`ComparatorChallenges/AlgorithmicThinTrees.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

38 declarations, 1 contain sorry, 0 declared as axioms

Deterministic polynomial-time construction of strong thin trees.

structure OAI.StrongThinTree.MultiGraph A finite loopless multigraph with individually indexed edge copies line 9
L4DEL explanation
every parallel copy counts separately
— OpenAI algorithmic strong-thin-tree manuscript
MultiGraph n m has vertices Fin n and edge copies Fin m. Each edge copy has a left and right endpoint, and loopless rules out an edge whose two endpoints coincide.
1. Parallel edges are allowed because two different members of Fin m may have the same endpoint pair.
2. The edge index is therefore an identity for a copy, not merely a pair of vertices. That distinction matters when cuts, trees, and binary multiplicities count parallel copies.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: ComparatorChallenges/StrongThinTree.lean:7 (same declaration text), OAI/Combinatorics/ThinTrees/Model.lean:7 (same declaration text).
```
structure MultiGraph (n m : ℕ) where
…
```
def OAI.StrongThinTree.MultiGraph.cut The selected edges crossing a vertex cut line 16
L4DEL explanation
every parallel copy counts separately
— OpenAI algorithmic strong-thin-tree manuscript
G.cut T S keeps exactly those edge copies in T with one endpoint in the vertex set S and the other endpoint outside it.
1. The two disjuncts cover both endpoint orientations, so the definition describes an undirected cut even though every edge stores left and right endpoints.
2. Supplying T = Finset.univ gives the whole graph cut. Supplying a proposed tree edge set gives the part of that cut used by the tree.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: ComparatorChallenges/StrongThinTree.lean:14 (same declaration text), OAI/Combinatorics/ThinTrees/Model.lean:14 (same declaration text).
```
def cut {n m : ℕ} (G : MultiGraph n m) (T : Finset (Fin m))
    (S : Finset (Fin n)) : Finset (Fin m) :=
…
```
Mentions: OAI.StrongThinTree.MultiGraph
def OAI.StrongThinTree.MultiGraph.Connected Connectivity of a chosen edge set line 21
L4DEL explanation
every parallel copy counts separately
— OpenAI algorithmic strong-thin-tree manuscript
G.Connected T says every nonempty proper set of vertices has at least one edge of T crossing to its complement.
1. This cut condition is the finite-graph characterization of connectedness; it is stated directly so it also applies to a selected subset of multigraph edge copies.
2. Empty and full vertex sets are excluded because their cuts are necessarily empty and say nothing about connectivity.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: ComparatorChallenges/StrongThinTree.lean:19 (same declaration text), OAI/Combinatorics/ThinTrees/Model.lean:19 (same declaration text).
```
def Connected {n m : ℕ} (G : MultiGraph n m) (T : Finset (Fin m)) : Prop :=
…
```
Mentions: OAI.StrongThinTree.MultiGraph
def OAI.StrongThinTree.MultiGraph.SpanningTree A minimally connected spanning edge set line 24
L4DEL explanation
The problem is to make the same tree thin on every cocircuit B.
— Luis Goddyn, Thin Trees conjecture page (archived)
G.SpanningTree T means T connects all vertices and deleting any one edge of T destroys that connectivity.
1. The vertex set is already all of Fin n; “spanning” is expressed by testing cuts of the entire vertex type rather than by storing a separate vertex subset.
2. Minimal connectedness supplies the acyclicity normally used in the definition of a finite spanning tree.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), Luis Goddyn, Thin Trees conjecture page (archived)
Also declared: ComparatorChallenges/StrongThinTree.lean:22 (same declaration text), OAI/Combinatorics/ThinTrees/Model.lean:22 (same declaration text).
```
def SpanningTree {n m : ℕ} (G : MultiGraph n m) (T : Finset (Fin m)) : Prop :=
…
```
Mentions: OAI.StrongThinTree.MultiGraph
def OAI.StrongThinTree.MultiGraph.EdgeConnected At least $k$ edge copies cross every nontrivial cut line 27
L4DEL explanation
any k-edge-connected unweighted graph has a polyloglog(n)/k-thin spanning tree.
— Nima Anari and Shayan Oveis Gharan, Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and Asymmetric TSP
G.EdgeConnected k says each nonempty proper vertex set has a whole-graph cut containing at least k edge copies.
1. The call G.cut Finset.univ S counts every indexed edge copy of G, including parallel copies.
2. This is the hypothesis that provides the denominator scale in the desired $C/k$ thinness bound.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), Nima Anari and Shayan Oveis Gharan, Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and Asymmetric TSP
Also declared: ComparatorChallenges/StrongThinTree.lean:25 (same declaration text), OAI/Combinatorics/ThinTrees/Model.lean:25 (same declaration text).
```
def EdgeConnected {n m : ℕ} (G : MultiGraph n m) (k : ℕ) : Prop :=
…
```
Mentions: OAI.StrongThinTree.MultiGraph
def OAI.CurrentKS.frame A self-delimiting Boolean-list frame line 36
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
frame encodes each payload bit as false followed by that bit, then terminates the framed list with true.
1. The empty payload becomes [true]; a nonempty payload begins with false, so a decoder can distinguish continuation from termination without an external length.
2. This is a serialization helper for the formal stack-machine input, not a mathematical ingredient of the thin-tree theorem.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Analysis/MatrixSigning/Model.lean:8 (same declaration text).
```
def frame : List Bool → List Bool
  | [] => [true]
  | b :: bs => false :: b :: frame bs
```
def OAI.CurrentKS.encodeNat A framed bit encoding of a natural number line 40
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
encodeNat n applies the file's self-delimiting frame to Mathlib's computability-oriented Boolean encoding of n.
1. Framing lets multiple natural-number encodings be concatenated while preserving their boundaries.
2. The theorem measures runtime against the length of the resulting encoded input, so this choice participates in the formal complexity statement.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Analysis/MatrixSigning/Model.lean:12 (same declaration text).
```
def encodeNat (n : ℕ) : List Bool := frame (Computability.encodeNat n)
```
Mentions: OAI.CurrentKS.frame
inductive OAI.CurrentKS.StackAction The three permitted updates to one Boolean stack line 42
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
A StackAction either leaves a stack alone, removes its head, or pushes one specified Boolean bit.
1. An instruction chooses one such action independently for each stack.
2. This deliberately small instruction language makes the existential algorithm a concrete finite machine rather than an unspecified Lean function.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Analysis/MatrixSigning/Model.lean:75 (same declaration text).
```
inductive StackAction where
…
```
def OAI.CurrentKS.StackAction.apply Execute one action on one Boolean stack line 48
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
stay returns the stack unchanged, pop takes its tail, and push b prepends b.
1. Lean's List.tail makes popping an empty stack return the empty stack, so this transition operation is total.
2. Machine.step applies these actions simultaneously to all indexed stacks.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Analysis/MatrixSigning/Model.lean:81 (same declaration text).
```
def StackAction.apply : StackAction → List Bool → List Bool
  | .stay, xs => xs
  | .pop, xs => xs.tail
  | .push b, xs => b :: xs
```
Mentions: OAI.CurrentKS.StackAction
structure OAI.CurrentKS.Instruction One transition of a finite multi-stack machine line 53
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
An Instruction k q contains an optional next state among q states and one stack action for each of k stacks.
1. A present next state continues computation; none represents a halted configuration.
2. Because both the state and stack indices are finite, a Machine can store a finite transition table for these instructions.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Analysis/MatrixSigning/Model.lean:86 (same declaration text).
```
structure Instruction (k q : ℕ) where
…
```
def OAI.CurrentKS.Instruction.halt The default halting instruction line 57
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
Instruction.halt k q sets the next state to none and leaves every stack unchanged.
1. Machine.step uses this instruction when a computed transition-table index is absent.
2. Thus an incomplete finite table has a defined, conservative behavior rather than an out-of-bounds transition.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Analysis/MatrixSigning/Model.lean:90 (same declaration text).
```
def Instruction.halt (k q : ℕ) : Instruction k q := ⟨none, fun _ => .stay⟩
```
Mentions: OAI.CurrentKS.Instruction
structure OAI.CurrentKS.Machine A finite Boolean-stack machine with designated input and output stacks line 59
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
Machine records finite numbers of stacks and states, an initial state, designated input and output stacks, and a finite instruction table.
1. The fields use Fin indices, so constructing a machine also supplies proofs that its initial state and designated stacks exist within the declared sizes.
2. This is the formalization's computational model for “deterministic algorithm”; the manuscript does not claim that its mathematical algorithm is literally presented as this Lean table.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Analysis/MatrixSigning/Model.lean:92 (same declaration text).
```
structure Machine where
…
```
structure OAI.CurrentKS.Config A machine state together with all current stacks line 67
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
Config k q stores either a current state among q states or none for halting, plus one Boolean list for each of k stacks.
1. A configuration is the complete instantaneous state consumed and produced by Machine.step.
2. Halting is represented inside the configuration, allowing the final proposition to assert result.state = none after bounded fuel.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Analysis/MatrixSigning/Model.lean:100 (same declaration text).
```
structure Config (k q : ℕ) where
…
```
def OAI.CurrentKS.topCode Encode a stack's visible head as one of three cases line 71
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
topCode returns 0 for an empty stack, 1 for a leading false, and 2 for a leading true.
1. Only the top cell is visible to a single machine transition.
2. The three-valued code is one base-$3$ digit used by headsCode to select a table instruction.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Analysis/MatrixSigning/Model.lean:104 (same declaration text).
```
def topCode : List Bool → ℕ
  | [] => 0
  | false :: _ => 1
  | true :: _ => 2
```
def OAI.CurrentKS.headsCode Combine all visible stack heads into one base-$3$ number line 76
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
headsCode k s recursively packs the empty/false/true status of each of the k stack heads into a natural number.
1. Stack zero is the least significant base-$3$ digit; subsequent stacks contribute after multiplication by 3.
2. The result ranges over the observation patterns used to index the machine's finite transition table.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Analysis/MatrixSigning/Model.lean:109 (same declaration text).
```
def headsCode : (k : ℕ) → (Fin k → List Bool) → ℕ
  | 0, _ => 0
  | k + 1, s => topCode (s 0) + 3 * headsCode k (fun j => s j.succ)
```
Mentions: OAI.CurrentKS.topCode
def OAI.CurrentKS.Machine.step Perform one deterministic machine transition line 80
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
A halted configuration stays fixed. Otherwise the current state and all visible stack heads select one instruction, whose next state and stack actions produce the next configuration.
1. The table index is 3 ^ stackCount * state + headsCode, separating each state's block of stack-head patterns.
2. getD chooses the halting instruction when that index lies beyond the finite table, and each selected stack action is applied to its corresponding stack.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Analysis/MatrixSigning/Model.lean:113 (same declaration text).
```
def Machine.step (M : Machine) (c : Config M.stackCount M.states) :
    Config M.stackCount M.states :=
…
```
Mentions: OAI.CurrentKS.Machine, OAI.CurrentKS.Config
def OAI.CurrentKS.Machine.init Initialize the machine from an encoded input line 90
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
M.init input starts in M.initial, places the entire Boolean input on M.inputStack, and makes every other stack empty.
1. The output stack is not granted any special initial contents; it is merely the stack read after the bounded run.
2. If input and output designations coincide, the structure permits that explicitly rather than assuming distinct stacks.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Analysis/MatrixSigning/Model.lean:123 (same declaration text).
```
def Machine.init (M : Machine) (input : List Bool) : Config M.stackCount M.states :=
…
```
Mentions: OAI.CurrentKS.Machine, OAI.CurrentKS.Config
def OAI.CurrentKS.Machine.run Run a machine for exactly the supplied fuel line 93
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
M.run input fuel initializes M and iterates its one-step transition exactly fuel times.
1. Because halted configurations are fixed points of step, extra fuel after halting does not change the result.
2. The final proposition supplies polynomial fuel and then explicitly requires the resulting state to be none; fuel alone does not assert termination.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Analysis/MatrixSigning/Model.lean:126 (same declaration text).
```
def Machine.run (M : Machine) (input : List Bool) (fuel : ℕ) :
    Config M.stackCount M.states := (M.step^[fuel]) (M.init input)
…
```
Mentions: OAI.CurrentKS.Machine, OAI.CurrentKS.Config
structure OAI.AlgorithmicThinTrees.ExplicitInput A thin-tree instance with every edge copy listed explicitly line 101
L4DEL explanation
every parallel copy counts separately
— OpenAI algorithmic strong-thin-tree manuscript
ExplicitInput packages the number of vertices n, number of edge copies m, connectivity parameter k, and an actual MultiGraph n m.
1. Parallel copies occupy separate positions of Fin m, so the input size grows with the total expanded multiplicity.
2. This representation is one branch of the common Input type handled by the claimed machine.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:13 (same declaration text).
```
structure ExplicitInput where
…
```
structure OAI.AlgorithmicThinTrees.BinaryInput A multiplicity-compressed multigraph input line 107
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
BinaryInput lists p distinct ordered endpoint pairs and assigns a natural-number multiplicity to each pair instead of enumerating every parallel edge copy.
1. ordered proves the endpoints differ and chooses a canonical orientation left < right; distinct prevents the same unordered pair from appearing twice.
2. Large multiplicities remain compact when BinaryInput.encode serializes those natural numbers in binary-oriented form.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:19 (same declaration text).
```
structure BinaryInput where
…
```
abbrev OAI.AlgorithmicThinTrees.BinaryInput.Edge One concrete edge copy in a compressed binary input line 117
L4DEL explanation
every parallel copy counts separately
— OpenAI algorithmic strong-thin-tree manuscript
x.Edge is a dependent pair: an endpoint-pair index i, followed by a copy index below x.multiplicity i.
1. A pair with multiplicity zero contributes no inhabitants because Fin 0 is empty.
2. This expands multiplicities mathematically while the serialized input can remain compressed.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:29 (same declaration text).
```
abbrev BinaryInput.Edge (x : BinaryInput) := Σ i : Fin x.p, Fin (x.multiplicity i)
```
Mentions: OAI.AlgorithmicThinTrees.BinaryInput
def OAI.AlgorithmicThinTrees.BinaryInput.edgeEquiv Number all expanded binary-input edge copies line 119
L4DEL explanation
every parallel copy counts separately
— OpenAI algorithmic strong-thin-tree manuscript
edgeEquiv chooses an equivalence between the finite type x.Edge and consecutive indices Fin (Fintype.card x.Edge).
1. It is noncomputable because Lean selects a finite equivalence abstractly rather than defining a particular executable ordering.
2. Later definitions use this equivalence consistently to build a MultiGraph and translate decoded output edges back to its indices.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:31 (same declaration text).
```
noncomputable def BinaryInput.edgeEquiv (x : BinaryInput) :
    x.Edge ≃ Fin (Fintype.card x.Edge) := Fintype.equivFin x.Edge
```
Mentions: OAI.AlgorithmicThinTrees.BinaryInput
def OAI.AlgorithmicThinTrees.BinaryInput.graph Expand compressed multiplicities into a loopless multigraph line 122
L4DEL explanation
every parallel copy counts separately
— OpenAI algorithmic strong-thin-tree manuscript
x.graph turns every inhabitant of x.Edge into its own indexed edge copy and assigns it the endpoints of its pair index.
1. The inverse of edgeEquiv recovers the dependent pair from a consecutive edge index.
2. The proof left < right supplied by BinaryInput.ordered establishes that every expanded copy is loopless.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:34 (same declaration text).
```
noncomputable def BinaryInput.graph (x : BinaryInput) :
    MultiGraph x.n (Fintype.card x.Edge) where
…
```
Mentions: OAI.AlgorithmicThinTrees.BinaryInput
inductive OAI.AlgorithmicThinTrees.Input Either an explicit or multiplicity-compressed graph instance line 128
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
Input is a tagged union whose explicit branch contains every edge copy and whose binary branch contains endpoint pairs with multiplicities.
1. Subsequent definitions pattern-match on this tag so validation, size, decoding, and correctness have one common interface.
2. The final algorithmic proposition requires one fixed machine to handle both representations.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:40 (same declaration text).
```
inductive Input where
…
```
def OAI.AlgorithmicThinTrees.ExplicitInput.encode Serialize an explicitly listed multigraph line 132
L4DEL explanation
every parallel copy counts separately
— OpenAI algorithmic strong-thin-tree manuscript
The encoding concatenates framed natural numbers for n, m, and k, then for every edge index concatenates its two endpoint indices.
1. List.ofFn traverses the finite edge-index type, and flatten joins the per-edge encodings into one Boolean list.
2. Since every parallel copy appears separately, the encoded length reflects the expanded edge count.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:44 (same declaration text).
```
def ExplicitInput.encode (x : ExplicitInput) : List Bool :=
…
```
Mentions: OAI.AlgorithmicThinTrees.ExplicitInput
def OAI.AlgorithmicThinTrees.BinaryInput.encode Serialize a multiplicity-compressed multigraph line 138
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
The encoding writes n, the number p of endpoint pairs, and k, followed by each pair's endpoints and multiplicity.
1. It does not emit one record per edge copy; a multiplicity is emitted once as a framed natural number.
2. The theorem's runtime bound uses this compressed Boolean-list length, so it is stronger than polynomial time in an expanded graph when multiplicities are huge.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:50 (same declaration text).
```
def BinaryInput.encode (x : BinaryInput) : List Bool :=
…
```
Mentions: OAI.AlgorithmicThinTrees.BinaryInput
def OAI.AlgorithmicThinTrees.Input.encode Tag and serialize either input representation line 144
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
Input.encode prefixes false for an explicit input and true for a binary-multiplicity input, then appends that branch's encoding.
1. The first bit makes the two formats distinguishable to the one machine required by AlgorithmicStrongThinTrees.
2. The branch encoders then provide self-delimiting framed naturals for the graph data.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:56 (same declaration text).
```
def Input.encode : Input → List Bool
  | .explicit x => false :: x.encode
  | .binary x => true :: x.encode
```
Mentions: OAI.AlgorithmicThinTrees.Input
def OAI.AlgorithmicThinTrees.Input.length The bit length used by the polynomial runtime bound line 148
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
x.length is exactly the length of x.encode, including the representation tag and all framing bits.
1. AlgorithmicStrongThinTrees supplies fuel polynomial in x.length + 1.
2. This is an encoding-dependent complexity measure; it is not the number of vertices, endpoint pairs, or expanded edge copies alone.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:60 (same declaration text).
```
def Input.length (x : Input) : ℕ := x.encode.length
```
Mentions: OAI.AlgorithmicThinTrees.Input
def OAI.AlgorithmicThinTrees.Input.n Read the vertex count from either input branch line 150
L4DEL explanation
in time polynomial in the total binary input length
— OpenAI algorithmic strong-thin-tree manuscript
Input.n returns the stored number of vertices whether the graph was represented explicitly or by binary multiplicities.
1. GoodOutput uses this common projection to impose the special empty output for a one-vertex graph.
2. No decoding or graph expansion is needed to obtain this value.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:62 (same declaration text).
```
def Input.n : Input → ℕ
  | .explicit x => x.n
  | .binary x => x.n
```
Mentions: OAI.AlgorithmicThinTrees.Input
def OAI.AlgorithmicThinTrees.Valid The admissible $k$-edge-connected inputs line 154
L4DEL explanation
any k-edge-connected unweighted graph has a polyloglog(n)/k-thin spanning tree.
— Nima Anari and Shayan Oveis Gharan, Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and Asymmetric TSP
An input is Valid when it has at least one vertex, has positive k, and its explicit or expanded multigraph is $k$-edge-connected.
1. The two branches impose the same mathematical conditions after the binary branch expands its multiplicities into edge copies.
2. The final proposition promises behavior only under this predicate; malformed or insufficiently connected encodings are outside its guarantee.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), Nima Anari and Shayan Oveis Gharan, Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and Asymmetric TSP
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:66 (same declaration text).
```
def Valid : Input → Prop
  | .explicit x => 1 ≤ x.n ∧ 1 ≤ x.k ∧ x.graph.EdgeConnected x.k
  | .binary x => 1 ≤ x.n ∧ 1 ≤ x.k ∧ x.graph.EdgeConnected x.k
```
Mentions: OAI.AlgorithmicThinTrees.Input
def OAI.AlgorithmicThinTrees.ThinTree A spanning tree uniformly thin across every cut line 158
L4DEL explanation
The problem is to make the same tree thin on every cocircuit B.
— Luis Goddyn, Thin Trees conjecture page (archived)
any k-edge-connected unweighted graph has a polyloglog(n)/k-thin spanning tree.
— Nima Anari and Shayan Oveis Gharan, Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and Asymmetric TSP
ThinTree C k G T requires T to be a spanning tree and to use at most a $C/k$ fraction of the whole graph's edges across every nontrivial vertex cut.
1. Both sides count indexed edge copies, so parallel multiplicity contributes to the tree cut and the graph cut in the same model.
2. The same T must satisfy the inequality for all cuts; it is not allowed to choose a different tree for each cut.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), Luis Goddyn, Thin Trees conjecture page (archived), Nima Anari and Shayan Oveis Gharan, Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and Asymmetric TSP
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:70 (same declaration text).
```
def ThinTree {n m : ℕ} (C : ℝ) (k : ℕ) (G : MultiGraph n m)
    (T : Finset (Fin m)) : Prop :=
…
```
def OAI.AlgorithmicThinTrees.ExplicitInput.encodeEdges Serialize a list of explicit edge indices line 163
L4DEL explanation
every parallel copy counts separately
— OpenAI algorithmic strong-thin-tree manuscript
encodeEdges concatenates the framed natural-number value of every explicit edge index in the proposed output list.
1. The output carries only edge identities; the input graph supplies their endpoints.
2. TreeOutput separately requires the decoded list to contain no duplicate edge indices.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:75 (same declaration text).
```
def ExplicitInput.encodeEdges (x : ExplicitInput) (edges : List (Fin x.m)) : List Bool :=
…
```
Mentions: OAI.AlgorithmicThinTrees.ExplicitInput
def OAI.AlgorithmicThinTrees.BinaryInput.encodeEdges Serialize selected copies from a compressed multigraph line 166
L4DEL explanation
every parallel copy counts separately
— OpenAI algorithmic strong-thin-tree manuscript
Each output edge is encoded by its endpoint-pair index and its copy index within that pair's multiplicity.
1. This pair of naturals identifies one inhabitant of the dependent edge-copy type x.Edge.
2. The encoding therefore distinguishes parallel copies without expanding the input representation itself.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:78 (same declaration text).
```
def BinaryInput.encodeEdges (x : BinaryInput) (edges : List x.Edge) : List Bool :=
…
```
Mentions: OAI.AlgorithmicThinTrees.BinaryInput
def OAI.AlgorithmicThinTrees.BinaryInput.selected Convert decoded binary-input edge copies to multigraph indices line 169
L4DEL explanation
every parallel copy counts separately
— OpenAI algorithmic strong-thin-tree manuscript
selected removes repetitions from a decoded edge list and maps each dependent edge copy through edgeEquiv into the Fin indices used by x.graph.
1. The result is a Finset, which is the representation expected by SpanningTree, cut, and ThinTree.
2. This definition is noncomputable for the same reason as edgeEquiv; it specifies output correctness rather than implementing the stack machine's decoder.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:81 (same declaration text).
```
noncomputable def BinaryInput.selected (x : BinaryInput) (edges : List x.Edge) :
    Finset (Fin (Fintype.card x.Edge)) := edges.toFinset.map x.edgeEquiv.toEmbedding
```
Mentions: OAI.AlgorithmicThinTrees.BinaryInput
def OAI.AlgorithmicThinTrees.TreeOutput The output bits encode a duplicate-free thin spanning tree line 172
L4DEL explanation
meeting every cut in at most a C/k fraction
— OpenAI algorithmic strong-thin-tree manuscript
TreeOutput C x bits means there exists a duplicate-free list of valid edge copies whose branch-specific serialization is exactly bits and whose selected edge set is ThinTree C.
1. In the explicit branch, edge indices already refer directly to x.graph; in the binary branch, dependent edge copies are translated by x.selected.
2. The existential decoded list is the certificate connecting raw output bits to the mathematical thin-tree property.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:84 (same declaration text).
```
def TreeOutput (C : ℝ) : Input → List Bool → Prop
  | .explicit x, bits => ∃ edges : List (Fin x.m), edges.Nodup ∧
      bits = x.encodeEdges edges ∧ ThinTree C x.k x.graph edges.toFinset
  | .binary x, bits => ∃ edges : List x.Edge, edges.Nodup ∧
      bits = x.encodeEdges edges ∧ ThinTree C x.k x.graph (x.selected edges)
```
Mentions: OAI.AlgorithmicThinTrees.Input, OAI.AlgorithmicThinTrees.ThinTree
def OAI.AlgorithmicThinTrees.GoodOutput A valid tree output with a canonical one-vertex case line 178
L4DEL explanation
meeting every cut in at most a C/k fraction
— OpenAI algorithmic strong-thin-tree manuscript
GoodOutput C x bits requires a TreeOutput and additionally requires bits = [] whenever the input has exactly one vertex.
1. A one-vertex graph's spanning tree has no edges, and the extra clause fixes its serialization to the empty Boolean list.
2. For larger inputs, all output requirements come from TreeOutput.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:90 (same declaration text).
```
def GoodOutput (C : ℝ) (x : Input) (bits : List Bool) : Prop :=
…
```
Mentions: OAI.AlgorithmicThinTrees.Input
def OAI.AlgorithmicThinTrees.AlgorithmicStrongThinTrees The complete deterministic polynomial-time strong thin-tree proposition line 181
L4DEL explanation
meeting every cut in at most a C/k fraction
— OpenAI algorithmic strong-thin-tree manuscript
any k-edge-connected unweighted graph has a polyloglog(n)/k-thin spanning tree.
— Nima Anari and Shayan Oveis Gharan, Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and Asymmetric TSP
There exist one positive real constant C, one fixed finite stack machine M, and natural numbers a and degree with a > 0 such that every valid input is solved within a * (length + 1) ^ degree steps.
1. The result after that exact fuel must be halted, and the designated output stack must satisfy GoodOutput C for the original input.
2. Quantifying over C, M, a, and degree before x makes them uniform across all graph sizes, connectivities, and both encodings.
3. This proposition specifies an algorithm and its certificate format. It does not itself construct the machine; the following theorem is intended to prove this proposition but currently has body by sorry.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript, Nima Anari and Shayan Oveis Gharan, Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and Asymmetric TSP
Also declared: OAI/Combinatorics/ThinTrees/Encoding/Specification.lean:93 (same declaration text).
```
def AlgorithmicStrongThinTrees : Prop :=
…
```
theorem OAI.AlgorithmicThinTrees.algorithmic_strong_thin_trees Challenge statement: A deterministic polynomial-time strong thin-tree construction sorry line 191
L4DEL explanation
meeting every cut in at most a C/k fraction
— OpenAI algorithmic strong-thin-tree manuscript
Every k-edge connected graph has O(1/k)-thin spanning tree.
— Nima Anari, Thin Trees and Interlacing Families lecture slides (Simons Institute)
There is one universal constant $C>0$ and one deterministic finite stack machine that, from either of the statement's two encodings of any finite loopless $k$-edge-connected multigraph, halts in polynomially many steps and outputs a spanning tree whose share of every nontrivial cut is at most $C/k$ times the graph's share of that cut.
1. Input has two cases. explicit lists every edge separately; binary lists distinct endpoint pairs and binary-encoded parallel-edge multiplicities. Valid requires at least one vertex, positive $k$, and at least $k$ graph edges across every nontrivial cut.
2. AlgorithmicStrongThinTrees supplies a positive real constant C, a concrete CurrentKS.Machine, a positive natural coefficient a, and a natural exponent degree that may be zero. On every valid encoding the machine must halt within a * (x.length + 1) ^ degree steps.
3. The output decodes to a duplicate-free edge list forming a spanning tree T, and every nonempty proper vertex set $S$ satisfies $|\delta_T(S)|\le (C/k)|\delta_G(S)|$. For the one-vertex case the bit output is required to be empty.
4. Nima Anari's Simons Institute lecture states the strong form associated with Luis Goddyn as an existence conjecture. The pinned manuscript's additional claim is constructive: one deterministic polynomial-time algorithm works even when parallel multiplicities are compressed in binary.
5. Trust boundary: this comparator declaration ends in by sorry. The pinned comparator configuration pairs it with OAI.Combinatorics.ThinTrees.AlgorithmicMain; the site's Lean trust result, not this prose or the manuscript, determines whether that separate solution module checks.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI algorithmic strong-thin-tree manuscript, Pinned OpenAI formalization note 174, Nima Anari, Thin Trees and Interlacing Families lecture slides (Simons Institute)
Distinct definitions of this name: OAI/Combinatorics/ThinTrees/Encoding/AlgorithmicMain.lean:65 (same statement text; different proof or body). Qualify references as OAI.AlgorithmicThinTrees.algorithmic_strong_thin_trees@path.

strong thin tree: a spanning tree that crosses every cut using at most an $O(1/k)$ fraction of the original cut in a $k$-edge-connected graph
edge connectivity: the minimum number of graph edges crossing any nonempty proper vertex cut
```
theorem algorithmic_strong_thin_trees : AlgorithmicStrongThinTrees := by
…
```
Mentions: OAI.AlgorithmicThinTrees.AlgorithmicStrongThinTrees