Finiteness of critical bridge mass and first length moments
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Theorem selected for verification:OAI.PolynomialVacuumBridge.Corridor.finite_bridge_sums
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
defFiniteBridgeSums : Prop := ∀ h : ℕ, 1 ≤ h → Summable (weight (h := h)) ∧ Summable (firstLengthWeight (h := h)) ∧ Summable (fun p : BridgePath h => firstLengthWeight p / bridgeMass h) ∧ Summable (fun p : ConfinedBridgePath h => weight p.val) ∧ Summable (fun p : ConfinedBridgePath h => firstLengthWeight p.val) ∧ Summable (fun p : ConfinedBridgePath h => firstLengthWeight p.val / confinedMass h)⋮theoremfinite_bridge_sums : FiniteBridgeSums := bysorry
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.
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.
defadjacent : Vertex → Vertex → Prop | .up i j, .down k l => (k = i ∧ l = j - 1) ∨ (k = i ∧ l = j) ∨ (k = i - 1 ∧ l = j) | .down k l, .up i j => (k = i ∧ l = j - 1) ∨ (k = i ∧ l = j) ∨ (k = i - 1 ∧ l = j) | _, _ => False
No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.
Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: bridge, confined, exponents, finite, height, horizontal, initial, length and five more.
Theorem 1.1 of Polynomial vacuum representations and bridge mass for honeycomb walks
As the integer layer height $h$ tends to infinity,
\[
b_h=h^{-1/4+o(1)},\qquad
\ell_h=h^{13/12+o(1)},\qquad
\mathbb E_h|\gamma|=h^{4/3+o(1)}.
\]
All three sums are finite. These exponents are unchanged if the paths
are also confined to horizontal distance $h\log^2h$ from their initial
port.
Other possible paper matches, closest wording first
Theorem 1.1 of Disk transfer representations and confined bridge mass. Words of the Lean statement, its names split into words, that also appear in the theorem: corridor, vertices, bridge, critical and mass. Found in the paper’s title, which its main theorem states: confined. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: visited, mean, law and 4/3.
Theorem 1.1 of Cap-selected amplitudes and triangle chords for honeycomb walks. Words of the Lean statement, its names split into words, that also appear in the theorem: layer, adjacent, weighted and critical. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: chords, mean, exponent, law and 4/3.
Corollary 10.2 of Renewal and changes of law for critical honeycomb walks. Words of the Lean statement, its names split into words, that also appear in the theorem: summed, source, vertices, vertex, first, bridge and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: visited, law and 4/3.
Proposition 4.1 of Renewal and changes of law for critical honeycomb walks. Words of the Lean statement, its names split into words, that also appear in the theorem: confined, terminal, path, strip, first, 1/2 and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: moment, law and 4/3.
Theorem 11.1 of Renewal and changes of law for critical honeycomb walks. Words of the Lean statement, its names split into words, that also appear in the theorem: summed, vertices, source, terminal, strip, bridge and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: mean, law and 4/3.
Theorem 8.1 of Renewal and changes of law for critical honeycomb walks. Words of the Lean statement, its names split into words, that also appear in the theorem: natural, vertex, first, bridges, weights, critical and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: moment, mean, exponents, law and 4/3.
How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.
Reported Comparator runs
three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.
For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.
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.
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
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 211 s; peak memory 6.2 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.
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Do family 237’s Lean statements express the advertised result?
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Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
Finds a supporting or a different result statedTheir term: “supporting-only”.
The challenge statements state a lemma or auxiliary result, not the headline. Their record
Their note: The four challenges are analytic inputs, not the headline: finiteness of the critical bridge sums and first-length moments at every strip height (HoneycombBridgeFiniteness), the strip estimates B_N ≍ N^{-1/4}, arch moment ≍ N^{3/4}, c*A_N+B_N=1 (CriticalStripMass, with the bridge-mass bound restated for h>=0 in HoneycombBridgeMassSupport), and existence of the force-tilted free-energy limit (HoneycombFreeEnergy). None mentions the uniform measure on n-step self-avoiding walks, their diameter, local mass or covering numbers, or any probability bound, so the headline theorem (diameter between n^{3/4-delta} and n^{3/4+delta} with probability at least 1-C n^{-k}) is not stated.
Definitions the group asks readers to compare with the paper: rho / kappa / criticalActivity: the four files use 1/sqrt(2+sqrt 2), (2cos(pi/8))^-1, (2+sqrt 2)^(-1/2) and (sqrt(2+sqrt 2))^-1, all equal to the honeycomb critical activity x_c (2cos(pi/8) = sqrt(2+sqrt 2)); a fixed concrete constant, standard ; Triangle (CriticalStripMass) / Vertex (BridgeFiniteness, FreeEnergy) / Center (BridgeMassSupport): the honeycomb lattice realised as up/down triangles of the triangular tiling with the same three-neighbour adjacency up(i,j) ~ down(i,j), down(i,j-1), down(i-1,j); CriticalStripMass uses rows in N (half-plane with a bottom wall), the others rows in Z; concrete lattice, standard ; CriticalStripMass.moment: sum over k >= 1 of k*K(k), a one-sided first displacement moment of arches (the paper's m_N); an unnormalised sum over all path lengths, not the displacement of a fixed-length walk ; CriticalStripMass: lambda, d, C, eta are defined but unused by the statement; tsum is 0 for non-summable families, guarded by the explicit StripSumsFinite conjunct ; HoneycombBridgeFiniteness.InLogCorridor / ConfinedBridgePath: confinement |x| <= h*(log h)^2 restricts to a finite vertex set for each h, so the three confined summability conjuncts are automatic (log 1 = 0 forces x = 0 at h = 1, so no bridge is confined there); the paper's content for the corridor (exponents unchanged) is not stated ; HoneycombBridgeFiniteness.FiniteBridgeSums: purely qualitative summability for each h >= 1, no rates; the normalised conjuncts divide by bridgeMass h / confinedMass h (Lean x/0 = 0, harmless here) ; HoneycombBridgeMassSupport.bridgeMass: ENNReal-valued tsum with B_0 := 1 by an if; Port and Port.position are defined but unused; the theorem's docstring says 'Trusted reference hole only; missing necessary main obligation' ; HoneycombFreeEnergy.freeEnergy: limUnder atTop (logPartition ..), a junk-valued limit, so logPartition_tendsto is exactly 'the limit exists'; e : C is not required to be a unit vector and s : R may have either sign (more general than the paper's unit e, s >= 0)
Finds a supporting or a different result statedTheir term: “DIFFERENT”.
The challenge statements state a different theorem from the headline. Their record
Their note: The headline (diameter n^{3/4+o(1)} and mass/covering exponent 4/3 for uniform fixed-length honeycomb SAW, the content of 'Mass-and-covering-exponents-for-fixed-length-honeycomb-walks') is stated by no challenge. The four challenges state ingredients from other manuscripts of the family: finiteness of critical bridge sums, existence of the forced free energy, the strip-crossing mass ≍ N^{-1/4} with displacement moment ≍ N^{3/4}, and strict-bridge mass ≍ (1+h)^{-1/4}. None implies the headline and the headline implies none of them.
OpenAI’s own note on what this Lean statement covers
“These are supporting summability statements. The paper's $h^{4/3+o(1)}$ mean-length law and its central-visit and hexagonal-chord exponents are outside them.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a supporting result”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
65 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, 16 to Mathlib’s documentation. 14 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
As the integer layer height $h$ tends to infinity,
\[
b_h=h^{-1/4+o(1)},\qquad
\ell_h=h^{13/12+o(1)},\qquad
\mathbb E_h|\gamma|=h^{4/3+o(1)}.
\]
All three sums are finite. These exponents are unchanged if the paths
are also confined to horizontal distance $h\log^2h$ from their initial
port.
Theorem(Hexagon visits)
The mass normalizing \mathbb P_R is R^{3/4+o(1)}. Uniformly at vertices in the concentric hexagon of side R/3,
\mathbb P_R(v\in\gamma)=R^{-2/3+o(1)}.
The corresponding upper bound holds at every vertex, and \mathbb E_R|\gamma|=R^{4/3+o(1)}.
The mass normalizing $\mathbb P_R$ is $R^{3/4+o(1)}$. Uniformly at
vertices in the concentric hexagon of side $R/3$,
\[
\mathbb P_R(v\in\gamma)=R^{-2/3+o(1)}.
\]
The corresponding upper bound holds at every vertex, and
$\mathbb E_R|\gamma|=R^{4/3+o(1)}$.
Proposition(Mixed exits)
The conditioned event has unnormalized mass h^{-1/4+o(1)} and first-length mass h^{13/12+o(1)}. Hence its normalized mean length is h^{4/3+o(1)}. Restricting horizontal distance to h\log^2h preserves these exponents. The total first-length mass of all exits, without the height condition, is at most h^{13/12+o(1)}.
The conditioned event has unnormalized mass $h^{-1/4+o(1)}$ and
first-length mass $h^{13/12+o(1)}$. Hence its normalized mean length is
$h^{4/3+o(1)}$. Restricting horizontal distance to $h\log^2h$ preserves
these exponents. The total first-length mass of all exits, without the
height condition, is at most $h^{13/12+o(1)}$.
Theorem(Normalized separator pairing)
For every compact set K\subset\{s\in\mathbb C:|\Im s|<\pi\} there is C_K<\infty such that, for all n\ge1 and s\in K,
There is a disk about \eta=0, independent of n, in which F_n has no zeros. For each fixed k, the order-k Taylor coefficient of \log F_n at zero is O_k(1+\log n). At the positive fugacity two,
For every compact set $K\subset\{s\in\mathbb C:|\Im s|<\pi\}$ there
is $C_K<\infty$ such that, for all $n\ge1$ and $s\in K$,
\[
|F_n(2\cosh s)|\le
C_K n^{1/6+2\Re(s^2)/(3\pi^2)}.
\]
There is a disk about $\eta=0$, independent of $n$, in which $F_n$ has
no zeros. For each fixed $k$, the order-$k$ Taylor coefficient of
$\log F_n$ at zero is $O_k(1+\log n)$. At the positive fugacity two,
\[
F_n(2)=n^{1/6+o(1)}.
\]
There are constants $c,C>0$ such that, for every sufficiently large integer $h$,
\[
c h^{-1/4}\le B_h\le C h^{-1/4},\qquad M_h\le C h^{13/12},\qquad
B_h\{L\ge c h^{4/3}\}\ge c h^{-1/4}.
\]
For all $h\ge1$ and $x\ge1$, one also has
\[
B_h\{D>x\}\le C x^{-1/4}.
\]
The same estimates hold in the symmetry-related pure directions.
Theorem(Infinite spatial law)
Concatenate independent irreducibles of law p, and enumerate the resulting vertices as \gamma_1,\gamma_2,\ldots. Almost surely,
Concatenate independent irreducibles of law $p$, and enumerate the resulting vertices as $\gamma_1,\gamma_2,\ldots$. Almost surely,
\[
|\gamma_n-\gamma_1|=n^{3/4+o(1)},\qquad
\#\{n:|\gamma_n-\gamma_1|\le R\}=R^{4/3+o(1)}.
\]
Theorem
One has a_{n+2}/a_n\to\rho^{-2}. The uniform half-plane laws converge on each fixed initial path to the infinite law of Theorem R:renew:infinite.
One has $a_{n+2}/a_n\to\rho^{-2}$. The uniform half-plane laws converge on each fixed initial path to the infinite law of Theorem~\ref{R:renew:infinite}.
For every integer $N\ge1$, all sums in
\eqref{eq:01-strip-quantities} are finite. With constants independent of
$N$,
\begin{equation}
c\mathcal A_N+\mathcal B_N=1,\qquad
m_{N+1}-m_N\asymp\mathcal B_N,\qquad
m_N\asymp N^{3/4},\qquad
\mathcal B_N\asymp N^{-1/4}.
\end{equation}
Moreover, $\mathcal B_N$ is nonincreasing in $N$.
Let D_R be convex pure lattice hexagons which, after scaling by R, converge to a regular hexagon. Let a_R,b_R be boundary ports whose scaled limits are distinct points in the relative interiors of sides. Then, for every \varepsilon>0, the critical chord law satisfies
The same conclusion holds in a full half-plane for prescribed boundary ports at distance R, in a strip of height comparable to R for one prescribed port on each of the two opposite boundary cuts, at mutual distance O(R), and in a wedge of angle \pi/3 or 2\pi/3 formed by two nonparallel pure cuts, with one prescribed port on each ray. In the wedge, each endpoint has clearance comparable to R from the other cut and distance O(R) from the apex. Constants are uniform on fixed compact sets of these nondegenerate scaled geometries. In each case the normalizing mass is at least cR^{-5/4}.
Let $D_R$ be convex pure lattice hexagons which, after scaling by $R$,
converge to a regular hexagon. Let $a_R,b_R$ be boundary ports whose
scaled limits are distinct points in the relative interiors of sides.
Then, for every $\varepsilon>0$, the critical chord law satisfies
\[
\mathbb P_{D_R;a_R,b_R}
\{R^{4/3-\varepsilon}\le L(\gamma)\le R^{4/3+\varepsilon}\}
\longrightarrow1.
\]
The same conclusion holds in a full half-plane for prescribed boundary
ports at distance $R$, in a strip of height comparable to $R$ for
one prescribed port on each of the two opposite boundary cuts, at mutual
distance $O(R)$, and in a wedge of angle
$\pi/3$ or $2\pi/3$ formed by two nonparallel pure cuts, with one prescribed port on each ray.
In the wedge, each endpoint has clearance comparable to $R$ from the other cut and
distance $O(R)$ from the apex. Constants are uniform on fixed compact
sets of these nondegenerate scaled geometries. In each case the
normalizing mass is at least $cR^{-5/4}$.
All three sums are finite. These exponents are unchanged if the paths are also confined to horizontal distance h\log^2h from their initial port.
Polynomial vacuum representations and bridge mass for honeycomb walks
TeX label thm:pv:bridge, build/sections/introduction.tex, line 34
The Lean statement says
defFiniteBridgeSums : Prop := ∀ h : ℕ, 1 ≤ h → Summable (weight (h := h)) ∧ Summable (firstLengthWeight (h := h)) ∧ Summable (fun p : BridgePath h => firstLengthWeight p / bridgeMass h) ∧ Summable (fun p : ConfinedBridgePath h => weight p.val) ∧ Summable (fun p : ConfinedBridgePath h => firstLengthWeight p.val) ∧ Summable (fun p : ConfinedBridgePath h => firstLengthWeight p.val / confinedMass h)⋮theoremfinite_bridge_sums : FiniteBridgeSums := bysorry
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
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.