`ComparatorChallenges/OneTapeSpace.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

55 declarations, 2 contain sorry, 0 declared as axioms

Finite-machine statements of the capped and unknown-time one-writable-tape simulation theorem. Work space counts the bits in every visited writable cell; public endmarked input tapes are read-only.

inductive OAI.Fifths.Dir line 11
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:22 (same declaration text).
instance OAI.Fifths.(instance at line 15) line 15
def OAI.Fifths.Dir.delta line 18
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:30 (same declaration text).
inductive OAI.Fifths.Mark line 24
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:40 (same declaration text).
def OAI.Fifths.Mark.see line 32
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:48 (same declaration text).
def OAI.Fifths.Mark.walk line 39
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:70 (same declaration text).
inductive OAI.Fifths.Runs line 45
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:89 (same declaration text).
structure OAI.Fifths.Inputs line 51
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:125 (same declaration text).
structure OAI.Fifths.Readers line 55
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:129 (same declaration text).
structure OAI.Fifths.Reads line 59
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:133 (same declaration text).
structure OAI.Fifths.Advances line 63
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:137 (same declaration text).
def OAI.Fifths.Readers.home line 69
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:143 (same declaration text).
def OAI.Fifths.Readers.see line 70
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:144 (same declaration text).
def OAI.Fifths.Readers.walk line 72
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:147 (same declaration text).
structure OAI.Fifths.Tape line 76
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:153 (same declaration text).
structure OAI.Fifths.WriteMove line 80
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:157 (same declaration text).
def OAI.Fifths.Tape.init line 86
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:163 (same declaration text).
def OAI.Fifths.Tape.see line 87
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:164 (same declaration text).
def OAI.Fifths.Tape.perform line 88
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:165 (same declaration text).
structure OAI.Fifths.TAction line 92
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:169 (same declaration text).
structure OAI.Fifths.TM line 97
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:176 (same declaration text).
structure OAI.Fifths.TM.Cfg line 113
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:192 (same declaration text).
def OAI.Fifths.TM.initial line 118
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:197 (same declaration text).
def OAI.Fifths.TM.step line 121
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:200 (same declaration text).
structure OAI.Fifths.TM.Window line 127
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:207 (same declaration text).
def OAI.Fifths.TM.width line 131
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:212 (same declaration text).
def OAI.Fifths.TM.Window.ok line 135
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:220 (same declaration text).
def OAI.Fifths.TM.Window.bits line 138
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:225 (same declaration text).
def OAI.Fifths.TM.Within line 142
Also declared: OAI/Computability/SpaceSimulation/Machine.lean:237 (same declaration text).
structure OAI.Fifths.SingleAct line 150
Also declared: OAI/Computability/SpaceSimulation/OneTape/Single.lean:12 (same declaration text).
structure OAI.Fifths.Single line 155
Also declared: OAI/Computability/SpaceSimulation/OneTape/Single.lean:17 (same declaration text).
instance OAI.Fifths.Single.(instance at line 170) line 170
instance OAI.Fifths.Single.(instance at line 171) line 171
abbrev OAI.Fifths.Single.Tag line 172
Also declared: OAI/Computability/SpaceSimulation/OneTape/Single.lean:35 (same declaration text).
structure OAI.Fifths.Single.At line 174
Also declared: OAI/Computability/SpaceSimulation/OneTape/Single.lean:37 (same declaration text).
def OAI.Fifths.Single.move line 179
Also declared: OAI/Computability/SpaceSimulation/OneTape/Single.lean:42 (same declaration text).
def OAI.Fifths.Single.initial line 185
Also declared: OAI/Computability/SpaceSimulation/OneTape/Single.lean:48 (same declaration text).
def OAI.Fifths.Single.next line 188
Also declared: OAI/Computability/SpaceSimulation/OneTape/Single.lean:51 (same declaration text).
def OAI.Fifths.Single.atTime line 192
Also declared: OAI/Computability/SpaceSimulation/OneTape/Single.lean:55 (same declaration text).
abbrev OAI.Fifths.Single.Outcome line 195
Also declared: OAI/Computability/SpaceSimulation/OneTape/Single.lean:58 (same declaration text).
def OAI.Fifths.Single.finish line 196
Also declared: OAI/Computability/SpaceSimulation/OneTape/Single.lean:60 (same declaration text).
def OAI.Fifths.Single.outcome line 198
Also declared: OAI/Computability/SpaceSimulation/OneTape/Single.lean:62 (same declaration text).
def OAI.Fifths.Single.Inside line 201
Also declared: OAI/Computability/SpaceSimulation/OneTape/Single.lean:79 (same declaration text).
def OAI.Fifths.Entry.code line 209
Also declared: OAI/Computability/SpaceSimulation/OneTape/Layout.lean:9 (same declaration text).
def OAI.Fifths.binaryIn line 213
Distinct definitions of this name: OAI/Computability/SpaceSimulation/Register/Output.lean:11 (different declaration text). Qualify references as OAI.Fifths.binaryIn@path.
def OAI.Fifths.files line 216
Distinct definitions of this name: OAI/Computability/SpaceSimulation/Register/Comp.lean:18 (different declaration text). Qualify references as OAI.Fifths.files@path.
inductive OAI.Fifths.Single.Arg line 223
Also declared: OAI/Computability/SpaceSimulation/OneTape/Supply.lean:57 (same declaration text).
instance OAI.Fifths.Single.(instance at line 227) line 227
abbrev OAI.Fifths.Single.Sym line 229
Also declared: OAI/Computability/SpaceSimulation/OneTape/Supply.lean:63 (same declaration text).
def OAI.Fifths.Single.arg line 231
Also declared: OAI/Computability/SpaceSimulation/OneTape/Supply.lean:65 (same declaration text).
def OAI.Fifths.Cost.Log line 239
Also declared: OAI/Computability/SpaceSimulation/OneTape/Growth.lean:7 (same declaration text).
def OAI.Fifths.Cost.exponent line 240
Also declared: OAI/Computability/SpaceSimulation/OneTape/Growth.lean:8 (same declaration text).
def OAI.Fifths.Single.Access line 246
Also declared: OAI/Computability/SpaceSimulation/Main.lean:20 (same declaration text).
theorem OAI.Fifths.Single.capped Challenge statement: One-tape time simulation in two-fifths-power space sorry line 258
L4DEL explanation
The formalization gives two-fifths-power space simulation of a fixed deterministic machine with one writable tape and finitely many read-only input heads. Heads move at most one cell per step, initial contents are independent of the time bound, and contents and input symbols must be accessible in polylogarithmic space. Given a binary time cap $T\ge2$, the simulator returns the source's finite-control outcome and halting flag using $O(T^{2/5}(\log(T+2))^C)$ space. A second simulator needs no time cap: on an input that halts at time $t$, it returns the halted outcome using $O((t+2)^{2/5}(\log(t+2))^C)$ space. Simulation time is unrestricted. General multitape simulation, hierarchy consequences, and bounded-space-halting results are outside these selected statements.
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.Computability.SpaceSimulation.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Computability/SpaceSimulation/Main.lean:39 (same statement text; different proof or body). Qualify references as OAI.Fifths.Single.capped@path.
```
theorem capped [Fintype X] (A : M.Sym) (p q : ℝ) (hp : 1 ≤ p) (_hq : 1 ≤ q) :
    ∃ (N : TM X Unit M.Outcome) (e : ℕ) (c : ℝ), 0 < c ∧
      ∀ (x : List X) (T : ℕ), 2 ≤ T → M.Access A p q x T →
        ∃ b : ℕ, N.Within (binaryIn x (fun _ => T)) (M.outcome x T) b ∧
          (b:ℝ) ≤ c*(T:ℝ)^exponent*(Log T)^e := by
…
```
Mentions: OAI.Fifths.Single.q, OAI.Fifths.TM, OAI.Fifths.Single.c
theorem OAI.Fifths.Single.unknown Challenge statement: One-tape time simulation in two-fifths-power space sorry line 265
L4DEL explanation
The formalization gives two-fifths-power space simulation of a fixed deterministic machine with one writable tape and finitely many read-only input heads. Heads move at most one cell per step, initial contents are independent of the time bound, and contents and input symbols must be accessible in polylogarithmic space. Given a binary time cap $T\ge2$, the simulator returns the source's finite-control outcome and halting flag using $O(T^{2/5}(\log(T+2))^C)$ space. A second simulator needs no time cap: on an input that halts at time $t$, it returns the halted outcome using $O((t+2)^{2/5}(\log(t+2))^C)$ space. Simulation time is unrestricted. General multitape simulation, hierarchy consequences, and bounded-space-halting results are outside these selected statements.
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.Computability.SpaceSimulation.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Computability/SpaceSimulation/Main.lean:64 (same statement text; different proof or body). Qualify references as OAI.Fifths.Single.unknown@path.
```
theorem unknown [Fintype X] (A : M.Sym) (p q : ℝ) (hp : 1 ≤ p) (_hq : 1 ≤ q) :
    ∃ (N : TM X Empty M.Outcome) (e : ℕ) (c : ℝ), 0 < c ∧
      ∀ (x : List X), (∀ T, 2 ≤ T → M.Access A p q x T) →
        ∀ (t : ℕ), (M.outcome x t).2 = true →
          ∃ b, N.Within (files x) (M.outcome x t) b ∧
            (b:ℝ) ≤ c*((t:ℝ)+2)^exponent*(Log t)^e := by
…
```
Mentions: OAI.Fifths.Single.q, OAI.Fifths.TM, OAI.Fifths.Single.c