Planar first-passage geometry and the absence of bigeodesics
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Proves that planar first-passage percolation has no doubly infinite geodesic for iid nonnegative nonatomic edge weights when the minimum of four weights has finite second moment. For exponential weights, the limit shape is strictly convex with C1 boundary. Differentiability also holds for every Gamma law with positive shape and rate. (OpenAI’s Lean note)
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Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
The other possible matches are Theorem 1.2(Differentiability for Gamma weights) in the same paper () and Theorem 1.1 in No bigeodesics in planar first-passage percolation ().
For independent exponential edge weights of any rate \lambda>0, the norm \mu is Fréchet differentiable on \R^2\setminus\{0\}, and its unit ball is strictly convex. In particular,
We prove that planar first-passage percolation with independent identically distributed nonnegative nonatomic edge weights has almost surely no doubly infinite geodesic, provided the minimum of four independent weights has finite second moment. This resolves the planar no-bigeodesics conjecture under that moment assumption. The conclusion rules out all bigeodesics simultaneously, without any regularity assumption on the limit shape.
We prove that the limit shape of undirected nearest-neighbor first-passage percolation on ℤ2 with independent exponential edge weights is strictly convex and has a C1 boundary. This resolves the strict convexity and differentiability conjectures for the planar exponential model. More generally, we prove differentiability of the time-constant norm for every Gamma edge-weight law with positive shape and rate.
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For independent exponential edge weights of any rate \lambda>0, the norm \mu is Fréchet differentiable on \R^2\setminus\{0\}, and its unit ball is strictly convex. In particular,
PlanarFirstPassage.lean, the release lists it as a main result.
theoremmanuscriptMain (Ω : Type) [MeasurableSpace Ω] (P : Measure Ω) (τ : Edge → Ω → ℝ) (henv : ExponentialEnvironment P τ) : ∃ μ : Plane → ℝ, TimeConstantNorm μ ∧ IsTimeConstant P τ μ ∧ DifferentiableAwayFromOrigin μ ∧ UniqueNormalizedSupports μ ∧ C1UnitSphere μ ∧ (∀ v ∈ frontier {z | μ z ≤ 1}, ∃! L : Set Plane, ∃ g : Plane →L[ℝ] ℝ, g ≠ 0 ∧ (∀ z, μ z ≤ 1 → g z ≤ g v) ∧ L={z | g z=g v}) ∧ (∀ v ∈ frontier {z | μ z ≤ 1}, HasC1CurveChart (frontier {z | μ z ≤ 1}) v) := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
For independent exponential edge weights of any rate \lambda>0, the norm \mu is Fréchet differentiable on \R^2\setminus\{0\}, and its unit ball is strictly convex. In particular,
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).
Their note:PRIMARY CLAIM NOT STATED (only a secondary result is formalized). Stated: exponential case manuscriptMain (norm μ with IsTimeConstant, DifferentiableAwayFromOrigin, UniqueNormalizedSupports, C1UnitSphere, unique supporting line, HasC1CurveChart on the frontier) and GammaFPP.gamma_differentiability for every shape/rate > 0. Not stated: the first headline claim (no doubly infinite geodesic for iid nonatomic weights with finite second moment of the min of four) and strict convexity of the exponential limit shape (docs: ‘Strict convexity is outside these selected differentiability statements’; bigeodesics are not mentioned in the docs scope).
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method: 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.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).
The gap they name:The no-bigeodesics theorem (the headline’s first sentence) is not stated; strict convexity of the exponential limit shape is not stated (declared).
“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
The formalization proves differentiability of the planar first-passage time-constant norm for independent Gamma edge weights of every positive shape and rate. The norm is differentiable away from the origin, and its unit sphere has a $C^1$ boundary. This includes the exponential model, for which the linked statement also gives a unique supporting line at every boundary point.
The paper also claims strict convexity for the exponential limit shape. Strict convexity is outside these selected differentiability statements.
AI-drafted comparisons, with no recorded human review
Differentiability of the exponential first-passage limit shapePlanarFirstPassage
Unreviewed AI draft: states part of a paper’s theorem. States the differentiability half of the exponential limit-shape theorem, with a unique supporting line at every boundary point. The strict convexity asserted in the same theorem is outside the statement.
Differentiability for every positive Gamma edge-weight lawGammaPassage
Unreviewed AI draft: states a paper’s theorem as the paper does. States the paper’s theorem on Gamma weights: the time-constant norm is differentiable away from the origin for every positive shape and rate.
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Differentiability of the exponential first-passage limit shapePlanarFirstPassage
Two reports, all at the release’s recorded version.
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Why this passage is shown
Theorem 1.2 (Differentiability for Gamma weights)
In Strict convexity and differentiability of the planar exponential first-passage limit shape, 24 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method
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.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method
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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.