Quantitative and superpolynomial sample lower bounds for trace reconstruction
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Theorems selected for verification:OAI.TraceReconstruction.quantitative_sample_lower_boundOAI.TraceReconstruction.superpolynomial_one_traceOAI.TraceReconstruction.superpolynomial_sample_complexity
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
theoremquantitative_sample_lower_bound (s c : ℝ) (hs : s ∈ Set.Ioc 01) (hc : c ∈ Set.Ioo 0 (1 / (4 * Real.log 2))) (n : ℕ → ℕ) (q : ℕ → ℝ) (hn : Tendsto n atTop atTop) (hq : ∀ k, q k ∈ Set.Ioo 01) (hscale : Tendsto (fun k => (q k) ^ 3 * Real.log (n k)) atTop atTop) : ∀ᶠ k in atTop, ENNReal.ofReal ((n k : ℝ) ^ (c * Real.log ((q k) ^ 3 * Real.log (n k)))) ≤ sampleComplexity (n k) (q k) s := bysorry⋮theoremsuperpolynomial_one_trace (q : ℝ) (hq : q ∈ Set.Ioo 01) (C : ℝ) (hC : 0 < C) : Tendsto (fun n : ℕ => (n : ℝ) ^ C * minTraceTV n q) atTop (𝓝 0) := bysorry⋮theoremsuperpolynomial_sample_complexity (q : ℝ) (hq : q ∈ Set.Ioo 01) (s : ℝ) (hs : s ∈ Set.Ioc 01) (C : ℝ) (hC : 0 < C) : Tendsto (fun n : ℕ => sampleComplexity n q s / ENNReal.ofReal ((n : ℝ) ^ C)) atTop (𝓝 ⊤) := bysorry
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Definitions needed to read it
Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.
defreconstructionSuccess {n m : ℕ} (q : ℝ) (A : (Fin m → Trace) → Word n → ℝ) (x : Word n) : ℝ := ∑ a : Fin m → Mask n, (∏ j, maskWeight q (a j)) * A (fun j => traceOfMask x (a j)) x
noncomputabledefminTraceTV (n : ℕ) (q : ℝ) : ℝ := sInf {t | ∃ x y : Word n, x ≠ y ∧ t = traceTV q x y}
Possible paper match, with no recorded review
Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.
Words of the Lean statement, its names split into words, that also appear in the theorem: quantitative, sample, natural, success, lower, bound, probability, deletion and one more. Found in the paper’s title, which its main theorem states: reconstruction and trace. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: required, logarithms, eventually, number, sequence and along.
Statements in Quantitative lower bounds for trace reconstruction come first because the release lists only that paper for this Lean statement.
Theorem 1.1 of Quantitative lower bounds for trace reconstruction
Theorem(Quantitative sample lower bound)
Fix s\in(0,1] and 0<c<1/(4\log2). Along every sequence of instances (n,q) with n\to\infty, q\in(0,1), and q^3\log n\to\infty,
T_{q,s}(n)\ge n^{c\log(q^3\log n)}
eventually. All logarithms are natural. In particular, for each fixed deletion probability q\in(0,1) and fixed positive success probability s, the required sample number is n^{\Omega(\log\log n)}.
Fix $s\in(0,1]$ and $0<c<1/(4\log2)$. Along every sequence of instances $(n,q)$ with $n\to\infty$, $q\in(0,1)$, and $q^3\log n\to\infty$,
\begin{equation}
T_{q,s}(n)\ge n^{c\log(q^3\log n)}
\end{equation}
eventually. All logarithms are natural. In particular, for each fixed deletion probability $q\in(0,1)$ and fixed positive success probability $s$, the required sample number is $n^{\Omega(\log\log n)}$.
Other possible paper matches, closest wording first
Theorem 1.2 of Quantitative lower bounds for trace reconstruction. Words of the Lean statement, its names split into words, that also appear in the theorem: superpolynomial, tv, min, probability, trace, deletion and one more.
Theorem 1.1 of Uniform quasipolynomial-time trace reconstruction. Words of the Lean statement, its names split into words, that also appear in the theorem: length, output, finite, bit, includes, independent and seven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: least and sequence.
Proposition 5.1 of Quantitative lower bounds for trace reconstruction. Words of the Lean statement, its names split into words, that also appear in the theorem: value, input, retained and one. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: along.
Proposition 2.3 of Quantitative lower bounds for trace reconstruction. Words of the Lean statement, its names split into words, that also appear in the theorem: depend, bounds, log and one. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sequence and along.
Proposition 4.1 of Quantitative lower bounds for trace reconstruction. Words of the Lean statement, its names split into words, that also appear in the theorem: may, absolute, log and one.
Proposition 3.1 of Uniform quasipolynomial-time trace reconstruction. Words of the Lean statement, its names split into words, that also appear in the theorem: lists, nonempty, min, absolute, log and one.
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Reported Comparator runs
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Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).
AcceptedTheir term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 123 s; peak memory 6.4 GiB).
If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.
This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.
Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.
Do family 122’s Lean statements express the advertised result?
These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
Finds part of it stated, or a narrower or weaker statementTheir term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: Only the lower bounds (superpolynomial and n^{c log(q^3 log n)}) are stated; the quasipolynomial uniform decoder and the polynomial small-deletion-probability regime are not.
Definitions the group asks readers to compare with the paper: sampleComplexity n q s := sInf {m | ∃ estimator A, IsEstimator A ∧ ∀ x, s ≤ reconstructionSuccess q A x} in ℝ≥0∞ (estimator = arbitrary map from m traces to a probability vector over words, success averaged exactly over deletion masks; standard)
Finds part of it stated, or a narrower or weaker statementTheir term: “NARROWER — UNLINKED”.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link. Their record
Their note: The headline's two upper-bound sentences (a uniform decoder with quasipolynomial sample and running-time bounds for fixed rational retention probability; polynomial bounds when the deletion probability is ≤ n^{-ε}) — the content of two of the family's three manuscripts — are not stated by any challenge. Only the lower bounds are.
OpenAI’s own note on what this Lean statement covers
“More quantitatively, along sequences with $q^3\log n\to\infty$, the required number of samples is at least $n^{c\log(q^3\log n)}$ eventually for every fixed $0<c<1/(4\log2)$. All logarithms are natural.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
91 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.
Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.
Links in the file: 34 to its own declarations, 41 to Mathlib’s documentation. 16 declared names also link to the same declaration in OpenAI’s library. 2 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
import Mathlibnamespace OAIopenscoped ENNRealnoncomputablesectionnamespace TraceReconstructionabbrevWord (n : ℕ) := Fin n → BoolabbrevMask (n : ℕ) := Fin n → BoolabbrevTrace := ListBool/-- `true` retains the bit. The observer gets only the resulting list. -/deftraceOfMask {n : ℕ} (x : Word n) (a : Mask n) : Trace := ((List.finRange n).filter fun i => a i).map x/-- The deletion-mask probability, with deletion probability `q`. -/defmaskWeight {n : ℕ} (q : ℝ) (a : Mask n) : ℝ := ∏ i, if a i then1 - q else q/-- Law of a single complete trace. -/deftraceMass {n : ℕ} (q : ℝ) (x : Word n) (w : Trace) : ℝ := ∑ a : Mask n, iftraceOfMask x a = w thenmaskWeight q a else0/-- All possible complete outputs from length-n inputs, with lengths included. -/deftraceSpace (n : ℕ) : FinsetTrace :=Finset.univ.biUnion fun k : Fin (n + 1) =>Finset.univ.image fun x : Word k.val => List.ofFn x/-- 1/2 sum of absolute differences, on the full finite output space. -/deftraceTV {n : ℕ} (q : ℝ) (x y : Word n) : ℝ := (1 / 2 : ℝ) * ∑ w ∈ traceSpace n, |traceMass q x w - traceMass q y w|/-- The estimator may depend arbitrarily on n, q and every complete sample. Its value is its conditional probability of reporting the given input. -/defIsEstimator {n m : ℕ} (A : (Fin m → Trace) → Word n → ℝ) : Prop := (∀ z x, 0 ≤ A z x) ∧ (∀ z, ∑ x : Word n, A z x = 1)/-- Independent masks generate independent complete traces; no retained positions, boundaries or other latent variables are revealed to A. -/defreconstructionSuccess {n m : ℕ} (q : ℝ) (A : (Fin m → Trace) → Word n → ℝ) (x : Word n) : ℝ := ∑ a : Fin m → Mask n, (∏ j, maskWeight q (a j)) * A (fun j => traceOfMask x (a j)) xdefSufficientBudget (n : ℕ) (q s : ℝ) (m : ℕ) : Prop := ∃ A : (Fin m → Trace) → Word n → ℝ,IsEstimator A ∧ ∀ x, s ≤ reconstructionSuccess q A x/-- Natural fixed sample budget; infimum of the empty set is infinity. -/noncomputabledefsampleComplexity (n : ℕ) (q s : ℝ) : ℝ≥0∞ :=sInf {t | ∃ m : ℕ, SufficientBudget n q s m ∧ t = (m : ℝ≥0∞)}/-- The real infimum is a minimum for n>0 (a nonempty finite set). -/noncomputabledefminTraceTV (n : ℕ) (q : ℝ) : ℝ :=sInf {t | ∃ x y : Word n, x ≠ y ∧ t = traceTV q x y}open Filteropenscoped Topology Classical ENNRealtheoremquantitative_sample_lower_bound (s c : ℝ) (hs : s ∈ Set.Ioc01) (hc : c ∈ Set.Ioo0 (1 / (4 * Real.log2))) (n : ℕ → ℕ) (q : ℕ → ℝ) (hn : Tendsto n atTopatTop) (hq : ∀ k, q k ∈ Set.Ioo01) (hscale : Tendsto (fun k => (q k) ^ 3 * Real.log (n k)) atTopatTop) : ∀ᶠ k inatTop,ENNReal.ofReal ((n k : ℝ) ^ (c * Real.log ((q k) ^ 3 * Real.log (n k)))) ≤sampleComplexity (n k) (q k) s := bysorrytheoremsuperpolynomial_one_trace (q : ℝ) (hq : q ∈ Set.Ioo01) (C : ℝ) (hC : 0 < C) :Tendsto (fun n : ℕ => (n : ℝ) ^ C * minTraceTV n q) atTop (𝓝 0) := bysorrytheoremsuperpolynomial_sample_complexity (q : ℝ) (hq : q ∈ Set.Ioo01) (s : ℝ) (hs : s ∈ Set.Ioc01) (C : ℝ) (hC : 0 < C) :Tendsto (fun n : ℕ => sampleComplexity n q s / ENNReal.ofReal ((n : ℝ) ^ C))atTop (𝓝 ⊤) := bysorryend TraceReconstructionendend OAI
Every extracted statement of the accompanying papers
The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.
Fix s\in(0,1] and 0<c<1/(4\log2). Along every sequence of instances (n,q) with n\to\infty, q\in(0,1), and q^3\log n\to\infty,
T_{q,s}(n)\ge n^{c\log(q^3\log n)}
eventually. All logarithms are natural. In particular, for each fixed deletion probability q\in(0,1) and fixed positive success probability s, the required sample number is n^{\Omega(\log\log n)}.
Fix $s\in(0,1]$ and $0<c<1/(4\log2)$. Along every sequence of instances $(n,q)$ with $n\to\infty$, $q\in(0,1)$, and $q^3\log n\to\infty$,
\begin{equation}
T_{q,s}(n)\ge n^{c\log(q^3\log n)}
\end{equation}
eventually. All logarithms are natural. In particular, for each fixed deletion probability $q\in(0,1)$ and fixed positive success probability $s$, the required sample number is $n^{\Omega(\log\log n)}$.
For every fixed deletion probability $q\in(0,1)$,
\begin{equation}
\lim_{n\to\infty}n^A
\min_{\substack{x,y\in\{0,1\}^n\\x\ne y}}
\TV\bigl(\Dcal_q(x),\Dcal_q(y)\bigr)=0
\qquad\text{for every }A>0.
\end{equation}
At this same channel $T_{q,s}(n)/n^A\to\infty$ for every fixed $A>0$ and $s\in(0,1]$.
OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 122 link it.
Solution module
OAI.Probability.TraceReconstruction
Allowed axioms
propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports
Mathlib
Imports from outside Mathlib
None
Local vocabulary
13 local vocabulary declarations
Lines of Lean, without comments or blank lines
49
Second proof-verification program (nanoda)
Off in OpenAI's configuration.
Statement comparison
A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.
Plectis Atlas
Quantitative and superpolynomial sample lower bounds for trace reconstruction
Family 122: Quantitative trace-reconstruction bounds with a uniform decoder
Candidate pairing
Theorem (Quantitative sample lower bound)
Fix s\in(0,1] and 0<c<1/(4\log2). Along every sequence of instances (n,q) with n\to\infty, q\in(0,1), and q^3\log n\to\infty,
T_{q,s}(n)\ge n^{c\log(q^3\log n)}
eventually. All logarithms are natural. In particular, for each fixed deletion probability q\in(0,1) and fixed positive success probability s, the required sample number is n^{\Omega(\log\log n)}.
Quantitative lower bounds for trace reconstruction
TeX label thm:main, build/sections/introduction.tex, line 7
The Lean statement says
theoremquantitative_sample_lower_bound (s c : ℝ) (hs : s ∈ Set.Ioc 01) (hc : c ∈ Set.Ioo 0 (1 / (4 * Real.log 2))) (n : ℕ → ℕ) (q : ℕ → ℝ) (hn : Tendsto n atTop atTop) (hq : ∀ k, q k ∈ Set.Ioo 01) (hscale : Tendsto (fun k => (q k) ^ 3 * Real.log (n k)) atTop atTop) : ∀ᶠ k in atTop, ENNReal.ofReal ((n k : ℝ) ^ (c * Real.log ((q k) ^ 3 * Real.log (n k)))) ≤ sampleComplexity (n k) (q k) s := bysorry⋮theoremsuperpolynomial_one_trace (q : ℝ) (hq : q ∈ Set.Ioo 01) (C : ℝ) (hC : 0 < C) : Tendsto (fun n : ℕ => (n : ℝ) ^ C * minTraceTV n q) atTop (𝓝 0) := bysorry⋮theoremsuperpolynomial_sample_complexity (q : ℝ) (hq : q ∈ Set.Ioo 01) (s : ℝ) (hs : s ∈ Set.Ioc 01) (C : ℝ) (hC : 0 < C) :⋮
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Recorded comparisons with the paper
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Contributed Comparator run reports
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Questions to resolve when comparing this statement with the paper
Both independent readers find this family's challenges state less than its headline, while the atlas's AI draft classes this file as stating a paper's theorem as the paper does; check which part of the headline it covers.