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Classical capacity of generalized amplitude damping

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Determines the unassisted classical capacity of every qubit generalized amplitude-damping channel, including all damping and thermal parameters. An explicit one-variable optimization gives the capacity, attained by independent two-state signal ensembles with collective decoding. Holevo capacity, minimum output entropy and regularized classical capacity are additive when tensoring with any finite-dimensional quantum channel. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement. Read the assessments

What the papers state

Theorem 1.1

In Classical capacity and entropy inequalities for generalized amplitude damping, 24 September 2026

The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.

For all \gamma,\nu\in[0,1] and every integer n\ge1,

\chi(\A_{\gamma,\nu}^{\otimes n}) =n\chi(\A_{\gamma,\nu}) =\frac n{\ln2}\max_{0\le p\le1} \{h((1-\gamma)p+\gamma\nu)-g(v_{\gamma,\nu}(p))\},

where

v_{\gamma,\nu}(p) =\gamma\nu(1-\nu)+\gamma(1-\gamma)(p-\nu)^2.

In particular, C(\A_{\gamma,\nu})=\chi(\A_{\gamma,\nu}). If p maximizes (1.6), independent products of the two equiprobable signals

\phi_\pm=\sqrt{1-p}\,\ket0\ \pm\sqrt p\,\ket1

attain the Holevo information at every block length.

Corollary 1.2 (Additivity with a finite-dimensional partner)

In Classical capacity and entropy inequalities for generalized amplitude damping, 24 September 2026

Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.

Let \gamma,\nu\in[0,1], put \A=\A_{\gamma,\nu}, and let \mathcal N:\mathcal B(\mathcal H_{\mathrm{in}})\to \mathcal B(\mathcal H_{\mathrm{out}}) be completely positive and trace preserving, where \mathcal H_{\mathrm{in}} and \mathcal H_{\mathrm{out}} are nonzero finite-dimensional complex Hilbert spaces. Then

\begin{aligned}\chi(\A\otimes\mathcal N) &=\chi(\A)+\chi(\mathcal N),\\ S_{\min}(\A\otimes\mathcal N) &=S_{\min}(\A)+S_{\min}(\mathcal N),\\ C(\A\otimes\mathcal N) &=C(\A)+C(\mathcal N).\end{aligned}

Here \chi is the unconstrained one-shot Holevo capacity in (1.3), and C is the regularized unassisted classical capacity in (1.4).

No Lean statement in the release is paired with this theorem.

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.

Reported proof verification
One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
Papers and explanations
One current paper, readable here as a page beside its PDF. Two explanations by others.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

Classical capacity and entropy inequalities for generalized amplitude damping

24 September 2026, current edition.

Read it herePDF

We determine the unassisted classical capacity of every qubit generalized amplitude-damping channel, for all damping strengths and thermal occupations. It equals the one-shot Holevo capacity and is given by a one-variable maximum. A binary pure-state ensemble attains the one-use optimum, and its independent products attain the optimum at every block length. We also prove that one-shot Holevo capacity, minimum output entropy, and regularized unassisted classical capacity are additive under tensor product with every finite-dimensional completely positive trace-preserving partner channel.

One Lean statement in this family

Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.

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

AmplitudeDamping

The paper states

Theorem 1.1

In Classical capacity and entropy inequalities for generalized amplitude damping, 24 September 2026

For all \gamma,\nu\in[0,1] and every integer n\ge1,

\chi(\A_{\gamma,\nu}^{\otimes n}) =n\chi(\A_{\gamma,\nu}) =\frac n{\ln2}\max_{0\le p\le1} \{h((1-\gamma)p+\gamma\nu)-g(v_{\gamma,\nu}(p))\},

where

v_{\gamma,\nu}(p) =\gamma\nu(1-\nu)+\gamma(1-\gamma)(p-\nu)^2.

In particular, C(\A_{\gamma,\nu})=\chi(\A_{\gamma,\nu}). If p maximizes (1.6), independent products of the two equiprobable signals

\phi_\pm=\sqrt{1-p}\,\ket0\ \pm\sqrt p\,\ket1

attain the Holevo information at every block length.

The words they share: equiprobable, maximizes, signals, eq, products, sqrt, holevo, integer, generalized, entropy, capacity, amplitude, damping, block, information, attain.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Additivity and capacity for generalized amplitude damping

AmplitudeDamping.lean, the release lists it as a main result.

theorem main (γ ν : ℝ) (hγ : γ ∈ Set.Icc (0 : ℝ) 1)    (hν : ν ∈ Set.Icc (0 : ℝ) 1) :    (∀ n : ℕ, 1 ≤ n → holevo γ ν n = (n : ℝ) * holevo γ ν 1) ∧    (∃ p : ℝ, Maximizes γ ν p ∧      ∀ n : ℕ, 1 ≤ n → holevo γ ν n = (n : ℝ) / Real.log 2 * objective γ ν p) ∧    capacity γ ν = holevo γ ν 1 ∧    (∀ p : ℝ, Maximizes γ ν p → ∀ n : ℕ, 1 ≤ n →      (∀ s : Basis n, IsState (phaseState p s)) ∧      holevo γ ν n = phaseValue γ ν p n) := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.
  • Neal McBurnett, openai-math-lean-tooling records it as “accepted”.

Read the complete Lean statement

Other mathematical statements in OpenAI’s Lean library

Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).

Their note: Stated: GAD.main: for all γ, ν ∈ [0,1], holevo γ ν n = n * holevo γ ν 1 (additivity across n uses of the same channel), ∃ p, Maximizes γ ν p ∧ holevo n = n / log 2 * objective γ ν p (explicit one-variable optimization), capacity γ ν = holevo γ ν 1 (operational capacity as sup of achievable rates with unrestricted collective decoding) and that every maximizing phase-pair product ensemble attains the block value. Not stated: the summary’s additivity of Holevo capacity, minimum output entropy and regularized capacity when tensored with ANY finite-dimensional channel (docs: ‘Additivity with an arbitrary different partner channel... not included’).

Definitions the group asks readers to compare with the paper: Hand-written information theory: entropy P := (trace (cfc Real.negMulLog P)).re, holevo γ ν n := sSup (range ensembleValue) over arbitrary finite ensembles, GAD Kraus operators kraus γ ν r, Achievable/capacity := sSup {R | Achievable γ ν R} with a Tendsto error and liminf-rate condition; standard definitions but not Mathlib’s.

“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.

Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).

The gap they name: Additivity with an arbitrary finite-dimensional partner channel (Holevo capacity, minimum output entropy, regularized capacity) is not stated.

“NARROWER — DECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note says so.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The formalized result determines the unassisted classical capacity of every generalized amplitude-damping channel with parameters in $[0,1]^2$. Its unrestricted Holevo information is additive across every number of repeated uses, equals an attained scalar maximum, and agrees with the operational capacity per use. Every maximizing equiprobable phase-pair product ensemble attains the corresponding block value. Additivity with an arbitrary different partner channel and decoding by separate measurements are not included.

AI-drafted comparisons, with no recorded human review

  • Additivity and capacity for generalized amplitude damping AmplitudeDamping

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: additivity of Holevo information and the classical capacity of every generalised amplitude-damping channel. Additivity with an arbitrary partner channel, the corollary, is not included.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Additivity and capacity for generalized amplitude damping AmplitudeDamping

Three reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report
Neal McBurnett, openai-math-lean-tooling“accepted”Re-run through Comparator with Lean’s kernel: accepted (the check took 612 s; peak memory 6.5 GiB).11 October 2026Original report

Explanations by others

Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.

  • seaworld008, seaworld008/openai-math-top30: thirty selected families explained in Chinese, 7 October 2026. Source

    Chinese-language explainer for general readers, with a beginner’s section, a specialist section, background on why the problem was hard and a diagram; it ends by saying it paraphrases OpenAI’s summary and does not represent consensus.

  • Yixin Ma, OpenAI mathematical physics, 25 families: a Chinese guided reading, 7 October 2026. Source

    Chinese guided reading of this family that compares the manuscripts’ main theorems with their abstracts and flags boundary conditions and limit orders that change the physical reading; drafted with Codex and not a re-proof.

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Papers elsewhere in the release that this family’s papers cite

This family’s papers cite no paper elsewhere in the release.

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

Explain a result, compare statements or propose a correction

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

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Read the Lean statement against the paper

  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form

Identify the theorem the catalogue describes

  • Family 276 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

Address a question raised in an independent assessment

Compare the proposed paper and Lean matches

  • Family 276 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form