`ComparatorChallenges/UniformKServer.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

32 declarations, 1 contain sorry, 0 declared as axioms

structure OAI.UniformKServer.RationalMetric line 12
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:15 (same declaration text).
abbrev OAI.UniformKServer.Configuration line 19
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:22 (same declaration text).
abbrev OAI.UniformKServer.History line 20
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:23 (same declaration text).
def OAI.UniformKServer.serve line 22
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:25 (same declaration text).
def OAI.UniformKServer.costAlong line 25
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:28 (same declaration text).
def OAI.UniformKServer.offlineCost All finite serving-label choices, knowing the entire request word line 31
Source documentation
All finite serving-label choices, knowing the entire request word.
L4DEL explanation
All finite serving-label choices, knowing the entire request word.
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/Combinatorics/UniformKServer/Basic.lean:34 (same declaration text).
```
noncomputable def offlineCost {n k : ℕ} (d : RationalMetric n) (s : Configuration n k)
    (w : List (Fin n)) : ℝ :=
…
```
Mentions: OAI.UniformKServer.RationalMetric, OAI.UniformKServer.Configuration
inductive OAI.UniformKServer.HeadMove line 35
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:60 (same declaration text).
structure OAI.UniformKServer.BitTape line 39
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:64 (same declaration text).
def OAI.UniformKServer.BitTape.blank line 46
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:71 (same declaration text).
def OAI.UniformKServer.BitTape.ofWord line 48
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:73 (same declaration text).
def OAI.UniformKServer.BitTape.shift line 53
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:78 (same declaration text).
structure OAI.UniformKServer.BitAction line 64
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:89 (same declaration text).
structure OAI.UniformKServer.BitMachine line 72
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:97 (same declaration text).
structure OAI.UniformKServer.MachineState line 79
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:104 (same declaration text).
def OAI.UniformKServer.BitMachine.initial line 88
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:113 (same declaration text).
def OAI.UniformKServer.BitMachine.step line 91
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:116 (same declaration text).
def OAI.UniformKServer.BitMachine.run line 102
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:127 (same declaration text).
def OAI.UniformKServer.BitMachine.install At a request boundary only the input/output interface is refreshed line 107
Source documentation
At a request boundary only the input/output interface is refreshed. Work tapes and control retain the complete internal state.
L4DEL explanation
At a request boundary only the input/output interface is refreshed. Work tapes and control retain the complete internal state.
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/Combinatorics/UniformKServer/Basic.lean:132 (same declaration text).
```
def install (M : BitMachine) (s : MachineState M) (input : List Bool) : MachineState M :=
…
```
Mentions: OAI.UniformKServer.BitMachine, OAI.UniformKServer.MachineState
def OAI.UniformKServer.natCode Self-delimiting binary code: ℓ ones, a zero, then ℓ low-to-high bits line 115
Source documentation
Self-delimiting binary code: ℓ ones, a zero, then ℓ low-to-high bits. This fixes one standard binary encoding rather than allowing a nonuniform advice string to be included in the input.
L4DEL explanation
Self-delimiting binary code: ℓ ones, a zero, then ℓ low-to-high bits. This fixes one standard binary encoding rather than allowing a nonuniform advice string to be included in the input.
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/Combinatorics/UniformKServer/Basic.lean:140 (same declaration text).
```
def natCode (a : ℕ) : List Bool :=
…
```
def OAI.UniformKServer.rationalCode line 118
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:143 (same declaration text).
def OAI.UniformKServer.instanceCode line 121
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:146 (same declaration text).
def OAI.UniformKServer.bitValue line 127
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:152 (same declaration text).
def OAI.UniformKServer.outputValue line 131
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:156 (same declaration text).
def OAI.UniformKServer.polynomialBudget A dominating polynomial; every fixed polynomial nonnegative on natural inputs is bounded by one of this form after increasing c and e line 136
Source documentation
A dominating polynomial; every fixed polynomial nonnegative on natural inputs is bounded by one of this form after increasing c and e.
L4DEL explanation
A dominating polynomial; every fixed polynomial nonnegative on natural inputs is bounded by one of this form after increasing c and 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/Combinatorics/UniformKServer/Basic.lean:161 (same declaration text).
```
def polynomialBudget (c e x : ℕ) : ℕ := c * (x + 1)^e
```
def OAI.UniformKServer.requestBudget line 138
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:163 (same declaration text).
def OAI.UniformKServer.boot line 141
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:166 (same declaration text).
def OAI.UniformKServer.requestStep line 146
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:171 (same declaration text).
inductive OAI.UniformKServer.Reachable Reachability includes every binary choice, not merely an almost-sure set line 153
Source documentation
Reachability includes every binary choice, not merely an almost-sure set. The index counts requests already served.
L4DEL explanation
Reachability includes every binary choice, not merely an almost-sure set. The index counts requests already served.
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/Combinatorics/UniformKServer/Basic.lean:178 (same declaration text).
```
inductive Reachable {n k : ℕ} (M : BitMachine) (c e : ℕ)
    (d : RationalMetric n) (s : Configuration n k) : ℕ → MachineState M → Prop
  | boot : Reachable M c e d s 0 (boot M c e d s)
  | next {t z} : Reachable M c e d s t z → (r : Fin n) →
      (bits : Fin (requestBudget c e (instanceCode d s).length (t+1)) → Bool) →
      Reachable M c e d s (t+1)
        (requestStep M c e (instanceCode d s).length (t+1) z r bits)

/-- Modulo makes the semantic recursion total even for unsuccessful machines.
    The main statement separately requires the output to be < k at *every*
    reachable outcome, so no invalid output is accepted or silently repaired. -/
```
Mentions: OAI.UniformKServer.BitMachine, OAI.UniformKServer.RationalMetric, OAI.UniformKServer.Configuration, OAI.UniformKServer.MachineState, OAI.UniformKServer.boot, OAI.UniformKServer.requestBudget, OAI.UniformKServer.instanceCode, OAI.UniformKServer.requestStep
def OAI.UniformKServer.selectedLabel Modulo makes the semantic recursion total even for unsuccessful machines line 164
Source documentation
Modulo makes the semantic recursion total even for unsuccessful machines. The main statement separately requires the output to be < k at every reachable outcome, so no invalid output is accepted or silently repaired.
L4DEL explanation
Modulo makes the semantic recursion total even for unsuccessful machines. The main statement separately requires the output to be < k at every reachable outcome, so no invalid output is accepted or silently repaired.
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/Combinatorics/UniformKServer/Basic.lean:189 (same declaration text).
```
def selectedLabel {M : BitMachine} {k : ℕ} (hk : 0 < k) (s : MachineState M) : Fin k :=
…
```
Mentions: OAI.UniformKServer.BitMachine, OAI.UniformKServer.MachineState
def OAI.UniformKServer.machineExpectedCost line 167
Also declared: OAI/Combinatorics/UniformKServer/Basic.lean:192 (same declaration text).
def OAI.UniformKServer.UniformAlgorithmClaim Full uniform main objective in the explicit bit model, not a theorem or assumption line 184
Source documentation
Full uniform main objective in the explicit bit model, not a theorem or assumption. Machine transition steps are bit operations. Installing the standard input and reading/erasing the output adds only O(L + budget) charged bit operations, hence is absorbed by a single larger polynomial. A single deterministic boot is sufficient for the source construction.
L4DEL explanation
Full uniform main objective in the explicit bit model, not a theorem or assumption. Machine transition steps are bit operations. Installing the standard input and reading/erasing the output adds only O(L + budget) charged bit operations, hence is absorbed by a single larger polynomial. A single deterministic boot is sufficient for the source construction.
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/Combinatorics/UniformKServer/Basic.lean:209 (same declaration text).
```
def UniformAlgorithmClaim : Prop :=
…
```
theorem OAI.UniformKServer.uniform_squared_logarithmic_main Challenge statement: Uniform bit algorithm for the squared-logarithmic $k$-server bound sorry line 200
Source documentation
A single uniform randomized bit machine has squared-logarithmic expected movement and a uniform polynomial bound on each request's bit operations.
L4DEL explanation
A single uniform randomized bit machine has squared-logarithmic expected movement and a uniform polynomial bound on each request's bit operations.
1. This explanation is a condensed rendering of this declaration's own doc comment at the pinned source commit.
2. The linked formalization note covers 2 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
3. The repository's comparator configuration selects OAI.Combinatorics.UniformKServer.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Combinatorics/UniformKServer/Main.lean:25 (same statement text; different proof or body). Qualify references as OAI.UniformKServer.uniform_squared_logarithmic_main@path.
```
theorem uniform_squared_logarithmic_main : UniformAlgorithmClaim := by
…
```
Mentions: OAI.UniformKServer.UniformAlgorithmClaim