Furstenberg No-Bigeodesics (Planar FPP, Standard Minimum Moment)

Furstenberg No-Bigeodesics (Planar FPP, Standard Minimum Moment)

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Planar iid nonatomic first-passage weights are claimed to admit no doubly infinite minimizing path when the minimum of four weights has a finite second moment.

Mathematical statement of the claimed result

Result 1.

Give the nearest-neighbor edges of $\mathbb Z^2$ independent nonnegative passage times with an atomless common law $G$, and let $T(x,y)$ be the minimal passage cost. Assume \[ \mathbb E[\min\{X_1,X_2,X_3,X_4\}^2]<\infty, \qquad X_i\text{ independent with law }G. \] The assertion is that almost surely no doubly infinite self-avoiding adjacent sequence $(v_i)_{i\in\mathbb Z}$ satisfies \[ \sum_{k=i}^{j-1}t_{\{v_k,v_{k+1}\}}=T(v_i,v_j) \quad\text{for all integers }i<j. \] Every finite segment would otherwise be geodesic.

Outline of the argument

Busemann labels reduce potential bigeodesics to compact sectors. Finite increases of passage times create marked paths with a definite extra cost; deterministic label bins and telescoping Busemann comparisons impose a finite budget that cannot support all the marks, including environment-selected exceptional directions.

Formal verification

For family 212, the documented entries give differentiability and continuously differentiable boundaries for planar Gamma shapes, including the exponential law. They do not supply strict convexity or the separate no-bigeodesic theorem. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

• OpenAI, No bigeodesics in planar first-passage percolation, 2026-09-24; citation key OAI:No-bigeodesics-in-planar-first-passage-percolation-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• Family 212: Lean scope.
• OpenAI catalogue entry.

Furstenberg No-Bigeodesics (Planar FPP, Standard Minimum Moment)

Definitions and mathematical statement

Assign independent edge times with a non-atomic law $G$ on $[0,\infty)$ to the unoriented nearest-neighbor edges of ${\mathbb{Z}}^2$. Assume \[ {\mathbb{E}}\left[\min(t_1,t_2,t_3,t_4)^2\right]<\infty \] for four independent samples from $G$. For vertices $x,y$, define $T(x,y)$ as the infimum of the sum of edge times over finite paths from $x$ to $y$. A bigeodesic is a doubly infinite self-avoiding nearest-neighbor path $(v_i)_{i\in{\mathbb{Z}}}$ satisfying \[ \sum_{k=i}^{j-1}t_{\{v_k,v_{k+1}\}}=T(v_i,v_j) \quad\text{for every }i<j. \] The assertion is that the product probability of existence of any bigeodesic is zero. The moment condition concerns the minimum of four times, not an individual edge time. No prescribed directions, density for $G$, curvature of a limit shape, or exponential moment is assumed. This selected version is not equated with a separately stated moment-free version.

Formulation and concept references: [S1].

Short English statement

Under the specified non-atomic minimum-moment assumption, is every doubly infinite path in planar first-passage percolation almost surely nonoptimal somewhere along its length?

Research attempt

Scope and primary-source audit (27 September 2026).

This GPT-6 Astra Ultra attempt uses independent, identically distributed nonnegative edge times with a non-atomic law. Non-atomicity does not mean that the law has a density. The only moment assumption in the target is the displayed second moment of the minimum of four samples. In particular, an individual edge may have infinite expectation.

The supplied Damron--Hanson manuscript [S1], Section 2.1, assumption A1', uses exactly that minimum moment. Its Theorem 1 also assumes geometric regularity at selected points of the limit shape and concludes absence of bigeodesics with an end in the resulting fixed sector. The simpler Theorem 1.1 in its introduction uses an individual first moment and a differentiable boundary. Neither form is an unconditional no-bigeodesic theorem under only this entry's assumptions.

Ahlberg--Hanson--Hoffman [E1], Theorem 1.1 and Section 2.4, assumption A1, prove that the probability a distant specified vertex belongs to an infinite geodesic from the origin tends to zero uniformly in that vertex. Their A1 again permits the present minimum moment. This is an important input for the discussion below, but it is a statement about a specified vertex; the rate needed by a naive union over a growing boundary is not part of its conclusion. No integrable directed last-passage theorem is substituted for undirected first-passage percolation here.

The main attack is to formulate an exact sequence of finite-box events whose probability limit is the probability of a bigeodesic through the origin. This avoids assuming directions for an unknown random path. The work also proves the basic existence and uniqueness facts directly, checks the moment distinction on an explicit heavy-tailed law, and rules out every straight coordinate line by independent local improvements. The remaining probability estimate is isolated after these reductions.

The minimum moment does not imply an individual first moment.

Let a nonnegative random variable $t$ have survival function \[ \mathbb P(t>s)=(1+s)^{-3/4},\qquad s\geq0. \] This is a continuous law with no atom, but $\mathbb E t=\int_0^\infty(1+s)^{-3/4}\,\mathrm ds=\infty$. For four independent copies and $M=\min(t_1,t_2,t_3,t_4)$, \[ \mathbb P(M>s)=(1+s)^{-3},\qquad \mathbb E M^2=2\int_0^\infty\frac{s}{(1+s)^3}\,\mathrm ds=1. \] More generally, the exponent $\alpha$ in $(1+s)^{-\alpha}$ gives \[ \mathbb E M^2=\frac{2}{(4\alpha-1)(4\alpha-2)} \quad\text{if }\alpha>1/2. \] Thus replacing the target's moment by $\mathbb E t<\infty$ would discard valid examples. The elementary arguments below mostly do not use any moment assumption at all; this makes their limits particularly transparent.

Geodesic foundations with no hidden moment input.

Since $G$ has no atom at zero, every edge time is strictly positive almost surely, simultaneously for the countably many lattice edges. It is finite because the law is supported on $[0,\infty)$, not on an extended interval including infinity. Finite loops can therefore be erased at a strict decrease in passage time.

Lemma 485.1 (long inexpensive paths are exponentially unlikely). There are constants $c>0$ and $0<\rho<1$ such that, for every vertex $x$ and positive integer $n$, \[ \mathbb P\left(\exists\text{ a self-avoiding $n$-edge path from $x$ with time at most }cn\right) \leq\frac43\rho^n. \]

Proof. Choose $\delta>0$ so small that $0<q=\mathbb P(t\leq\delta)<1/36$. This is possible because the distribution function is continuous, starts at zero and tends to one. Set $c=\delta/2$ and $\rho=6\sqrt q<1$. If a path of $n$ distinct edges has time at most $\delta n/2$, at least $\lceil n/2\rceil$ of its edges have time at most $\delta$. For a fixed self-avoiding path, independence and a union over subsets of its edges bound this probability by $2^n q^{\lceil n/2\rceil}\leq2^n q^{n/2}$. There are at most $4\cdot3^{n-1}$ self-avoiding paths of length $n$ starting at $x$: four possibilities for the first step, and at most three thereafter because immediate reversal is forbidden. A union bound gives the stated estimate.$\square$

By summability and Borel--Cantelli, for each $x$ there is an almost surely finite $N_x$ such that every self-avoiding path from $x$ with at least $N_x$ edges has time greater than $c$ times its length. Intersect these probability-one events over all vertices. This controls excessive excursions of a prospective minimizer without any expectation of $t$.

Corollary 485.2. Almost surely, for every pair of vertices there exists a unique finite geodesic between them; and from every vertex there exists at least one singly infinite geodesic.

Proof. For fixed $x,y$, take any deterministic finite path between them and call its finite time $C$. A self-avoiding competing path of time at most $C$ has fewer than $\max\{N_x,C/c+1\}$ edges by the preceding bound. There are finitely many such paths. Their minimum is attained, and loop erasure shows it is the infimum over all finite paths.

For uniqueness, two different simple paths with the same fixed endpoints have different undirected edge sets. Condition on all edge times except one edge in their symmetric difference. Equality of the two passage times then specifies a single value of that remaining edge. Independence and non-atomicity make its conditional probability zero. There are only countably many finite paths, so almost surely no such tie occurs for any pair simultaneously. This applies to finite induced boxes as well.

To obtain an infinite geodesic from $x$, let endpoints tend to infinity and consider their unique finite geodesics from $x$. Their lengths tend to infinity. There are finitely many first steps; retain an infinite subsequence with the same first step. Repeat for the second step, and so on, taking the diagonal limit. Every finite segment in the resulting infinite path eventually occurs in a finite geodesic and is minimizing. The limiting path is self-avoiding because any repeated vertex would already appear in one of its finite prefixes.$\square$

The last result produces one-sided paths, not bigeodesics. Taking two independent choices of one-sided geodesics at the origin does not imply that their concatenation is minimizing across the origin. That missing cross-origin optimality is exactly what the following finite events test.

An exact finite-box reformulation of a root bigeodesic.

Let \[ B_n=\{z\in\mathbb Z^2:\|z\|_\infty\leq n\},\qquad \partial B_n=\{z:\|z\|_\infty=n\}. \] Write $T_n$ for the distance using only edges whose endpoints lie in $B_n$. Define $F_n$ to be the event that there are distinct $a,b\in\partial B_n$ and a $T_n$-geodesic from $a$ to $b$ passing through the origin. This event depends on finitely many edge times. By positivity and uniqueness, it can be checked by \begin{equation}\label{wsref-0-72-61-34-38-35-3a-66-69-6e-69-74-65-65-76-65-6e-74} F_n=\bigcup_{\{a,b\}\subset\partial B_n} \{T_n(a,b)=T_n(a,0)+T_n(0,b)\}. \end{equation} If equality holds, concatenating the two shortest paths gives a shortest walk. It cannot have a positive-cost loop, so it is a path through zero.

Proposition 485.3 (nested crossing criterion). For positive finite edge times, \[ F_{n+1}\subseteq F_n, \qquad \{\text{there is a global bigeodesic through }0\} =\bigcap_{n\geq1}F_n. \] Uniqueness of finite geodesics is not needed for this proposition.

Proof. Suppose a box geodesic witnessing $F_{n+1}$ is rooted at its visit to zero. Follow each of its two arms from zero up to its first visit to $\partial B_n$. The concatenated truncated segment lies in $B_n$. It is shortest among paths in $B_{n+1}$ between its new endpoints, because a shorter competitor could replace that segment in the original geodesic. Hence it is also shortest among the smaller collection of paths contained in $B_n$, and witnesses $F_n$.

A global bigeodesic through zero supplies the same truncated segment in every box. Each arm eventually reaches the box boundary, since a self-avoiding infinite path cannot remain in a finite set. A global minimizer is also a minimizer when competitors are restricted to a box, so all $F_n$ occur.

Conversely suppose every $F_n$ occurs. Choose one witnessing path in each box and root it at zero. Each arm has at least $n$ edges because its endpoint has sup norm $n$. Orient the pair of arms by any fixed ordering of the two neighbors of zero used by the path. For each $k$ there are only finitely many possible pairs of $k$-edge initial arms. Repeated subsequence extraction gives a limiting doubly infinite path $(v_i)_{i\in\mathbb Z}$ with $v_0=0$. Every fixed finite rooted segment agrees with segments of witnesses in boxes of arbitrarily large size. In particular the limit is self-avoiding.

Fix $i<j$ and any finite competing path from $v_i$ to $v_j$ in the whole lattice. That competitor lies in some box $B_m$. For a sufficiently large selected witness, its segment indexed from $i$ to $j$ agrees with the limit, and both it and the competitor lie in its box. The witness is shortest there, so the passage time of the limiting segment is no greater than the competitor's time. Taking the infimum over all finite competitors proves the bigeodesic property.$\square$

Consequently \begin{equation}\label{wsref-0-72-61-34-38-35-3a-70-72-6f-62-61-62-69-6c-69-74-79-74-61-72-67-65-74} \mathbb P(\text{a bigeodesic visits }0) =\lim_{n\to\infty}\mathbb P(F_n). \end{equation} By translation invariance, vanishing of this one probability implies absence of every bigeodesic: the existence event is the union over countably many vertices of the event that one visits that vertex. Conversely absence certainly forces the probability at zero to vanish. Thus (ra485:probabilitytarget) is an exact reduction of the target, without directions, shape curvature, or moments stronger than those given.

The use of box distances is legitimate only because the exhaustion argument eventually tests every finite global competitor. For a fixed $n$, a $T_n$-geodesic need not be a global geodesic: leaving the box may produce a shortcut. Neither one finite witness nor witnesses whose centers escape to infinity establish a bigeodesic through a fixed vertex.

Trying to close the reduction with available sparsity.

Let $\mathcal T_x$ denote the union of all singly infinite global geodesics starting at $x$. The theorem in [E1] gives \[ q_n:=\sup_{\|v\|_\infty=n}\mathbb P(v\in\mathcal T_0) \longrightarrow0. \] If a bigeodesic visits zero, one of its arms first reaches some $v\in\partial B_n$. Starting at $v$ and following the other direction along that bigeodesic gives an infinite geodesic containing zero. Thus \[ \{0\text{ lies on a bigeodesic}\} \subseteq\bigcup_{v\in\partial B_n}\{0\in\mathcal T_v\}. \] Translation invariance changes the probability in each term to $\mathbb P(-v\in\mathcal T_0)$. Since $|\partial B_n|=8n$, \begin{equation}\label{wsref-0-72-61-34-38-35-3a-75-6e-69-6f-6e-6c-6f-73-73} \mathbb P(0\text{ lies on a bigeodesic})\leq8nq_n. \end{equation} The verified result $q_n\to0$ does not make this upper bound tend to zero. It would suffice to prove $q_n=o(n^{-1})$, but that stronger rate is not established here or supplied by the quoted theorem.

Equivalently, the expected number of boundary vertices $v$ whose infinite geodesic tree contains zero is $o(n)$. A bigeodesic only requires a bounded number of suitable arms at each scale; sublinear expectation does not exclude that possibility. Turning this sparsity into absence requires more geometric information than the direct union estimate.

The finite formulation has a similar obstruction. There are $\binom{8n}{2}$ possible endpoint pairs in (ra485:finiteevent). A probability tending to zero for each specified pair is not enough to bound their union. A uniform $o(n^{-2})$ bound would suffice by the union bound, but is again a stronger proposed input, not a consequence established in this attempt. The finite and infinite formulations involve different events, so their probabilities are not silently substituted for one another.

A complete restricted exclusion by local detours.

Proposition 485.4. Almost surely no horizontal or vertical coordinate line is a bigeodesic.

Proof. Non-atomicity ensures that the law is not concentrated at a single value. Choose finite $0<a<b$ with $r=\mathbb P(t\leq a)>0$ and $s=\mathbb P(t\geq b)>0$. Take an integer $m$ so large that $mb>(m+2)a$. Consider a segment of $m$ horizontal edges on a fixed horizontal line, and the parallel segment one row above, joined to its endpoints by two vertical edges. Require all $m$ original edges to have time at least $b$, and all $m+2$ detour edges to have time at most $a$. The two edge sets are disjoint, so this event has probability $s^m r^{m+2}>0$. On it, the original line segment is not minimizing.

Place infinitely many such rectangles with disjoint edge sets along the line, separated by at least one unused column. Their improvement events are independent. The probability that none of the first $N$ improvements occurs is $(1-s^m r^{m+2})^N$, which tends to zero. Hence the fixed line fails the bigeodesic condition almost surely. Apply the same argument to vertical lines and take the countable intersection over all coordinate lines.$\square$

This proof works even when the support is a very short interval: one chooses $m$ large enough to offset the two extra detour edges. It does not require zero weights, large gaps in the support, or a density for the law. Its limitation is that the tested line was fixed before the environment was sampled. A random optimizing path preferentially avoids bad rectangles, so those independent events cannot simply be reused along a path selected from the same weights.

Direction and quantifier stress tests.

Even a theorem excluding one prescribed direction at a time need not exclude a path with a random exceptional direction. The logical issue already occurs for a uniform random variable $U\in[0,1]$: $\mathbb P(U=\theta)=0$ for every deterministic $\theta$, while the random set $\{U\}$ is nonempty with probability one. An uncountable union of null events need not be null. Nor has this attempt proved that an arbitrary infinite geodesic possesses a limiting direction. The finite-box criterion avoids both assumptions but still needs its own probability estimate.

There is also no inference from unique finite geodesics to the absence of bigeodesics. On an infinite tree with strictly positive edge lengths, the unique combinatorial path between two vertices is always minimizing, and every doubly infinite simple path is a bigeodesic. This example is outside the planar square-lattice model; it shows precisely that local uniqueness alone cannot be the missing principle. The ability to make competing detours in the square lattice must be exploited on the random path itself.

Finite verification and outcome.

The companion exact-integer program checks (ra485:finiteevent) by shortest-path computations in coupled boxes of radii one through four. It uses explicitly generated positive rational weights with distinct sums for distinct simple edge sets, implemented by an integer part and binary perturbations. These deterministic finite environments are not asserted to be samples from the infinite non-atomic product law. Among the 24 tested environments, the numbers with a root crossing were respectively $24,22,21,19$ at radii $1,2,3,4$. All 72 successive nestedness comparisons passed, and each distance-additivity witness was checked to contain zero in its reconstructed shortest path. The full output is recorded separately; no observed finite frequency is extrapolated to (ra485:probabilitytarget).

The script also verifies the heavy-tail moment formula and the detour inequality using rational parameters. All stochastic claims used in the proofs come from the explicit independence, countability, and summability arguments above, not from those finite computations.

Outcome: UNRESOLVED. The work proves finite geodesic existence and uniqueness, existence of one-sided geodesics, the exact nested finite-box criterion, and exclusion of every coordinate line. These are elementary reductions and benchmarks, not a new no-bigeodesics theorem. The infinite-geodesic sparsity theorem [E1] is an external established input whose limitation in the direct union argument is quantified by (ra485:unionloss).

The first missing statement is $\mathbb P(F_n)\to0$ under the original minimum-moment hypothesis. Concrete possible next steps are a multiscale argument that bounds this root-crossing event without paying for every boundary pair, a stronger control of the random set of geodesic ancestors than its expected size, or a detour construction that remains valid after the candidate path has been chosen from the environment. None is established here. No prescribed-direction, finite-mean, or shape-regularity condition is added to the unresolved target.

Sources

• S1 Furstenberg No-Bigeodesics (Planar FPP, Standard Minimum Moment); exact editorial selection using the primary formulations and hypotheses recorded in the source receipts. https://arxiv.org/pdf/1512.00804. Formulation source.
• E1 D. Ahlberg, J. Hanson, and C. Hoffman, The number of geodesics in planar first-passage percolation grows sublinearly, arXiv:2208.11576v1, 24 August 2022, Theorem 1.1 and Section 2.4, assumption A1. https://arxiv.org/pdf/2208.11576. Primary author manuscript; exact minimum-moment scope and uniform specified-vertex conclusion verified 27 September 2026.
• E2 M. Damron and J. Hanson, Bigeodesics in first-passage percolation, arXiv:1512.00804v2, 1 July 2016, Theorem 1.1, assumption A1', and Theorem 1. https://arxiv.org/abs/1512.00804v2. Bibliographic identification of supplied source [S1]; distinguishes the selected minimum moment from the extra geometric hypotheses in the proved directional theorem. Consulted 27 September 2026.