ) where  Q : Fin H → ℕ  cofactor : ℕ

definition OAI.TotientAsymptotic.tailPrime line 39

def tailPrime {H : ℕ} (η : TailDatum H) (h : ℕ) : ℕ :=  if hH : h < H then η.Q ⟨h, hH⟩ else 1

definition OAI.TotientAsymptotic.tailLog line 42

def tailLog {H : ℕ} (η : TailDatum H) (h : ℕ) : ℝ :=  Real.log (Real.log (tailPrime η h : ℝ))

definition OAI.TotientAsymptotic.IsWitness line 45

def IsWitness (H : ℕ) (s : ℝ) (η : TailDatum H) : Prop :=

Show the remaining 10 lines

  (∀ h : Fin H, h.val < P H → η.Q h = 1) ∧  0 < η.cofactor ∧  (∀ h ∈ Finset.Ico (P H) H, (tailPrime η h).Prime ∧    (9 / 10 : ℝ) * alpha s * h * (rho^h)⁻¹ ≤ tailLog η h ∧    tailLog η h ≤ (11 / 10 : ℝ) * alpha s * h * (rho^h)⁻¹ ∧    (∑ l ∈ Finset.Ico (P H) h, a (h-l) * tailLog η l) ≤      (1 + (1 / 10000 : ℝ) * Real.exp (-(h : ℝ) / 40)) * tailLog η h) ∧  largestPrimeFactor η.cofactor ≤ tailPrime η (P H) ∧  Real.log (η.cofactor : ℝ) ≤    Real.exp (2 * alpha s * (P H : ℝ) * (rho^(P H))⁻¹)

definition OAI.TotientAsymptotic.w line 57

def w {H : ℕ} (η : TailDatum H) : ℕ :=  η.cofactor * ∏ h ∈ Finset.Ico (P H) H, tailPrime η h

definition OAI.TotientAsymptotic.witnesses line 60

def witnesses (H : ℕ) (s : ℝ) (d : ℕ) : Set (TailDatum H) :=  {η | IsWitness H s η ∧ (w η).totient = d}

definition OAI.TotientAsymptotic.D line 63

def D {H : ℕ} (h : ℕ) (η : TailDatum H) : ℝ :=  ∑ l ∈ Finset.Ico (P H) H, a (h-l) * tailLog η l

definition OAI.TotientAsymptotic.maxD line 66

def maxD {H : ℕ} (h : ℕ) (T : Finset (TailDatum H)) : ℝ := by  classical  exact if ht : T.Nonempty then T.sup' ht (D h) else 0

definition OAI.TotientAsymptotic.AH line 70

def AH (H : ℕ) (f : ℝ → ℝ) (s : ℝ) : ℝ := by

Show the remaining 12 lines

  classical  exact rho^(H*(H-1)/2) * (gamma / alpha s)^H *    ∑ᶠ d : ℕ,      if IsTotient d then        f ((ell d : ℝ) / d) / d *          ∑ᶠ T : Finset (TailDatum H),            if T.Nonempty ∧ (↑T : Set (TailDatum H)) ⊆ witnesses H s d then              (-1 : ℝ)^(T.card-1) *               Real.exp (-(gamma / alpha s) *                 ∑' n : ℕ, rho^(H+n) * maxD (H+n) T)            else 0      else 0

definition OAI.TotientAsymptotic.A line 84

def A (f : ℝ → ℝ) (s : ℝ) : ℝ := limUnder atTop (fun H : ℕ => AH H f s)

definition OAI.TotientAsymptotic.N line 86

def N (k : ℕ) (x : ℝ) : ℝ := by  classical  exact (((Finset.Icc 1 ⌊x⌋₊).filter fun v =>    IsTotient v ∧ (k : ℝ) * x < ell v ∧ (ell v : ℝ) ≤ (k+1 : ℝ) * x).card : ℝ)

definition OAI.TotientAsymptotic.fk line 91

def fk (k : ℕ) (r : ℝ) : ℝ := min 1 ((k+1 : ℝ) / r) - min 1 ((k : ℝ) / r)

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: arithmetic, finite, fixed, function and positive.

Theorem 2.1 of An asymptotic formula for the number of totients

Theorem

For sufficiently large positive integers H, there are nonnegative functions A_H(1;\cdot):[0,1)\to\mathbb R, given by the finite arithmetic formula (eq:AH) below, which converge uniformly as H\to\infty to a function A(1;\cdot) satisfying

0<\inf_{0\le s<1}A(1;s) \le\sup_{0\le s<1}A(1;s)<\infty.

With the parameters (eq:renewal)–(eq:Gj),

V(x)\sim\frac{x}{\log x}\,G_m A(1;\theta).

Moreover, for every fixed real c>0,

\frac{V(cx)}{V(x)}\longrightarrow c.

Section 2, The explicit main term Read it in the paper TeX source, line 45

Show the TeX

For sufficiently large positive integers $H$, there are nonnegative
functions $A_H(1;\cdot):[0,1)\to\mathbb R$, given by the finite
arithmetic formula \eqref{eq:AH} below,
which converge uniformly as $H\to\infty$ to a function $A(1;\cdot)$
satisfying
\[
 0<\inf_{0\le s<1}A(1;s)
   \le\sup_{0\le s<1}A(1;s)<\infty.
\]
With the parameters \eqref{eq:renewal}--\eqref{eq:Gj},
\[
 V(x)\sim\frac{x}{\log x}\,G_m A(1;\theta).
\]
Moreover, for every fixed real $c>0$,
\[
 \frac{V(cx)}{V(x)}\longrightarrow c.
\]

Other possible paper matches, closest wording first

Proposition 5.3 of An asymptotic formula for the number of totients. Words of the Lean statement, its names split into words, that also appear in the theorem: factor, values, totient, preimages, ell and log.
Proposition 4.2 of An asymptotic formula for the number of totients. Words of the Lean statement, its names split into words, that also appear in the theorem: exceeds, zero, count, primes, real, integer and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: conditions and positive.
Proposition 3.3 of An asymptotic formula for the number of totients. Words of the Lean statement, its names split into words, that also appear in the theorem: rho, exp, preimage, witness and log.
Proposition 6.1 of An asymptotic formula for the number of totients. Words of the Lean statement, its names split into words, that also appear in the theorem: max, prime and log.
Proposition 6.5 of An asymptotic formula for the number of totients. Words of the Lean statement, its names split into words, that also appear in the theorem: coefficient, weights and values.
Theorem 2.2 of An asymptotic formula for the number of totients. Words of the Lean statement, its names split into words, that also appear in the theorem: ell, integer and log. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: kd and positive.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

One independent group reports that Comparator accepted a run for this Lean statement file: Carlos Toledo’s cert-machine. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 024’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: None for the headline (V(cx)/V(x) -> c is stated in terms of V alone). The explicit main term rests on a large custom apparatus that cannot be checked against a standard object from the statement.
Definitions the group asks readers to compare with the paper: Main term built from custom rho := sInf RenewalRootSet, lam, gamma, g (renewal recursion), AH (uses ∑ᶠ finsums over d and over Finset (TailDatum H), whose junk value 0 on infinite support is not excluded by the statement) and A f s := limUnder atTop (fun H => AH H f s) (junk value if no limit; the statement separately asserts TendstoUniformlyOn so it is not vacuous)
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The associated coefficient is positive under the stated existence condition; when no totient $d$ satisfies $kd<\ell(d)$, the count and coefficient are identically zero.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states part of a paper’s theorem”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

104 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 65 to its own declarations, 33 to Mathlib’s documentation. 23 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves companion_zero_case exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 22 other declarations of the file; with the theorem itself they span 64 lines. 5 of them are named in the statement itself.

Names in the namespace OAI.TotientAsymptotic are listed without it.

def IsTotientline 13in the statementlibrary
def ellline 15in the statementlibrary
def aline 19library
def RenewalRootSetline 21library
def rholine 24library
def lamline 26library
def gammaline 28library
def Pline 30library
def alphaline 31library
def largestPrimeFactorline 33library
structure TailDatumline 35library
def tailPrimeline 39library
def tailLogline 42library
def IsWitnessline 45library
def wline 57library
def witnessesline 60library
def Dline 63library
def maxDline 66library
def AHline 70library
def Aline 84in the statementlibrary
def Nline 86in the statementlibrary
def fkline 91in the statementlibrary
theorem companion_zero_caseline 95selected theoremlibrary

import Mathlibnamespace OAI/-! The no-seed alternative of the weighted totient companion theorem. -/noncomputable sectionopen scoped BigOperators Topologyopen Filternamespace TotientAsymptoticdef IsTotient (v : ℕ) : Prop := ∃ n : ℕ, 0 < n ∧ n.totient = vdef ell (v : ℕ) : ℕ := by  classical  exact if h : IsTotient v then Nat.find h else 0def a (j : ℕ) : ℝ := (j + 1 : ℝ) * Real.log (j + 1) - j * Real.log j - 1def RenewalRootSet : Set ℝ :=  {z | 0 < z ∧ z < 1 ∧ (∑' j : ℕ, a (j+1) * z^(j+1)) = 1}def rho : ℝ := sInf RenewalRootSetdef lam : ℝ := Real.log (1 / rho)def gamma : ℝ := (∑' j : ℕ, (j+1 : ℝ) * a (j+1) * rho^(j+1))⁻¹def P (H : ℕ) : ℕ := ⌊Real.log (Real.log (H : ℝ))⌋₊def alpha (s : ℝ) : ℝ := lam * Real.exp (lam * s)def largestPrimeFactor (n : ℕ) : ℕ := max 1 (n.primeFactors.sup id)structure TailDatum (H : ℕ) where  Q : Fin H → ℕ  cofactor : ℕdef tailPrime {H : ℕ} (η : TailDatum H) (h : ℕ) : ℕ :=  if hH : h < H then η.Q ⟨h, hH⟩ else 1def tailLog {H : ℕ} (η : TailDatum H) (h : ℕ) : ℝ :=  Real.log (Real.log (tailPrime η h : ℝ))def IsWitness (H : ℕ) (s : ℝ) (η : TailDatum H) : Prop :=  (∀ h : Fin H, h.val < P H → η.Q h = 1) ∧  0 < η.cofactor ∧  (∀ h ∈ Finset.Ico (P H) H, (tailPrime η h).Prime ∧    (9 / 10 : ℝ) * alpha s * h * (rho^h)⁻¹ ≤ tailLog η h ∧    tailLog η h ≤ (11 / 10 : ℝ) * alpha s * h * (rho^h)⁻¹ ∧    (∑ l ∈ Finset.Ico (P H) h, a (h-l) * tailLog η l) ≤      (1 + (1 / 10000 : ℝ) * Real.exp (-(h : ℝ) / 40)) * tailLog η h) ∧  largestPrimeFactor η.cofactor ≤ tailPrime η (P H) ∧  Real.log (η.cofactor : ℝ) ≤    Real.exp (2 * alpha s * (P H : ℝ) * (rho^(P H))⁻¹)def w {H : ℕ} (η : TailDatum H) : ℕ :=  η.cofactor * ∏ h ∈ Finset.Ico (P H) H, tailPrime η hdef witnesses (H : ℕ) (s : ℝ) (d : ℕ) : Set (TailDatum H) :=  {η | IsWitness H s η ∧ (w η).totient = d}def D {H : ℕ} (h : ℕ) (η : TailDatum H) : ℝ :=  ∑ l ∈ Finset.Ico (P H) H, a (h-l) * tailLog η ldef maxD {H : ℕ} (h : ℕ) (T : Finset (TailDatum H)) : ℝ := by  classical  exact if ht : T.Nonempty then T.sup' ht (D h) else 0def AH (H : ℕ) (f : ℝ → ℝ) (s : ℝ) : ℝ := by  classical  exact rho^(H*(H-1)/2) * (gamma / alpha s)^H *    ∑ᶠ d : ℕ,      if IsTotient d then        f ((ell d : ℝ) / d) / d *          ∑ᶠ T : Finset (TailDatum H),            if T.Nonempty ∧ (↑T : Set (TailDatum H)) ⊆ witnesses H s d then              (-1 : ℝ)^(T.card-1) *               Real.exp (-(gamma / alpha s) *                 ∑' n : ℕ, rho^(H+n) * maxD (H+n) T)            else 0      else 0def A (f : ℝ → ℝ) (s : ℝ) : ℝ := limUnder atTop (fun H : ℕ => AH H f s)def N (k : ℕ) (x : ℝ) : ℝ := by  classical  exact (((Finset.Icc 1 ⌊x⌋₊).filter fun v =>    IsTotient v ∧ (k : ℝ) * x < ell v ∧ (ell v : ℝ) ≤ (k+1 : ℝ) * x).card : ℝ)def fk (k : ℕ) (r : ℝ) : ℝ := min 1 ((k+1 : ℝ) / r) - min 1 ((k : ℝ) / r)/-- Without a totient value whose least preimage exceeds its prescribed multiple,both the weighted count and its limiting coefficient vanish. -/theorem companion_zero_case (k : ℕ) (_hk : 0 < k)    (hno : ¬ ∃ d : ℕ, IsTotient d ∧ k*d < ell d) :    (∀ x : ℝ, 0 < x → N k x = 0) ∧    (∀ s ∈ Set.Ico (0 : ℝ) 1, A (fk k) s = 0) := by  sorryend TotientAsymptoticendend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

An asymptotic formula for the number of totients

Theorem

0<\inf_{0\le s<1}A(1;s) \le\sup_{0\le s<1}A(1;s)<\infty.

With the parameters (eq:renewal)–(eq:Gj),

V(x)\sim\frac{x}{\log x}\,G_m A(1;\theta).

Moreover, for every fixed real c>0,

\frac{V(cx)}{V(x)}\longrightarrow c.

Section 2, The explicit main term Read it in the paper TeX source, line 45

Show the TeX

For sufficiently large positive integers $H$, there are nonnegative
functions $A_H(1;\cdot):[0,1)\to\mathbb R$, given by the finite
arithmetic formula \eqref{eq:AH} below,
which converge uniformly as $H\to\infty$ to a function $A(1;\cdot)$
satisfying
\[
 0<\inf_{0\le s<1}A(1;s)
   \le\sup_{0\le s<1}A(1;s)<\infty.
\]
With the parameters \eqref{eq:renewal}--\eqref{eq:Gj},
\[
 V(x)\sim\frac{x}{\log x}\,G_m A(1;\theta).
\]
Moreover, for every fixed real $c>0$,
\[
 \frac{V(cx)}{V(x)}\longrightarrow c.
\]

Theorem

For each fixed positive integer k, the functions A_H(f_k;s) converge uniformly for 0\le s<1, and

N_k(x)=\frac{x}{\log x}\,G_m\bigl(A(f_k;\theta)+o(1)\bigr).

There are two possibilities.

(i)
If some d\in\mathcal V satisfies \ell(d)>kd, then
\inf_{0\le s<1}A(f_k;s)>0,\qquad N_k(x)\sim\frac{x}{\log x}\,G_m A(f_k;\theta) \asymp_k V(x).
(ii)
If no such d exists, then N_k(x)=0 for every x>0, and A(f_k;s)=0 for every s\in[0,1).

The first alternative holds for k=1,2.

Section 2, The explicit main term Read it in the paper TeX source, line 176

Show the TeX

For each fixed positive integer $k$, the functions $A_H(f_k;s)$ converge uniformly for $0\le s<1$, and
\[
 N_k(x)=\frac{x}{\log x}\,G_m\bigl(A(f_k;\theta)+o(1)\bigr).
\]
There are two possibilities.
\begin{enumerate}[label=\textup{(\roman*)}]
 \item If some $d\in\mathcal V$ satisfies $\ell(d)>kd$, then
 \[
  \inf_{0\le s<1}A(f_k;s)>0,\qquad
  N_k(x)\sim\frac{x}{\log x}\,G_m A(f_k;\theta)
       \asymp_k V(x).
 \]
 \item If no such $d$ exists, then $N_k(x)=0$ for every $x>0$, and $A(f_k;s)=0$ for every $s\in[0,1)$.
\end{enumerate}
The first alternative holds for $k=1,2$.

Configuration and file facts

File: lean/ComparatorChallenges/TotientCompanionZero.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 024 link it.
Solution module: OAI.NumberTheory.TotientAsymptotic.Main
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 22 local vocabulary declarations
Lines of Lean, without comments or blank lines: 64
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

Plectis Atlas