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The entropy photon-number inequality

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

Proves the entropy photon-number inequality for beam-splitter mixing of two independent finite-energy bosonic inputs in any finite number of modes: the output’s entropy photon number is at least the transmissivity-weighted average of the inputs'. Arbitrary entanglement within each input is allowed, and product thermal inputs attain equality even when their entropies differ. (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 two independent assessments disagree on how much of the result described above is expressed by the Lean statements. Read the assessments

What the papers state

Theorem 1.1 (Entropy photon-number inequality)

In The entropy photon-number inequality, 24 September 2026

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

Let n\ge1 be finite, and let \rho_A,\rho_B be states on n bosonic modes, with annihilation operators a_1,\ldots,a_n and b_1,\ldots,b_n, respectively, satisfying

\Tr\rho_A\sum_{j=1}^n a_j^\dagger a_j<\infty, \qquad \Tr\rho_B\sum_{j=1}^n b_j^\dagger b_j<\infty.

The joint input is \rho_A\otimes\rho_B; no independence assumption is made among the modes within either state. For 0\le\eta\le1, let \rho_C be the reduced state of the modes

c_j=\sqrt\eta\,a_j+\sqrt{1-\eta}\,b_j,\qquad 1\le j\le n,

after passive beam-splitter mixing. Then

g^{-1}\!\left(\frac{S(\rho_C)}n\right) \ge \eta\,g^{-1}\!\left(\frac{S(\rho_A)}n\right) +(1-\eta)\,g^{-1}\!\left(\frac{S(\rho_B)}n\right),

where S(\rho)=-\Tr\rho\log\rho.

Corollary 1.2 (Constrained entropy minimum for identical thermal attenuators)

In The entropy photon-number inequality, 24 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

Let n\ge1 be finite, let s,N_B\ge0 be finite, and let 0\le\eta\le1. Every state \rho on n modes with \Tr\rho\sum_{j=1}^n a_j^\dagger a_j<\infty satisfies

\frac1n S\!\left(\mathcal E_{\eta,N_B}^{\otimes n}(\rho)\right) \ge g\!\left(\eta g^{-1}\!\left(\frac{S(\rho)}n\right) +(1-\eta)N_B\right).

In particular, the exact constrained minimum is

\begin{gathered}\min_{\substack{\rho\text{ a state on }n\text{ modes}\\ \Tr\rho\sum_{j=1}^n a_j^\dagger a_j<\infty\\ S(\rho)=ns}} \frac1n S\!\left(\mathcal E_{\eta,N_B}^{\otimes n}(\rho)\right) =g\!\left(\eta g^{-1}(s)+(1-\eta)N_B\right).\end{gathered}

It is attained by the product thermal input \tau_{g^{-1}(s)}. No independence assumption is imposed among the modes of \rho; only the environment is required to be independent of that input.

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. One explanation 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.

The entropy photon-number inequality

24 September 2026, current edition.

Read it herePDF

We prove the entropy photon-number inequality for two independent bosonic inputs with finite mean energy, for every finite number of modes. This resolves the entropy photon-number conjecture in this finite-energy setting while allowing arbitrary entanglement among the modes within either input. As a consequence, we determine the exact minimum output entropy at fixed input entropy for every finite tensor power of an identical thermal attenuator, over finite-energy inputs that may be entangled across modes. We also obtain the exact classical capacity region of the degraded two-receiver pure-loss bosonic broadcast channel under a mean photon constraint.

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.

EntropyPhotonNumber

The paper states

Theorem 1.1 (Entropy photon-number inequality)

In The entropy photon-number inequality, 24 September 2026

Let n\ge1 be finite, and let \rho_A,\rho_B be states on n bosonic modes, with annihilation operators a_1,\ldots,a_n and b_1,\ldots,b_n, respectively, satisfying

\Tr\rho_A\sum_{j=1}^n a_j^\dagger a_j<\infty, \qquad \Tr\rho_B\sum_{j=1}^n b_j^\dagger b_j<\infty.

The joint input is \rho_A\otimes\rho_B; no independence assumption is made among the modes within either state. For 0\le\eta\le1, let \rho_C be the reduced state of the modes

c_j=\sqrt\eta\,a_j+\sqrt{1-\eta}\,b_j,\qquad 1\le j\le n,

after passive beam-splitter mixing. Then

g^{-1}\!\left(\frac{S(\rho_C)}n\right) \ge \eta\,g^{-1}\!\left(\frac{S(\rho_A)}n\right) +(1-\eta)\,g^{-1}\!\left(\frac{S(\rho_B)}n\right),

where S(\rho)=-\Tr\rho\log\rho.

The words they share: inequality, photon, reduced, beam, passive, operators, splitter, bosonic, number, entropy, tr, mixing, finite, sqrt, modes, log, input, state, eta, rho.

Possible Lean counterpart

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

Entropy photon-number inequality for finite-energy bosonic inputs

EntropyPhotonNumber.lean, the release lists it as a supporting statement.

theorem entropy_photon_number_inequality    (n : ℕ) (hn : 1 ≤ n) (ρA ρB ρC : State n)    (hA : FiniteEnergy ρA) (hB : FiniteEnergy ρB)    (η : ℝ) (hη : η ∈ Set.Icc 0 1)    (hC : IsBeamSplitterOutput η ρA ρB ρC) :    η * gInv (entropy ρA / n) + (1 - η) * gInv (entropy ρB / n) ≤      gInv (entropy ρC / 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

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

MathVet and cert-machine disagree on how much of the advertised result the Lean statements express. MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”); cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).

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 or imply the whole advertised result (their term: “full”).

Their note: entropy_photon_number_inequality (n) (hn : 1 ≤ n) (ρA ρB ρC : State n) (hA : FiniteEnergy ρA) (hB : FiniteEnergy ρB) (η ∈ [0,1]) (hC : IsBeamSplitterOutput η ρA ρB ρC) : η * gInv (S ρA / n) + (1 - η) * gInv (S ρB / n) ≤ gInv (S ρC / n) on the untruncated Fock space ℓ²(ℕⁿ) with arbitrary (entangled) inputs and gInv the inverse of the thermal entropy g t = (t+1) log(t+1) - t log t. The summary’s remark that product thermal inputs attain equality is not separately stated. Caveat: the output state is specified only by the hypothesis IsBeamSplitterOutput (explicit matrix-element formula), so the theorem is vacuous if that formula were wrong or unsatisfiable.

Definitions the group asks readers to compare with the paper: Hand-written quantum optics: State n = positive operator with HasSum diagonal 1 on lp ℂ 2 over ℕⁿ, entropy ρ := ∑' k, (entry (cfc (-t log t) ρ.op) k k).re (real tsum, junk 0 if not summable; finite energy ensures it is), gInv s := sInf {t ≥ 0 | s ≤ g t}, and the beam-splitter output defined by explicit coefficients oneModeBeamCoefficient η k e a b (binomial/factorial formula) and IsBeamSplitterOutput: ∀ k l, HasSum (outputBlock..) (entry ρC k l).

“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.

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 — UNDECLARED”).

The gap they name: The equality (sharpness) clause, that product thermal inputs attain equality, is not stated. It is a direct computation rather than a hard theorem, but it is a headline claim.

“NARROWER — UNDECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note does not say 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 formalization proves the entropy photon-number inequality for two independent bosonic inputs with finite mean energy and any finite positive number $n$ of modes. Write $N(\rho)=g^{-1}(S(\rho)/n)$, where $S$ is von Neumann entropy and $g(x)=(x+1)\log(x+1)-x\log x$ is the thermal entropy per mode. For a beam splitter of transmissivity $0\le\eta\le1$, its output satisfies $N(\rho_C)\ge\eta N(\rho_A)+(1-\eta)N(\rho_B)$.

Entanglement among modes within either input is allowed. The thermal-attenuator minimum-output-entropy and broadcast-capacity consequences in the paper are outside this selected inequality.

AI-drafted comparisons, with no recorded human review

  • Entropy photon-number inequality for finite-energy bosonic inputs EntropyPhotonNumber

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: the entropy photon-number inequality for independent finite-energy bosonic inputs on any finite number of modes. The thermal-attenuator and broadcast consequences are outside it.

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.

Entropy photon-number inequality for finite-energy bosonic inputs EntropyPhotonNumber

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 381 s; peak memory 6.6 GiB).10 October 2026Original report

Explanations by others

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

  • 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

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

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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 “Major math journal”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

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 273 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 273 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form