`ComparatorChallenges/EntropyPhotonNumber.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 120 curated AI-written annotations and 1273 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 9 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. Those 9 annotations were revised on 2026-10-14 to bind all of them to exact page-one quotations from 6 pinned OpenAI manuscripts, the matching pinned formalization notes, and named historical primary sources; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. The 3 manuscript-grounded matrix-multiplication annotations were composed from pinned OpenAI manuscripts and named historical sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Six additional target-specific annotations for Catalan's constant, the irrationality exponent of pi, dyadic Erdos similarity, Unique Games, quintic Lienard systems, and Yau nodal growth were composed from pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for Seymour's second-neighborhood conjecture, Cannon's conjecture, Koebe circle-domain uniformization, and circle-domain rigidity were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical primary sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for simple Lebesgue spectrum on the three-torus, Rokhlin multiple mixing, the nine-dimensional Borsuk counterexample, and Tingley's sphere-isometry problem were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical primary sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for the Abhyankar-Sathaye counterexample, the circulant Hadamard conjecture, the Eilenberg-Ganea counterexample, and short Egyptian fractions were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and quotations from Sathaye, Turyn with Storer, Eilenberg with Ganea, and Erdos in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for the complete Crouzeix inequality, unrestricted Euclidean five-coloring, the general Mahler conjecture, and Thompson F nonamenability were composed on 2026-10-14 from exact Lean statements, 4 pinned OpenAI manuscripts, the matching formalization notes, and linked historical sources by Crouzeix, de Grey, Mahler, Cannon, Floyd, and Parry; no human review is claimed. Four additional target-specific annotations for algorithmic strong thin trees, Barnette's Hamiltonian-cycle conjecture, the explicit Artin CAT(0) counterexample, and generalized amplitude-damping capacity were composed on 2026-10-14 from exact Lean statements, 4 pinned OpenAI manuscripts, matching formalization notes, and linked historical sources by Anari with Goddyn, Arts with Hopman and Timmermans, Charney, and Khatri with Sharma and Wilde; no human review is claimed. Fourteen solution-level annotations for Seymour's second-neighborhood theorem were added on 2026-10-14 from the pinned 15-page manuscript, covering the exact-neighborhood model, minimal-counterexample reduction, subset deficit, transitive blowup, pruning, extremal contradiction, and proof-bearing final theorem. The subset and blowup annotations also bind exact quotations from Tyler Seacrest (2019) and Brantner, Brockman, Kay, and Snively (2009); no human review is claimed. Thirty-seven declaration-level annotations were added on 2026-10-14 for the rest of ComparatorChallenges/AlgorithmicThinTrees.lean, covering its multigraph semantics, stack-machine cost model, explicit and binary-multiplicity encodings, output certificate, and complete algorithmic proposition. They are bound to the pinned OpenAI manuscript and linked earlier thin-tree sources by Luis Goddyn, Nima Anari, and Shayan Oveis Gharan; the annotation explicitly preserves that the concluding comparator theorem still has body by sorry, and no human review is claimed. Thirty-one supporting declarations in ComparatorChallenges/UniqueGamesTheorem.lean were upgraded or added on 2026-10-14, completing curated coverage of all thirty-two declarations in that file. The annotations cover the finite game model, Boolean formula and parser, binary-vector translations, explicit output serialization, finite-alphabet machine contract, and gap theorem; they distinguish manuscript-backed mathematics from Lean-specific canonical parsing and unary game output encoding, bind exact OpenAI manuscript quotations, and credit Subhash Khot through his 2010 survey. The comparator theorem remains explicitly labeled as having body by sorry; no human review is claimed. The source-derived generator was hardened on 2026-10-14 so a formalization note covering multiple comparator results no longer copies its combined Scope prose onto each declaration; those targets now use declaration-specific source comments when available and otherwise remain explicitly bounded to the exact Lean statement. The remaining source-derived annotations were generated mechanically from pinned formalization notes and source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the source prose.

Credit: OpenAI's pinned repository says an unreleased internal model produced the manuscripts and supporting proof artifacts. The collection builds on the mathematics community's accumulated problems, results, language, and formal libraries; each manuscript's BibTeX and bibliography are the authority for individual scholarly credit.

Project sources: Pinned openai/math documentation, Pinned formalization catalogue

18 declarations, 1 contain sorry, 0 declared as axioms

abbrev OAI.EntropyPhotonNumber.NumberIndex The number-basis indices for n bosonic modes line 12
Source documentation
The number-basis indices for n bosonic modes.
L4DEL explanation
The number-basis indices for n bosonic modes.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/InformationTheory/PhotonNumber/Basic.lean:12 (same declaration text).
```
abbrev NumberIndex (n : ℕ) := Fin n → ℕ
```
abbrev OAI.EntropyPhotonNumber.Fock The full, untruncated bosonic Fock space ℓ²(ℕⁿ, ℂ) line 15
Source documentation
The full, untruncated bosonic Fock space ℓ²(ℕⁿ, ℂ).
L4DEL explanation
The full, untruncated bosonic Fock space ℓ²(ℕⁿ, ℂ).
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/InformationTheory/PhotonNumber/Basic.lean:15 (same declaration text).
```
abbrev Fock (n : ℕ) := lp (fun _ : NumberIndex n => ℂ) 2
```
Mentions: OAI.EntropyPhotonNumber.NumberIndex
def OAI.EntropyPhotonNumber.numberKet A number-basis vector line 18
Source documentation
A number-basis vector.
L4DEL explanation
A number-basis vector.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/InformationTheory/PhotonNumber/Basic.lean:18 (same declaration text).
```
def numberKet {n : ℕ} (k : NumberIndex n) : Fock n := lp.single 2 k 1
```
Mentions: OAI.EntropyPhotonNumber.NumberIndex, OAI.EntropyPhotonNumber.Fock
def OAI.EntropyPhotonNumber.entry Matrix entries of an operator in the number basis line 21
Source documentation
Matrix entries of an operator in the number basis.
L4DEL explanation
Matrix entries of an operator in the number basis.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/InformationTheory/PhotonNumber/Basic.lean:21 (same declaration text).
```
def entry {n : ℕ} (T : Fock n →L[ℂ] Fock n) (k l : NumberIndex n) : ℂ :=
…
```
Mentions: OAI.EntropyPhotonNumber.Fock, OAI.EntropyPhotonNumber.NumberIndex
structure OAI.EntropyPhotonNumber.State Positive trace-one operators line 27
Source documentation
Positive trace-one operators. For a positive bounded operator on this Hilbert space, summability of its diagonal is the positive trace-class criterion; HasSum ... 1 includes summability as well as normalization.
L4DEL explanation
Positive trace-one operators. For a positive bounded operator on this Hilbert space, summability of its diagonal is the positive trace-class criterion; HasSum ... 1 includes summability as well as normalization.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/InformationTheory/PhotonNumber/Basic.lean:27 (same declaration text).
```
structure State (n : ℕ) where
…
```
def OAI.EntropyPhotonNumber.totalNumber Total number of quanta in a number-basis vector line 33
Source documentation
Total number of quanta in a number-basis vector.
L4DEL explanation
Total number of quanta in a number-basis vector.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/InformationTheory/PhotonNumber/Basic.lean:33 (same declaration text).
```
def totalNumber {n : ℕ} (k : NumberIndex n) : ℕ := ∑ j, k j
```
Mentions: OAI.EntropyPhotonNumber.NumberIndex
def OAI.EntropyPhotonNumber.FiniteEnergy Exactly finiteness of the expectation of the nonnegative number operator line 36
Source documentation
Exactly finiteness of the expectation of the nonnegative number operator.
L4DEL explanation
Exactly finiteness of the expectation of the nonnegative number operator.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/InformationTheory/PhotonNumber/Basic.lean:36 (same declaration text).
```
def FiniteEnergy {n : ℕ} (ρ : State n) : Prop :=
…
```
Mentions: OAI.EntropyPhotonNumber.State
def OAI.EntropyPhotonNumber.entropy −Tr ρ log ρ, using the continuous extension of −t log t at zero and continuous functional calculus for the positive operator ρ line 41
Source documentation
−Tr ρ log ρ, using the continuous extension of −t log t at zero and continuous functional calculus for the positive operator ρ.
L4DEL explanation
−Tr ρ log ρ, using the continuous extension of −t log t at zero and continuous functional calculus for the positive operator ρ.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/InformationTheory/PhotonNumber/Basic.lean:41 (same declaration text).
```
def entropy {n : ℕ} (ρ : State n) : ℝ :=
…
```
Mentions: OAI.EntropyPhotonNumber.State
def OAI.EntropyPhotonNumber.g The thermal entropy per mode, with the convention 0 log 0 = 0 line 45
Source documentation
The thermal entropy per mode, with the convention 0 log 0 = 0.
L4DEL explanation
The thermal entropy per mode, with the convention 0 log 0 = 0.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/InformationTheory/PhotonNumber/Basic.lean:46 (same declaration text).
```
def g (t : ℝ) : ℝ := (t + 1) * Real.log (t + 1) - t * Real.log t
```
def OAI.EntropyPhotonNumber.gInv The inverse of increasing g : [0,∞) → [0,∞), expressed by its lower level cut line 49
Source documentation
The inverse of increasing g : [0,∞) → [0,∞), expressed by its lower level cut. Only its values on [0,∞) occur in the main theorem.
L4DEL explanation
The inverse of increasing g : [0,∞) → [0,∞), expressed by its lower level cut. Only its values on [0,∞) occur in the main theorem.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/InformationTheory/PhotonNumber/Basic.lean:50 (same declaration text).
```
def gInv (s : ℝ) : ℝ := sInf {t : ℝ | 0 ≤ t ∧ s ≤ g t}
```
Mentions: OAI.EntropyPhotonNumber.g
def OAI.EntropyPhotonNumber.oneModeBeamCoefficient One-mode passive beam-splitter coefficient, in number bases line 55
Source documentation
One-mode passive beam-splitter coefficient, in number bases. It is the coefficient of (c†)^k (d†)^e in (√η c† − √(1−η) d†)^a (√(1−η) c† + √η d†)^b, with the normalized number-vector factorials.
L4DEL explanation
One-mode passive beam-splitter coefficient, in number bases. It is the coefficient of (c†)^k (d†)^e in (√η c† − √(1−η) d†)^a (√(1−η) c† + √η d†)^b, with the normalized number-vector factorials.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/InformationTheory/PhotonNumber/Basic.lean:56 (same declaration text).
```
def oneModeBeamCoefficient (η : ℝ) (k e a b : ℕ) : ℝ :=
…
```
Mentions: OAI.EntropyPhotonNumber.b
def OAI.EntropyPhotonNumber.beamCoefficient The same passive rotation is performed on all mode indices line 67
Source documentation
The same passive rotation is performed on all mode indices.
L4DEL explanation
The same passive rotation is performed on all mode indices.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/InformationTheory/PhotonNumber/Basic.lean:68 (same declaration text).
```
def beamCoefficient {n : ℕ} (η : ℝ) (k e a b : NumberIndex n) : ℝ :=
…
```
Mentions: OAI.EntropyPhotonNumber.b, OAI.EntropyPhotonNumber.NumberIndex
abbrev OAI.EntropyPhotonNumber.SectorSplit Choices of the first input number in the total-number sector of (k,e) line 71
Source documentation
Choices of the first input number in the total-number sector of (k,e).
L4DEL explanation
Choices of the first input number in the total-number sector of (k,e).
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/InformationTheory/PhotonNumber/Basic.lean:72 (same declaration text).
```
abbrev SectorSplit {n : ℕ} (k e : NumberIndex n) := ∀ j, Fin (k j + e j + 1)
```
Mentions: OAI.EntropyPhotonNumber.NumberIndex
def OAI.EntropyPhotonNumber.splitLeft line 73
Also declared: OAI/InformationTheory/PhotonNumber/Basic.lean:74 (same declaration text).
def OAI.EntropyPhotonNumber.splitRight line 76
Also declared: OAI/InformationTheory/PhotonNumber/Basic.lean:77 (same declaration text).
def OAI.EntropyPhotonNumber.outputBlock A single discarded-number diagonal block of U(ρA ⊗ ρB)U† line 82
Source documentation
A single discarded-number diagonal block of U(ρA ⊗ ρB)U†. The sums are finite because passive mixing preserves number in each paired mode. The two unrestricted density matrices allow all internal entanglement.
L4DEL explanation
A single discarded-number diagonal block of U(ρA ⊗ ρB)U†. The sums are finite because passive mixing preserves number in each paired mode. The two unrestricted density matrices allow all internal entanglement.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/InformationTheory/PhotonNumber/Basic.lean:83 (same declaration text).
```
def outputBlock {n : ℕ} (η : ℝ) (ρA ρB : State n)
    (k l e : NumberIndex n) : ℂ :=
…
```
Mentions: OAI.EntropyPhotonNumber.State, OAI.EntropyPhotonNumber.NumberIndex
def OAI.EntropyPhotonNumber.IsBeamSplitterOutput Reduced state after passive beam-splitter mixing of independent ports line 94
Source documentation
Reduced state after passive beam-splitter mixing of independent ports. The second row of the orthogonal rotation is (-√(1−η),√η); changing its sign has no effect on this reduced state. HasSum records the actual partial trace, not a default value for a possibly divergent series.
L4DEL explanation
Reduced state after passive beam-splitter mixing of independent ports. The second row of the orthogonal rotation is (-√(1−η),√η); changing its sign has no effect on this reduced state. HasSum records the actual partial trace, not a default value for a possibly divergent series.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/InformationTheory/PhotonNumber/Basic.lean:95 (same declaration text).
```
def IsBeamSplitterOutput {n : ℕ} (η : ℝ) (ρA ρB ρC : State n) : Prop :=
…
```
Mentions: OAI.EntropyPhotonNumber.State
theorem OAI.EntropyPhotonNumber.entropy_photon_number_inequality Challenge statement: The entropy photon-number inequality sorry line 99
Source documentation
The entropy photon-number inequality for arbitrary finite-energy inputs and any positive finite number of modes.
L4DEL explanation
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.
1. This is an exact comparator target linked by the pinned repository's formalization note for this result.
2. The repository's comparator configuration selects OAI.InformationTheory.PhotonNumber.Inequality as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/InformationTheory/PhotonNumber/Inequality.lean:689 (same statement text; different proof or body). Qualify references as OAI.EntropyPhotonNumber.entropy_photon_number_inequality@path.
```
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
…
```
Mentions: OAI.EntropyPhotonNumber.State, OAI.EntropyPhotonNumber.FiniteEnergy, OAI.EntropyPhotonNumber.IsBeamSplitterOutput, OAI.EntropyPhotonNumber.gInv, OAI.EntropyPhotonNumber.entropy