内容来源：https://www.quantamagazine.org/stunning-percolation-proof-solves-decades-old-puzzle-about-phase-transitions-20260831/

内容总结：

五数学家破解渗流理论数十年难题，证明“尖锐相变”普适性

2025年圣诞节前夕，五位数学家聚集在苏黎世联邦理工学院的一间教室里，完成了一项困扰领域数十年的重大突破。他们成功证明了渗流理论中关于相变“尖锐性”的核心猜想，该成果被誉为“令人惊叹的瑰宝”。

渗流理论研究流体在网络中的流动，其原型可追溯至20世纪40年代。当时，因DNA研究而闻名的科学家罗莎琳德·富兰克林在研究煤炭的孔隙结构时，提出了相关基础问题。此后，西蒙·布罗德本特和约翰·哈默斯利于1957年左右建立了数学模型，用概率方式模拟流体能否在多孔介质中长距离流动。

该理论的核心概念是“临界概率”。当连接概率低于此阈值时，流体只能形成孤立小水洼；一旦超越该值，系统会突变为一个贯通的大型流体网络，类似水结成冰的相变。自20世纪80年代在规则晶格上证明“尖锐性”后，数学界一直期待将此性质推广至更广泛的“传递图”（即所有顶点看起来都相同的图）。

1996年，数学家伊泰·本杰米尼和奥德·施拉姆开启了这一探索方向，但证明进展缓慢。尤其对于临界点以上的“超临界半部分”，即证明流体一旦贯通，孤立大水池的出现概率极低，被视为“最后的堡垒”。施拉姆于2008年不幸离世，研究一度停滞。

此次，由博士后萨哈尔·迪斯金、菲利普·埃阿索，研究生里特维克·拉马南·拉达克里希南，以及教授本尼·苏达科夫和文森特·塔西翁组成的团队，最初并未瞄准此目标。在学期截止压力下，他们调整方向，却意外发现了一个极其简洁的证明思路。团队采用并改良了概率论中的“喷洒”技巧，通过先分析预先留出的开放边，再研究大流体池的“海岸线”结构，最终证明：只要概率略高于临界值，出现孤立大型流体池的可能性趋近于零，流体几乎覆盖整个无限传递图。

该证明统一适用于所有无限传递图，具有极强的普适性。同行专家评价其“每一个句子都熟悉简单，但组合方式真正新颖”。这一成果不仅解决了本杰米尼和施拉姆提出的长期猜想，也为研究更复杂系统（如三维晶格临界点行为、量子材料等）提供了新工具。本杰米尼感叹：“我们等待了十年，它是一颗宝石。”

中文翻译：

“惊艳”的渗流证明解开关于相变的数十年谜题

引言

2025年圣诞节前一周，五位数学家齐聚苏黎世联邦理工学院的一间教室。气氛激动人心：他们离一项足以定义职业生涯的突破仅一步之遥。

这个团队——由当时的博士后萨哈尔·迪斯金和菲利普·埃索、研究生里特维克·拉马南·拉德哈克里希南、本尼·苏达科夫和文森特·塔西翁组成——正在完善渗流理论中最大开放问题之一的解决方案。渗流理论研究的是网络中的流动。

渗流涵盖了极为广泛的现象，但其典型例子涉及流体，比如热水渗过咖啡粉层。迪斯金、埃索、拉德哈克里希南、苏达科夫和塔西翁试图解决一个关于图（由线或边连接的点构成的网络）如何被大型连通区域——相当于流体汇聚的“水潭”——所覆盖的基本问题。这个团队瞥见了一个能够同时处理大量不同类型图的简单论证。

“我们一开始几乎不敢相信，”拉德哈克里希南说。

他们争分夺秒地核实每一个细节，急于在想法消散之前将其记录下来——完全顾不上假期。“我不确定女朋友和家人像我们一样开心。但那一刻我们所有人都非常开心，”迪斯金说。“能碰到一个感觉如此宏大、如此有意义的东西，真的是非常罕见的。”

他们通宵工作。到12月17日早晨，筋疲力尽却兴奋不已的他们确信自己的想法是正确的。到圣诞节时，他们已经敲定了一个证明。

他们回答了一个困扰数十年的问题：当你将渗流网络开放给流体时，它填满的速度有多快？“这个证明让我感到极大的喜悦，”特拉维夫大学的阿萨夫·纳赫米亚斯说，他研究渗流理论和概率论。“它令人惊艳。”

本尼·苏达科夫

富兰克林的流体

渗流可以描述多种流动：病毒在城市中的传播、气体通过过滤器、或野火的蔓延。但它最初的灵感来自煤炭。

20世纪40年代，科学家罗莎琳德·富兰克林——如今因其在DNA结构方面的工作而闻名——受雇于英国煤炭利用研究协会（BCURA），试图理解煤、木炭和石墨的复杂性质。科学家们知道煤中布满了微小的孔洞，但他们不知道为什么某些类型的煤允许流体通过，而另一些则不可渗透。

通过将煤浸入各种流体中，富兰克林能够测量其孔洞的典型尺寸以及变异程度。大约十年后，研究人员西蒙·布罗德本特和约翰·哈默斯利——他们想理解防毒面具中的碳过滤器——开发了一个数学模型。

他们的想法很简单。取一个均匀分布点的网格（也称为格点）和一枚硬币。对于每对相邻点，抛掷硬币。如果正面朝上，就用一条边连接这两个点，流体可以在这些点之间流动。如果反面朝上，流动就被阻断。对每对点重复这一过程。流体能流多远？

答案取决于硬币正面朝上的概率，这个概率可以从0%到100%。当概率较低时，流体只能通过少数渠道流动，因此它会聚集在小型孤立的水洼中。

但一旦概率超过一个称为临界概率的阈值，格点突然打开。流体可以在系统中广泛传播。

临界概率的确切值取决于格点的形状——正方形格点的临界概率不同于三角形格点，三维格点的临界概率也不同于二维格点。但当你超过临界值向上移动时，你会看到相变，就像液态水变成冰。在有限图上，跨越临界概率意味着网络将被一个巨大的流体海洋所主导。在无限图上——一种抽象，其中图在所有方向上无限延伸——一个或几个无限海洋将占据主导地位。物理学家很快意识到，通过渗流，他们可以了解融化和冻结以及其他相变，如磁化。

“物理学中的相变很难严格研究，”埃索说。“渗流就像它的漫画版。所以人们通常先研究它，然后工具再逐渐渗透下去。”

对于那些长期被复杂的现实世界相变所困扰的科学家来说，“它在一个非常简单的设置中捕捉到了本质，”魏茨曼科学研究所的伊泰·本杰米尼说。“很多可能使问题模糊不清的东西都被移除了。”

几十年来，研究人员致力于量化渗流相变。他们想知道，当你增加硬币正面朝上的概率时，流体水潭能多快地增长。许多人预测水潭增长得非常快——在临界概率以下，水洼很小；在临界概率以上，一个单一的海洋几乎覆盖了一切。这个预测被称为“尖锐性猜想”。

当尖锐性在20世纪80年代在格点上被证明时——由两个独立团队，一个在新泽西，一个在莫斯科——它是“奠基性的，”普林斯顿大学和加州理工学院的汤姆·哈奇克罗夫特说，他是埃索的博士导师。知道了尖锐性，数学家可以推断出网络中被淹没部分的大量结构信息——特别是，它看起来与底层格点非常相似。

因此，当本杰米尼和他的同事奥德·施拉姆规划了一次将渗流带到新型网络的远征时，很自然地会想知道尖锐性是否会随之一同延续。

脱离网格

1996年，本杰米尼和施拉姆想研究更大一类图中的渗流，这类图称为传递图。要理解什么是传递图，可以把图想象成平坦荒凉景观上的道路网络。如果你想在这些道路上确定自己的位置，唯一的地标是交叉路口。但如果图是传递的，所有交叉路口看起来都相似——要弄清楚你在哪里，你需要GPS或指南针。

正方形格点是传递图的一个例子：每个交叉路口由四条边以直角相交组成。但传递图有很多种——简单环（下图左）和无限扩展的“树”（下图右），以及几乎无法可视化的类型。

许多传递图代表了其他数学子领域中的对象，如代数或几何。本杰米尼对这些跨学科的可能性很感兴趣——他希望渗流过程能揭示图本身的洞见。“你有一个舞台，即几何，和一个舞者，即随机过程，”他说。通过观察舞者，他希望更多地了解舞台。

在接下来的十年里，本杰米尼、施拉姆及其同事发表了一系列关于传递图渗流的结果。他们证明了，对于一类无限传递图，当你开放边供流体通过时，渗流表现出相变：小而孤立的流体水潭突然凝聚成一个无限的连通河流网络。

但他们仍然不知道这种转变发生得有多快。在临界点以下，水潭有多大、有多少？在临界点以上，无限网络是一片溪流纵横的草地——还是更像海洋，淹没整个图？

本杰米尼和施拉姆怀疑尖锐性猜想的一个版本在所有无限传递图上都成立。这个猜想可以分解为两个独立的问题。“亚临界”部分——处理临界点以下发生的情况——于2007年由通奇·安图诺维奇和伊万·韦塞利奇完成。他们的工作表明，在临界点以下，流体水潭很小且相距很远。即使在低于临界点一根头发丝的距离，系统看起来也更像亚利桑那州而不是明尼苏达州。

猜想的“超临界”部分——处理临界阈值以上的概率——看起来更难。在这里，景观应该由可能多个海洋组成，每个海洋都无限大。在这种情况下，不与无限海洋相连的大型水潭变得极为罕见。这是因为一个大型孤立水潭只有在周围有大量干燥土地——即闭合边——的情况下才能保持孤立。

但超临界尖锐性的证明似乎难以企及。首先，之前的工作没有帮助：格点上超临界尖锐性的证明冗长而复杂，无法推广到更一般的情况。虽然其他基础结果在过去十年中被简化了，“这是唯一剩下的堡垒，”纳赫米亚斯说。

研究这个问题的数学家们“做了非常漂亮的工作，开创了理论，摘取了所有低垂的果实，”本杰米尼说。“然后我们开始撞墙。”

2008年，随着非格点渗流的进展放缓，施拉姆在徒步时坠落去世，年仅46岁。“我们失去了一位天才，奥德·施拉姆，因为一场悲剧性的事故，”本杰米尼说。“然后我们需要等待一些新天才的到来。”

大约十年前，这个领域开始再次加速。但证明超临界尖锐性仍然困难。

然后，苏黎世的团队产出了一个简单证明。

一个尖锐的转折

迪斯金、埃索、拉德哈克里希南、苏达科夫和塔西翁本来无意证明超临界尖锐性。在2025年秋季的大部分时间里，他们在试图理解临界概率如何随图中的边数变化。

但这五位数学家在学期结束前想要得到结果。随着截止日期临近，他们仍然没有任何像证明的东西。于是埃索建议转向尖锐性。他和迪斯金、拉德哈克里希南取得了一些进展，并把结果带给苏达科夫和塔西翁。当塔西翁消化他们的工作时，一个想法——也许是一个出格的想法——在他脑海中形成。

他认为，经过一些调整，他们的策略可能足够强大，能够证明所有无限传递图的尖锐性。“从那以后，它日日夜夜都在我脑子里，”塔西翁说。

“文森特对它疯魔了，”迪斯金说。“我觉得他至少两周没睡觉。”

不只是塔西翁。在那几周里，合作变得狂热。数学家们不断交流想法，经常深夜发短信。“我们真的都有这种直觉，觉得这里面可能有东西，”迪斯金说。“一开始我们半开玩笑……也许同一个想法能解决这个巨大的猜想。我们都互相笑，但万一呢，万一呢？”

激进的简洁

满怀兴奋，加上假期临近，他们决定是时候认真起来，写他们的论文了。

为了证明在临界概率以上一个大型孤立流体水潭是不可能的，数学家们假设存在这样一个水潭，并研究了周围的岸线。沿着岸线，有溪流汇入水潭，但也有溪流连接回其中一个无限海洋。如果这些溪流在任何地方交汇，数学家们就会得到一个矛盾——他们所谓的有限流体水潭实际上会是无限海洋的一部分。

如果水潭很大，岸线就长——这意味着有更大的区域可能让水潭连接到其中一个无限海洋。这五位数学家证明了这使得几乎不可能避免这个矛盾。

当他们敲定论文的最后一个细节时，他们突然看到，通过一个简单的改变，他们的论证可以得到大幅改进。

他们一直在使用概率论中一种常见的技术，称为“撒布”：他们预留几条开放边，相当于略微降低临界概率。然后他们在其余边中寻找大型水潭，并分析其周围的开放路径。由于预留的边与水潭无关，它们可以被独立分析。这使得更容易证明，一旦与图的其余部分结合，它们几乎总会创建一条通向某个无限海洋的路径。

但在讨论中，他们想到了一个非正统的改进方案。如果先分析撒布的边，证明会大大简化。更重要的是，它增强了论证，使其对所有无限传递图都成立。“我们进行了这种想法的乒乓对打，”迪斯金说。“每次你把想法互相抛来抛去，突然这堵墙就变得更加模糊，直到它完全消失。然后有点可怕，因为你可能真的得到了它。”

最终，他们确信自己证明了：如果概率在临界阈值以上的任何位置，哪怕只高一点点，流体就会覆盖几乎整个传递图。

两个月后，他们发布了一篇论文。他们的论证适用于任何无限传递图上的渗流。“如果你放大证明中的每一句话，它感觉非常熟悉和简单，但他们将其组合在一起的方式是真正新颖的，”纳赫米亚斯说。

他们这项技术可以应用的研究不足的渗流系统并不缺乏——比如节点并非全部相同的图，或者描述结冰水或量子材料的更复杂模型。

不过仍有一个重大问题：在三维格点上——最贴近物理系统的图——在临界概率处究竟会发生什么？是否存在无限海洋？

这个问题的进展对本杰米尼来说尤为重要，他等待了十年才让他的远征重新启动。“对于这个社区，对于我们来说，这是一个非常深刻而有意义的定理，它是拼图的一部分，”他说。

对于这个证明，本杰米尼说：“它是一颗宝石。它是一颗宝石。”

英文来源：

‘Stunning’ Percolation Proof Solves Decades-Old Puzzle About Phase Transitions
Introduction
The week before Christmas 2025, five mathematicians were holed up in a classroom at ETH Zurich. The mood was electric: They were this close to a career-defining breakthrough.
The group — consisting of then-postdocs Sahar Diskin and Philip Easo, graduate student Ritvik Ramanan Radhakrishnan, Benny Sudakov, and Vincent Tassion — was perfecting a solution to one of the biggest open problems in percolation theory, the study of flow in a network.
Percolation captures a vast array of phenomena, but the prototypical examples involve fluids, like hot water seeping through a bed of coffee grounds. Diskin, Easo, Radhakrishnan, Sudakov, and Tassion were trying to work out something fundamental about how graphs — networks of points connected by lines, or edges — can be taken over by large connected areas, the equivalent of pools of fluid. The group had glimpsed a simple argument that could deal with a huge variety of graphs at once.
“We almost didn’t believe it at first,” Radhakrishnan said.
They raced to confirm each detail, eager to get their idea down before it shimmered away — and heedless of the holiday. “I’m not sure the girlfriends and the families were as happy as we were. But we were all very happy at that moment,” Diskin said. “It’s really rare that you’re able to hit something that feels so big and so meaningful.”
They worked through the night. By the morning of December 17, exhilarated from the effort, they were convinced their idea was correct. By Christmas, they’d nailed down a proof.
They had answered a decades-old question about how fast a percolation network floods as you open it up to fluid flow. “I find great joy in this proof,” said Asaf Nachmias of Tel Aviv University, who studies percolation theory and probability. “It’s stunning.”
Benjamin Sudakov
Franklin’s Fluids
Percolation can describe many kinds of flow: the spread of a virus through a city, gas passing through a filter, or the propagation of a wildfire. But its original inspiration was coal.
In the 1940s, the scientist Rosalind Franklin — now famous for her work on the structure of DNA — was employed at the British Coal Utilization Research Association (BCURA), trying to understand the intricate properties of coal, charcoals, and graphite. Scientists knew that coal was studded with tiny holes, but they didn’t know why some types of coal allowed fluids to pass through them, while others were impermeable.
By submerging coal in a variety of fluids, Franklin was able to measure the typical size of its holes, as well as the amount of variation. About a decade later, the researchers Simon Broadbent and John Hammersley — wanting to understand the carbon filters in gas masks — developed a mathematical model.
Their idea was simple. Take a grid of evenly spaced points (also called a lattice) and a coin. For each pair of neighboring points, flip the coin. If it lands on heads, connect the points with an edge. Fluid can flow between these points. If the coin lands on tails, the flow is blocked. Repeat this procedure for every pair of points. How far does the fluid go?
Mark Belan/Quanta Magazine
The answer depends on the probability that your coin lands on heads, which can range from 0% to 100%. When the probability is low, fluid can flow through only a few channels, and so it collects in small, isolated puddles.
But once the probability passes a threshold called the critical probability, the lattice suddenly opens up. Fluid can travel extensively through the system.
The exact value of the critical probability changes depending on the shape of the lattice — a square lattice has a different critical probability than a triangular one, and a 3D lattice has a different critical probability than a 2D one. But as you move above that critical value, you’ll see a phase transition, like liquid water turning to ice. On a finite graph, crossing the critical probability means the network will become dominated by one large sea of fluid. On an infinite graph — an abstraction where the graph extends forever in all directions — one or several infinite seas will dominate. Physicists quickly realized that through percolation, they could learn about melting and freezing, as well as other phase transitions like magnetization.
“Phase transitions in physics are very hard to study rigorously,” Easo said. “Percolation is like the caricature. So people often try to study that first, and then tools trickle down.”
For scientists who had long been stymied by complicated real-world phase transitions, “it was catching the essence in a very simple setup,” said Itai Benjamini of the Weizmann Institute for Science. “A lot of things that could cloud the issue were removed.”
Courtesy of Itai Benjamini
For decades, researchers worked to quantify the percolation phase transition. They wanted to know exactly how quickly pools of fluid can grow as you increase the probability that your coin lands on heads. Many predicted that the pools grow very fast — that below the critical probability, puddles are tiny, and above it, a single ocean covers almost everything. This prediction is called the sharpness conjecture.
When sharpness was proved on lattices in the 1980s — by two independent groups, one in New Jersey and one in Moscow — it was “foundational,” said Tom Hutchcroft of Princeton University and the California Institute of Technology, who was Easo’s doctoral adviser. Knowing sharpness, mathematicians can deduce a lot about the structure of the flooded portion of the network — in particular, that it looks very similar to the underlying lattice.
So when Benjamini and his colleague Oded Schramm plotted an expedition to bring percolation to new types of networks, it seemed natural to wonder if sharpness would go with them.
Off the Grid
In 1996, Benjamini and Schramm wanted to study percolation in a much larger class of graphs, called transitive graphs. To understand what a transitive graph is, imagine the graph as a network of roads on a flat, desolate landscape. If you want to know where you are on these roads, the only landmarks are the intersections. But if the graph is transitive, all the intersections look similar — to figure out where you are, you’ll need GPS or a compass.
A square lattice is one example of a transitive graph: Every intersection consists of four edges meeting at right angles. But there are many kinds of transitive graphs — simple loops (below left) and infinitely expanding “trees” (below right), as well as ones that are almost impossible to visualize.
Many transitive graphs represent objects from other mathematical subfields, like algebra or geometry. Benjamini was intrigued by these interdisciplinary possibilities — he hoped the percolation process would reveal insights into the graph itself. “You have a stage, which is geometry, and a dancer, which is the random process,” he said. By watching the dancer, he hoped to learn more about the stage.
Over the next decade, Benjamini, Schramm, and their colleagues published a flurry of results on the percolation of transitive graphs. They proved that, for a class of infinite transitive graphs, percolation exhibits a phase transition as you open up edges to flow: Small, isolated pools of fluid suddenly coalesce into an infinite web of connected rivers.
But they still didn’t know how fast that transition happened. Below the critical point, how big and how numerous were the pools? Above it, was the infinite web a meadow crisscrossed with streams — or was it more like an ocean, swamping the entire graph?
Benjamini and Schramm suspected that a version of the sharpness conjecture was true on all infinite transitive graphs. That conjecture could be broken down into two separate problems. The “subcritical” half — addressing what happens below the critical point — was completed in 2007, by Tonći Antunović and Ivan Veselić. Their work showed that here, pools of fluid are tiny and far apart. Even a hair below the critical point, the system looks more like Arizona than Minnesota.
The “supercritical” half of the conjecture — which deals with probabilities above the critical threshold — seemed harder. Here, the landscape should be made up of possibly many seas, each infinitely large. In this scenario, large pools that are not connected to the infinite seas become exceedingly rare. That’s because a large, isolated pool can only stay separate if there is a lot of dry land — or closed edges — around it.
But a proof of supercritical sharpness seemed unattainable. For one thing, the previous work was no help: A proof of supercritical sharpness on lattices was long and complicated, and it couldn’t be adapted to the more general case. While other foundational results were simplified in the last decade, “this was the one remaining fortress,” Nachmias said.
Mathematicians working on this problem “did some very beautiful things, initiated the theory, picked all the low-hanging fruit,” Benjamini said. “And then we started hitting the wall.”
In 2008, as progress on non-lattice percolation slowed, Schramm died at age 46 in a fall while hiking. “We lost a genius, Oded Schramm, to a tragic accident,” Benjamini said. “And then we needed to wait for some new geniuses to come.”
About a decade ago, the field began to accelerate again. But proving supercritical sharpness remained difficult.
Then, the team in Zurich produced a simple proof.
A Sharp Turn
Diskin, Easo, Radhakrishnan, Sudakov, and Tassion didn’t intend to prove supercritical sharpness. For most of fall 2025, they were trying to understand how critical probability scales with the number of edges in graphs.
But the five mathematicians wanted results by the end of the semester. As that deadline neared, they still had nothing resembling a proof. So Easo suggested pivoting to sharpness. He, Diskin, and Radhakrishnan made some progress and brought their results to Sudakov and Tassion. As Tassion took in their work, an idea — perhaps an outrageous one — formed in his mind.
He thought that, with some tweaks, their strategy might be strong enough to prove sharpness for all infinite transitive graphs. “From there, it was in my head day and night,” Tassion said.
“Vincent went crazy with it,” Diskin said. “I think he didn’t sleep for two weeks at least.”
It wasn’t only Tassion. Over those weeks, the collaboration became frenzied. The mathematicians traded ideas constantly, often texting late at night. “We really all had this hunch that there might be something to it,” Diskin said. “We were half joking at the beginning … maybe the same idea could resolve this huge conjecture. We were all laughing at each other, but what if, what if?”
Radical Simplicity
Brimming with excitement, and with the holidays looming, they decided it was time to get serious and write their paper.
To prove that a large isolated pool of fluid is unlikely above the critical probability, the mathematicians assumed they had such a pool and studied the surrounding shoreline. Along that shoreline, there were streams emptying into the pool, but there were also streams that linked back to one of the infinite seas. If those streams coincided anywhere, the mathematicians would have a contradiction — their so-called finite pool of fluid would actually be part of an infinite sea.
If the pool was big, the shoreline was long — meaning a larger area where the pool might connect to one of the infinite seas. The fivesome showed that this made it nearly impossible to avoid the contradiction.
As they hammered out the last details of their paper, they suddenly saw that with a simple change, their argument could be drastically improved.
They had been using a common technique in probability theory called sprinkling: They set aside a few of their open edges, corresponding to a slight lowering of the critical probability. They then looked for a large pool among the rest of the edges and analyzed the open paths around it. Since the set-aside edges had nothing to do with the pool, they could be analyzed independently. That made it easier to prove that, once combined with the rest of the graph, they almost always created a path to one of the infinite seas.
But as they talked, they hit upon an unorthodox improvement to this strategy. If they analyzed the sprinkles first, the proof got a lot simpler. What’s more, it strengthened the argument enough that it worked for all infinite transitive graphs. “We had this ping-pong of ideas,” Diskin said. “Every time you throw ideas one at another, suddenly this wall becomes more blurry, until it vanishes completely. Then it’s a bit scary, because you might actually have it.”
Finally, they were sure they had proved it: If the probability is anywhere above the critical threshold, even just a smidge, then fluid covers nearly the entire transitive graph.
Two months later, they posted a paper. Their argument applies to percolation on any infinite transitive graph. “If you zoom into every sentence in the proof, it feels very familiar and simple, but the way they put it all together is genuinely novel,” Nachmias said.
There is no shortage of unstudied percolation systems that their technique could apply to — like graphs where the nodes don’t all look identical, or more complicated models that describe freezing water or quantum materials.
A major question remains, though: On three-dimensional lattices — the graphs that most closely mirror physical systems — what happens exactly at the critical probability? Is there an infinite sea?
The progress on the problem is especially significant to Benjamini, who waited a decade for his expedition to start up again. “For the community, for us, it’s a very deep and meaningful theorem, and it’s a part of the puzzle,” he said.
Of the proof, Benjamini said, “it’s a gem. It’s a gem.”