`ComparatorChallenges/AmplitudeDamping.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.

GAD: generalized amplitude damping
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

23 declarations, 1 contain sorry, 0 declared as axioms

abbrev OAI.GAD.Basis line 13
Also declared: OAI/InformationTheory/AmplitudeDamping/Model.lean:13 (same declaration text).
abbrev OAI.GAD.QMatrix line 14
Also declared: OAI/InformationTheory/AmplitudeDamping/Model.lean:14 (same declaration text).
def OAI.GAD.entropy The spectral definition of -Tr(P log P), with 0 log 0 = 0 line 17
Source documentation
The spectral definition of -Tr(P log P), with 0 log 0 = 0.
L4DEL explanation
The spectral definition of -Tr(P log P), with 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/AmplitudeDamping/Model.lean:17 (same declaration text).
```
def entropy {ι : Type u} [Fintype ι] [DecidableEq ι] (P : Matrix ι ι ℂ) : ℝ :=
…
```
def OAI.GAD.IsState line 20
Also declared: OAI/InformationTheory/AmplitudeDamping/Model.lean:20 (same declaration text).
def OAI.GAD.kraus line 24
Also declared: OAI/InformationTheory/AmplitudeDamping/Model.lean:24 (same declaration text).
def OAI.GAD.tensorKraus Tensor products of the single-site Kraus operators, with the full Kraus index line 36
Source documentation
Tensor products of the single-site Kraus operators, with the full Kraus index.
L4DEL explanation
Tensor products of the single-site Kraus operators, with the full Kraus index.
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/AmplitudeDamping/Model.lean:36 (same declaration text).
```
def tensorKraus (γ ν : ℝ) (n : ℕ) (r : Fin n → Fin 4) : QMatrix n :=
…
```
Mentions: OAI.GAD.QMatrix
def OAI.GAD.channel The memoryless n-fold generalized amplitude-damping channel line 40
Source documentation
The memoryless n-fold generalized amplitude-damping channel.
L4DEL explanation
The memoryless n-fold generalized amplitude-damping channel.
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/AmplitudeDamping/Model.lean:40 (same declaration text).
```
def channel (γ ν : ℝ) (n : ℕ) (P : QMatrix n) : QMatrix n :=
…
```
Mentions: OAI.GAD.QMatrix
structure OAI.GAD.Ensemble All finite ensembles, without any product or purity restriction line 45
Source documentation
All finite ensembles, without any product or purity restriction.
L4DEL explanation
All finite ensembles, without any product or purity restriction.
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/AmplitudeDamping/Model.lean:45 (same declaration text).
```
structure Ensemble (n : ℕ) where
…
```
def OAI.GAD.ensembleValue line 53
Also declared: OAI/InformationTheory/AmplitudeDamping/Model.lean:53 (same declaration text).
def OAI.GAD.holevo line 58
Also declared: OAI/InformationTheory/AmplitudeDamping/Model.lean:58 (same declaration text).
structure OAI.GAD.Code An unrestricted n-use quantum code with one collective output POVM line 62
Source documentation
An unrestricted n-use quantum code with one collective output POVM.
L4DEL explanation
An unrestricted n-use quantum code with one collective output POVM.
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/AmplitudeDamping/Model.lean:62 (same declaration text).
```
structure Code (n : ℕ) where
…
```
def OAI.GAD.averageError line 71
Also declared: OAI/InformationTheory/AmplitudeDamping/Model.lean:71 (same declaration text).
def OAI.GAD.Achievable The liminf rate condition written as its eventual lower-bound characterization line 77
Source documentation
The liminf rate condition written as its eventual lower-bound characterization. There is no preshared entanglement, feedback, or restriction on encodings/decoding.
L4DEL explanation
The liminf rate condition written as its eventual lower-bound characterization. There is no preshared entanglement, feedback, or restriction on encodings/decoding.
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/AmplitudeDamping/Model.lean:77 (same declaration text).
```
def Achievable (γ ν R : ℝ) : Prop :=
…
```
def OAI.GAD.capacity Operational unassisted classical capacity, not a definition by regularization line 84
Source documentation
Operational unassisted classical capacity, not a definition by regularization.
L4DEL explanation
Operational unassisted classical capacity, not a definition by regularization.
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/AmplitudeDamping/Model.lean:84 (same declaration text).
```
def capacity (γ ν : ℝ) : ℝ := sSup {R : ℝ | Achievable γ ν R}
```
Mentions: OAI.GAD.Achievable
def OAI.GAD.g line 86
Also declared: OAI/InformationTheory/AmplitudeDamping/Model.lean:86 (same declaration text).
def OAI.GAD.v line 88
Also declared: OAI/InformationTheory/AmplitudeDamping/Model.lean:88 (same declaration text).
def OAI.GAD.objective line 90
Also declared: OAI/InformationTheory/AmplitudeDamping/Model.lean:90 (same declaration text).
def OAI.GAD.Maximizes line 93
Also declared: OAI/InformationTheory/AmplitudeDamping/Model.lean:93 (same declaration text).
def OAI.GAD.phaseSignal line 97
Also declared: OAI/InformationTheory/AmplitudeDamping/Model.lean:97 (same declaration text).
def OAI.GAD.phaseProduct line 101
Also declared: OAI/InformationTheory/AmplitudeDamping/Model.lean:101 (same declaration text).
def OAI.GAD.phaseState line 104
Also declared: OAI/InformationTheory/AmplitudeDamping/Model.lean:104 (same declaration text).
def OAI.GAD.phaseValue Holevo value of the explicit 2^n equiprobable product signals line 108
Source documentation
Holevo value of the explicit 2^n equiprobable product signals.
L4DEL explanation
Holevo value of the explicit 2^n equiprobable product signals.
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/AmplitudeDamping/Model.lean:108 (same declaration text).
```
def phaseValue (γ ν p : ℝ) (n : ℕ) : ℝ :=
…
```
theorem OAI.GAD.main Challenge statement: Exact unassisted classical capacity of generalized amplitude damping sorry line 113
L4DEL explanation
It equals the one-shot Holevo capacity and is given by a one-variable maximum.
— OpenAI generalized-amplitude-damping manuscript (abstract)
We then establish several upper bounds on its classical, quantum, and private capacities.
— Sumeet Khatri, Kunal Sharma, and Mark M. Wilde, Information-theoretic aspects of GAD (2020)
For every damping strength and thermal occupation in $[0,1]$, the unrestricted $n$-use Holevo information is exactly $n$ times its one-use value, an explicit one-variable maximum attains that value, and the operational unassisted classical capacity equals it.
1. holevo γ ν n optimizes over all finite ensembles of $n$-qubit density matrices; inputs need not be pure, product, or equiprobable. The first conjunct proves additivity across every positive block length.
2. The second conjunct supplies a maximizer $p\in[0,1]$ for the scalar objective γ ν p and identifies every block optimum with $n/\log 2$ times that objective.
3. The third conjunct identifies the operational capacity, defined through codes with collective output measurements and vanishing average error, with the one-use Holevo value. The fourth verifies that every maximizing equiprobable phase-pair product ensemble attains every block optimum.
4. Scope boundary: unlike the stronger pinned manuscript abstract, this comparator theorem does not state additivity with an arbitrary different partner channel or minimum-output-entropy additivity. It covers repeated uses of the same generalized amplitude-damping channel.
5. Khatri, Sharma, and Wilde previously established parameter ranges and upper bounds for generalized amplitude-damping capacities. The cited OpenAI result claims an exact unrestricted classical-capacity formula beyond those bounds.
6. Trust boundary: this comparator declaration ends in by sorry. The pinned comparator configuration pairs it with OAI.InformationTheory.AmplitudeDamping.Main; the site's Lean trust result, not this annotation, determines whether that separate solution module checks.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI generalized-amplitude-damping manuscript (abstract), Pinned OpenAI formalization note 276, Sumeet Khatri, Kunal Sharma, and Mark M. Wilde, Information-theoretic aspects of GAD (2020)
Distinct definitions of this name: OAI/InformationTheory/AmplitudeDamping/Main.lean:38 (same statement text; different proof or body). Qualify references as OAI.GAD.main@path.

generalized amplitude-damping channel: a qubit noise channel modeling energy relaxation in contact with a thermal environment
Holevo information: the entropy of an ensemble's average output minus the ensemble-average output entropy
```
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
…
```
Mentions: OAI.GAD.holevo, OAI.GAD.Maximizes, OAI.GAD.objective, OAI.GAD.capacity, OAI.GAD.Basis, OAI.GAD.IsState, OAI.GAD.phaseState, OAI.GAD.phaseValue