`ComparatorChallenges/DilutedSpin.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

22 declarations, 1 contain sorry, 0 declared as axioms

abbrev OAI.DilutedSpinGlass.Spin line 12
Also declared: OAI/Probability/DilutedSpin/Model.lean:10 (same declaration text).
def OAI.DilutedSpinGlass.spin line 14
Also declared: OAI/Probability/DilutedSpin/Model.lean:12 (same declaration text).
abbrev OAI.DilutedSpinGlass.Interaction line 16
Also declared: OAI/Probability/DilutedSpin/Model.lean:14 (same declaration text).
abbrev OAI.DilutedSpinGlass.InteractionSample One interaction together with its factorization witness: theta,a,b,f line 19
Source documentation
One interaction together with its factorization witness: theta,a,b,f.
L4DEL explanation
One interaction together with its factorization witness: theta,a,b,f.
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/Probability/DilutedSpin/Model.lean:17 (same declaration text).
```
abbrev InteractionSample (p : ℕ) :=
…
```
structure OAI.DilutedSpinGlass.Model line 22
Also declared: OAI/Probability/DilutedSpin/Model.lean:20 (same declaration text).
structure OAI.DilutedSpinGlass.Admissible Exactly the standing hypotheses (model:integrability through positivity) line 29
Source documentation
Exactly the standing hypotheses (model:integrability through positivity). norm on the finite function space is max_s |theta(s)|.
L4DEL explanation
Exactly the standing hypotheses (model:integrability through positivity). norm on the finite function space is max_s |theta(s)|.
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/Probability/DilutedSpin/Model.lean:27 (same declaration text).
```
structure Admissible {p : ℕ} (M : Model p) : Prop where
…
```
Mentions: OAI.DilutedSpinGlass.Model
def OAI.DilutedSpinGlass.logWeight Minus the Hamiltonian, with all index choices including repetitions line 54
Source documentation
Minus the Hamiltonian, with all index choices including repetitions.
L4DEL explanation
Minus the Hamiltonian, with all index choices including repetitions.
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/Probability/DilutedSpin/Model.lean:52 (same declaration text).
```
noncomputable def logWeight {p N k : ℕ} (theta : Fin k → InteractionSample p)
    (h : Fin N → ℝ) (indices : Fin k → Fin p → Fin N) (σ : Fin N → Spin) : ℝ :=
…
```
Mentions: OAI.DilutedSpinGlass.InteractionSample, OAI.DilutedSpinGlass.Spin
def OAI.DilutedSpinGlass.logPartition line 58
Also declared: OAI/Probability/DilutedSpin/Model.lean:56 (same declaration text).
def OAI.DilutedSpinGlass.pressure F_N, using an explicit uniform average over the independent index labels line 64
Source documentation
F_N, using an explicit uniform average over the independent index labels. The value at N=0 is irrelevant to the limit; valid system sizes are N>=1.
L4DEL explanation
F_N, using an explicit uniform average over the independent index labels. The value at N=0 is irrelevant to the limit; valid system sizes are N>=1.
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/Probability/DilutedSpin/Model.lean:62 (same declaration text).
```
noncomputable def pressure {p : ℕ} (M : Model p) (N : ℕ) : ℝ :=
…
```
Mentions: OAI.DilutedSpinGlass.Model
def OAI.DilutedSpinGlass.q line 74
Also declared: OAI/Probability/DilutedSpin/Model.lean:72 (same declaration text).
def OAI.DilutedSpinGlass.edge line 77
Also declared: OAI/Probability/DilutedSpin/Model.lean:75 (same declaration text).
def OAI.DilutedSpinGlass.appendSpin Append the distinguished cavity spin in the last position line 81
Source documentation
Append the distinguished cavity spin in the last position.
L4DEL explanation
Append the distinguished cavity spin in the last position.
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/Probability/DilutedSpin/Model.lean:79 (same declaration text).
```
def appendSpin {p : ℕ} (s : Fin (p - 1) → Spin) (ε : Spin) (l : Fin p) : Spin :=
…
```
Mentions: OAI.DilutedSpinGlass.Spin
def OAI.DilutedSpinGlass.message line 84
Also declared: OAI/Probability/DilutedSpin/Model.lean:82 (same declaration text).
def OAI.DilutedSpinGlass.siteLog line 89
Also declared: OAI/Probability/DilutedSpin/Model.lean:87 (same declaration text).
def OAI.DilutedSpinGlass.Hierarchy Nested probability laws: H_0=R and H_{j+1}=P(H_j), with weak topologies line 100
Source documentation
Nested probability laws: H_0=R and H_{j+1}=P(H_j), with weak topologies.
L4DEL explanation
Nested probability laws: H_0=R and H_{j+1}=P(H_j), with weak topologies.
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/Probability/DilutedSpin/Model.lean:98 (same declaration text).
```
noncomputable def Hierarchy : ℕ → TopCat
  | 0 => TopCat.of ℝ
  | r + 1 => TopCat.of (ProbabilityMeasure (Hierarchy r))

/-- Logarithmic form of the conditional power means T_0 using independent
message labels. The current eta's are held fixed; only their descendants
are integrated. Physical disorder is a fixed argument of g throughout. -/
```
Mentions: OAI.DilutedSpinGlass.message
def OAI.DilutedSpinGlass.logMean Logarithmic form of the conditional power means T_0 using independent message labels line 107
Source documentation
Logarithmic form of the conditional power means T_0 using independent message labels. The current eta's are held fixed; only their descendants are integrated. Physical disorder is a fixed argument of g throughout.
L4DEL explanation
Logarithmic form of the conditional power means T_0 using independent message labels. The current eta's are held fixed; only their descendants are integrated. Physical disorder is a fixed argument of g throughout.
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/Probability/DilutedSpin/Model.lean:105 (same declaration text).
```
noncomputable def logMean {ι : Type} [Fintype ι] :
    (r : ℕ) → ((ι → ℝ) → ℝ) → (Fin r → ℝ) → (ι → Hierarchy r) → ℝ
  | 0, g, _, x => g x
  | r + 1, g, m, eta =>
    Real.log (∫ x : ι → Hierarchy r,
      Real.exp (m 0 * logMean r g (fun i => m i.succ) x)
      ∂Measure.pi (fun i => (eta i).toMeasure)) / m 0

/-- Expectation over the root laws eta_0 of each independent message label.
This also gives the r=0 convention (iid real messages, no power means). -/
```
Mentions: OAI.DilutedSpinGlass.Hierarchy, OAI.DilutedSpinGlass.message
def OAI.DilutedSpinGlass.trialLog Expectation over the root laws eta_0 of each independent message label line 117
Source documentation
Expectation over the root laws eta_0 of each independent message label. This also gives the r=0 convention (iid real messages, no power means).
L4DEL explanation
Expectation over the root laws eta_0 of each independent message label. This also gives the r=0 convention (iid real messages, no power means).
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/Probability/DilutedSpin/Model.lean:115 (same declaration text).
```
noncomputable def trialLog {ι : Type} [Fintype ι] (r : ℕ)
    (ζ : Hierarchy (r + 1)) (m : Fin r → ℝ) (g : (ι → ℝ) → ℝ) : ℝ :=
…
```
Mentions: OAI.DilutedSpinGlass.Hierarchy
def OAI.DilutedSpinGlass.Exponents Admissible exponents 0<m_1<...<m_r<1; vacuous at r=0 line 123
Source documentation
Admissible exponents 0<m_1<...<m_r<1; vacuous at r=0.
L4DEL explanation
Admissible exponents 0<m_1<...<m_r<1; vacuous at r=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/Probability/DilutedSpin/Model.lean:121 (same declaration text).
```
def Exponents {r : ℕ} (m : Fin r → ℝ) : Prop :=
…
```
def OAI.DilutedSpinGlass.functional The exact finite-depth cavity functional B_r line 127
Source documentation
The exact finite-depth cavity functional B_r.
L4DEL explanation
The exact finite-depth cavity functional B_r.
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/Probability/DilutedSpin/Model.lean:125 (same declaration text).
```
noncomputable def functional {p : ℕ} (M : Model p) (r : ℕ)
    (ζ : Hierarchy (r + 1)) (m : Fin r → ℝ) : ℝ :=
…
```
Mentions: OAI.DilutedSpinGlass.Model, OAI.DilutedSpinGlass.Hierarchy
def OAI.DilutedSpinGlass.phi line 139
Also declared: OAI/Probability/DilutedSpin/Model.lean:137 (same declaration text).
def OAI.DilutedSpinGlass.variationalValue line 143
Also declared: OAI/Probability/DilutedSpin/Model.lean:141 (same declaration text).
theorem OAI.DilutedSpinGlass.mezard_parisi Challenge statement: The Mézard–Parisi formula for diluted spin glasses sorry line 148
Source documentation
main:theorem. No boundedness of interactions, fields, or messages is added. Existence of the thermodynamic limit is part of this convergence assertion.
L4DEL explanation
The formalization proves the Mézard–Parisi hierarchical cavity formula for diluted even-arity Ising models in the Panchenko–Talagrand class. Under the class's factorization, independence, integrability, and positivity assumptions, the finite-system pressure converges to the infimum of the trial functional over all finite hierarchy depths and trial laws. The arity is any even integer at least two and the interaction density is positive.
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.Probability.DilutedSpin.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Probability/DilutedSpin/Main.lean:12 (same statement text; different proof or body). Qualify references as OAI.DilutedSpinGlass.mezard_parisi@path.
```
theorem mezard_parisi {p : ℕ} (M : Model p) (hM : Admissible M) :
    Tendsto (pressure M) atTop (𝓝 (variationalValue M)) := by
…
```
Mentions: OAI.DilutedSpinGlass.Model, OAI.DilutedSpinGlass.Admissible, OAI.DilutedSpinGlass.pressure, OAI.DilutedSpinGlass.variationalValue