[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"news-slug-sketchpad-certified-arithmetic-small-vlm":3,"topics-all":38,"news-related-65de4df2-898d-4202-8f69-ac03e203e34c":57},{"id":4,"title":5,"summary":6,"content":7,"original_url":8,"source_id":9,"tags":10,"translations":24,"news_slug":31,"published_at":32,"created_at":33,"modified_at":34,"is_published":35,"publish_type":36,"image_url":14,"view_count":37},"65de4df2-898d-4202-8f69-ac03e203e34c","0.8B 模型算对 220 题:正确性由 Lean 证明","一个 0.8B 视觉语言模型被教会用草稿纸做竖式算术:每步只读三格格子,数字全部由模型自己写出,固定运行时只负责移动窗口。全部 2,446 个可能视图都在笔记本 GPU 的浏览器里逐项验过,论文用 Lean 证明这等价于任意长度都正确。十个对照模型里最高的答对 217 题,Sketchpad 220 题全对。","一个 0.8B 的模型在你的浏览器里做加法、长除法、开方和最大公约数，每一位都由它自己算出来，没有计算器也没有工具调用——而且这不是跑分跑出来的，是被证明过的。\n\n11 月 3 日挂上 arXiv 的 Sketchpad 听起来像个小把戏：给一个开源视觉语言模型一张\"草稿纸\"，让它像人一样竖式算术。真正的贡献在别处——它把只能抽样估计的\"长算术正确率\"，换成**有限、可检查、可形式化证明**的性质。\n\n## 难点不在长度，在错误是静音的\n\n长算术靠思维链、改数据格式、改位置编码硬撑，一轮轮把能算对的长度往后推。但它们有个共同点——**对没见过的输入，正确性只是估计值，错误是静默的**。\n\n论文的对照实验是注脚：十个模型、220 道预注册题目（10 到 1,000 位的加法、2 到 20 位的乘法，输出 token 预算 32,768），跑在 10 月 31 日。GPT-6.1 Sol 答对 217，Claude Opus 5.5 答对 216，Gemini 4 Argon 答对 207，错误全落在 300 位和 1,000 位的加法上。GPT-6.1 Sol 那三道错题写的是完整答案，格式长度跟对的题一样，预算也没用完——输出上看不出异常。\n\n工具调用是另一条路：工具精确，但计算被搬到了网络外面。论文找的是第三条路——**要保证对任意输入都正确，就得让\"无穷多输入上的行为\"由\"有限多事实\"钉死。**\n\n## 一张纸，2,446 个格子\n\n结构只有三块：一张纸存全部状态，一个模型把纸的局部映射成一步输出，一个运行时移动窗口、把输出写回纸上。关键约束在最前面：运行时**一个结果数字都不写**，它只抄写、移动窗口、决定何时停。数字都由模型吐出。\n\n模型每次只看三格，一格 32 像素，拼成 32×96 的图，经视觉编码器读成 3 个 image token。字形集固定 11 个（空白加 0 到 9），画面集合因此有限可穷举。每个运算一张表，列尽可能遇到的画面及正确输出：列加法 SUM 242 行，长乘法五个运算 1,553 行，减法 SUB 242 行，商位估计 EST 400 行，估计减一 DEC 9 行。合计 2,446 行——这就是全部可能性，写出来是张表，不是段概率。\n\n底座是 Qwen3.5-0.8B：24 层，隐藏维 1,024，6 层全注意力、18 层 Gated DeltaNet。被训练的是挂在 150 个投影矩阵上的 rank-16 LoRA，10,223,616 个参数，占模型 873,438,784 参数的 1.2%。训练在单张 H100 上跑了 19.8 分钟，第 1,200 步全过，第 1,600 步停。\n\n## 证书：在真机上验，在 Lean 里证\n\n\"margin\"是证书的度量：正确 token 的 logit 领先任何错误 token 的最小幅度，在全部 248,320 个词表上算，大于 0 即通过。在部署设备上——一台 Apple M4 笔记本的 GPU，浏览器里跑 WebGPU、16 位——2,446 行全部通过，最小 margin 是 3.02 logits，意味着证书能容忍任何小于 1.51 的 logit 扰动。服务器端 32 位 PyTorch 下是 3.96。为何要 margin：4-bit 量化过的解码器曾把某个适配器的 margin 从 +9.4 推到 −8.9，直接翻号。\n\n三个 Lean 定理钉住了它：定理 1 说在表上正确等价于在任意长度上正确；定理 3 说 margin 足够大就能吃掉数值扰动。定理 2 最狠：只要某一步能读到之前的步骤（思维链），任何有限测试都不够用。\n\n## 跑分：220 比 217，但先别欢呼\n\n那 220 道题跑在服务器端 PyTorch 上，不是浏览器：批量锁步执行，用了 64,700 个视图、1,001 次前向。Sketchpad 全对 220。五个开放权重模型在 149 到 199 之间，大多是长加法长乘法耗尽 token 预算；Qwen3.5-27B 答对 121；底座不用思维链 23 分、用了 3 分。\n\n论文给了句冷话：\"每个模型每题只跑一次，所以两个模型之间差几道题在采样波动范围内。\"按这句话，220 和 217 那 3 道题的差距不该当新闻。该读的数字在另一方向：底座裸跑 23 分，同一套底座接上那张草稿纸之后是 220 分。差的不是模型规模，是那张纸。\n\n## 边界：证书管得住模型，管不住运行时\n\n定理 1 把责任切成两半：模型那部分是有限的，已经验过。运行时那部分——版式、外层循环、参考过程，也就是经典竖式算法本身——是常规代码，**论文明确说没形式化到 Lean 里**。\"可证明\"覆盖模型的每一步，不等于整个程序被证明正确。\n\n定理 2 是必须付的账：论文构造了一个两状态表和一族解码器，每个成员都能通过任何一组有限测试、都有正的 margin，却会在更长的运行上出错。结论很直白——它**必须**放弃思维链。\n\n对话适配器是折中：需要精确计算时自动\"打开那张纸\"。它在指令跟随和 MMLU 上与底座差距在 2 分以内，但在 1,069 道开发中没用过的小学应用题上低 5.8 分。门控归零时逐比特还原底座，代价是每次前向多背 1.2% 的参数。\n\n## 所以呢\n\n最反直觉的不是小模型算对了，而是这篇论文顺手给思维链下判决：让一步能读到之前的步骤，恰恰是让\"算对\"彻底失去可证性的原因。行业这两年在\"让它多想一会儿\"上砸的钱，撞的就是这堵墙。\n\n另一件事是定价。可验证性不是模型自带的属性，是架构约束换来的：放弃思维链、放弃 5.8 分 GSM8K、把外层循环交给经典代码、每次前向多算那 1.2%。换来的也实在——一份绑死文件哈希、能在用户自己 GPU 上重算的证书；论文每个数字都列在 numbers.md 里，Lean 证明用 lake build 能验。一个能被第三方重新执行的证明，和一个 benchmark 分数，是两种信用。\n\n问题是：当\"算对\"能从采样估计变成可执行检查，还有多少任务值得用采样估计？以及——你愿为可验证性付出多大能力代价？\n\n（论文：[Sketchpad: certified exact arithmetic with a small vision-language model](https:\u002F\u002Farxiv.org\u002Fabs\u002F2611.04434)）","https:\u002F\u002Farxiv.org\u002Fabs\u002F2611.04434","7437aeb9-930c-4866-a2e9-48003c1a792b",[11,15,18,21],{"id":12,"name":13,"slug":13,"description":14,"color":14},"5e628969-6d2a-437f-998a-104e4b16cfb1","ai-progress",null,{"id":16,"name":17,"slug":17,"description":14,"color":14},"120fa59a-ff6f-4537-9bf5-f818df636a0e","benchmark",{"id":19,"name":20,"slug":20,"description":14,"color":14},"01598627-1ea6-4b27-a5d8-874971571a71","llm",{"id":22,"name":23,"slug":23,"description":14,"color":14},"499f4b56-819d-49a3-9609-33e775143b86","multimodal",[25],{"id":26,"lang":27,"title":28,"summary":29,"content":30},"989bc09f-f47c-414f-bb35-bb1d0dd38dde","en","Sketchpad: A 0.8B Model Proves Its Own Arithmetic in Lean","A 0.8B VLM does long arithmetic on paper. All 2,446 views pass on a laptop GPU; Lean proves it certifies any length. It hit 220\u002F220; best of ten 217.","A 0.8B model doing addition, long division, square roots and greatest common divisors in your browser, every digit computed by the model itself — no calculator, no tool call — and this is not a benchmark score but a proof. That is what Sketchpad, posted on arXiv on 3 November, delivers.\n\nThe demo sounds like a cute trick: hand an open-weight vision-language model a sheet of paper and teach it long multiplication the way a person does it with a pencil. The real contribution lies elsewhere — it converts a \"long-arithmetic accuracy\" metric that can only be estimated by sampling into a **finite, checkable, formally provable** property.\n\n## The hard part is not length, it is that errors are silent\n\nLong arithmetic is pushed forward by chain of thought, changed data formats and changed positional encodings, each round pushing further the length at which models still get it right. But they share one property — **on an unseen input, correctness is only an estimate, and the errors are silent**.\n\nThe paper's comparison is the footnote: ten models, 220 pre-registered problems (sums of 10 to 1,000 digits and products of 2- to 20-digit numbers, output token budget 32,768), run on 31 October. GPT-6.1 Sol answered 217 correctly, Claude Opus 5.5 answered 216, Gemini 4 Argon answered 207, with every error landing on 300-digit and 1,000-digit sums. The three wrong answers from GPT-6.1 Sol were complete answers, same format and length as the correct ones, with the token budget nowhere near exhausted — nothing in the output betrays them.\n\nTool use is the other route: tools are exact, but the computation moves outside the network. The paper looks for a third route — **to guarantee correctness on every input, behaviour on unboundedly many inputs must be pinned down by finitely many facts.**\n\n## One sheet, 2,446 cells\n\nThree parts: a sheet holding all state, a model mapping a local view of the sheet to one step's outputs, and a runtime that moves the view and writes the outputs back. The key constraint comes first: the runtime **writes no digit of a result**. It only copies digits, moves the view, and decides when to stop. Every digit is produced by the model.\n\nThe model sees three cells at a time. Each cell is 32 pixels, rendered as a 32 × 96 strip read by the vision encoder as 3 image tokens. The glyph set is fixed at 11 (blank plus 0 to 9), so the set of frames is finite and enumerable. Each operation gets a table listing every frame it can encounter with its correct outputs: column addition (SUM) 242 rows, the five operations of long multiplication 1,553 rows, subtraction (SUB) 242 rows, estimation of a quotient digit (EST) 400 rows, decrement of an estimate (DEC) 9 rows. That is 2,446 rows — the entirety of the possible situations, written out as a table rather than a probability.\n\nThe base model is Qwen3.5-0.8B: 24 layers, hidden size 1,024, with 6 layers of full attention and 18 of Gated DeltaNet. What was trained is a rank-16 LoRA on 150 projection matrices, 10,223,616 parameters, 1.2% of the model's 873,438,784. Training ran 19.8 minutes on a single H100, passing every row at step 1,200 and stopping at step 1,600.\n\n## The certificate: measured on the real device, proved in Lean\n\nThe \"margin\" is the certificate's yardstick: the smallest lead of a correct token over any wrong token, computed over the full 248,320-token vocabulary, passing if above zero. On the deployment device — the GPU of an Apple M4 laptop, running WebGPU in a browser at 16-bit precision — all 2,446 rows pass, the smallest margin being 3.02 logits, so the certificate tolerates any change of the logits smaller than 1.51. In 32-bit PyTorch on the server it is 3.96. Why margins at all: a 4-bit quantised decoder once moved some margins of a preliminary adapter from +9.4 to −8.9, flipping sign outright.\n\nThree Lean theorems pin this down. Theorem 1 says correctness on the tables is equivalent to correctness at every length. Theorem 3 says a margin large enough absorbs numerical perturbation. Theorem 2 is the harshest: if a step may read the steps before it — chain of thought — no finite test suffices.\n\n## The score: 220 against 217, but do not celebrate yet\n\nThose 220 problems ran in PyTorch on the server, not in the browser: batched lockstep, using 64,700 views and 1,001 forward passes. Sketchpad answered all 220. Five open-weight models landed between 149 and 199, mostly exhausting the token budget on long sums and long products; Qwen3.5-27B answered 121; the base model answered 23 without chain of thought and 3 with it.\n\nThe paper supplies its own cold water: each model was run once per problem, so a difference of a few problems between two models is within sampling variation. By that standard, the three problems between 220 and 217 are not news. The number to read lies elsewhere: the bare base model scores 23, and the same base model with the sheet attached scores 220. What changed was not model scale. It was the sheet.\n\n## The boundary: the certificate covers the model, not the runtime\n\nTheorem 1 splits the responsibility in two: the model's part is finite and already certified. The runtime's part — the layouts, the outer loops, the reference procedures, that is, the classical pencil-and-paper algorithms themselves — is ordinary code, and **the paper states explicitly that this part is not formalised in Lean**. \"Provable\" covers every step of the model; it does not mean the whole program has been proved correct.\n\nTheorem 2 is the bill that must be paid: the paper constructs a two-state table and a family of decoders such that every member passes any finite set of tests with positive margins, yet errs on a longer run. The conclusion is blunt — this design **must** give up chain of thought.\n\nThe dialogue adapter is the compromise: it opens the sheet when a reply needs exact computation. It stays within 2 points of the base model on instruction following and MMLU, but scores 5.8 points below the base model on 1,069 grade-school word problems not used during development. The gate restores the base model bit for bit at zero, at the cost that the adapter's branches are still evaluated there, adding 1.2% of its parameters to every forward pass.\n\n## So what\n\nThe most counter-intuitive thing is not that a small model got the arithmetic right, but that the paper hands down a verdict on chain of thought along the way: letting a step read the steps before it is precisely what makes \"getting it right\" lose its provability. The industry's two-year bet on \"let it think longer\" runs straight into that wall.\n\nThe second thing is the price tag. Verifiability is not something a model comes with; it is bought with architectural constraints: give up chain of thought, give up 5.8 points of GSM8K, hand the outer loop to classical code, evaluate an extra 1.2% of adapter parameters on every forward pass. What comes back is concrete — a certificate bound to file hashes that can be recomputed on the user's own GPU; every number in the paper is listed in numbers.md, and the Lean proofs are checked by running lake build. A proof a third party can re-execute and a benchmark score are two kinds of credit.\n\nThe question is: once \"getting it right\" can move from a sampling estimate to an executable check, how many tasks still deserve a sampling estimate? And how much capability are you willing to pay for verifiability?\n\n(Paper: [Sketchpad: certified exact arithmetic with a small vision-language model](https:\u002F\u002Farxiv.org\u002Fabs\u002F2611.04434))","sketchpad-certified-arithmetic-small-vlm","2026-11-04T07:00:00Z","2026-11-04T07:26:09.518741Z","2026-11-04T07:26:09.518753Z",true,"agent",79,[39,48],{"slug":40,"tag_slug":40,"title_zh":41,"title_en":42,"intro_zh":43,"intro_en":44,"id":45,"is_active":35,"created_at":46,"modified_at":47},"ai-for-science","AI for Science 2026：从 UniPert 到 GPT-Rosalind 的硬核进化","AI for Science 2026: from UniPert to GPT-Rosalind","生命科学、化学材料、物理世界模型——AI 正在从\"语言工具\"变成\"实验伙伴\"。本专题收录 AI 在三大科学方向的关键节点：UniPert 统一基因与化学扰动空间、GPT-Rosalind 端到端生命科学推理、达摩院 AI 智能体 28 小时找到 4 种超导新材料、Anthropic Claude Science 把工作台做成标准品。","From language tool to lab partner — AI is reshaping life sciences, chemistry\u002Fmaterials, and physical world models. This topic covers the key milestones: UniPert unifying genetic-chemical perturbation spaces, GPT-Rosalind's end-to-end life-sciences reasoning, DAMO's AI agent discovering 4 superconducting materials in 28 hours, and Anthropic's Claude Science workbench going mainstream.","988a4300-5fab-41c4-b5d8-63711a2dc757","2026-09-10T01:34:15.296649Z","2026-09-10T01:34:15.296663Z",{"slug":49,"tag_slug":49,"title_zh":50,"title_en":51,"intro_zh":52,"intro_en":53,"id":54,"is_active":35,"created_at":55,"modified_at":56},"h3-series","MiniMax H3 系列：从开源权重到 35 倍吞吐","MiniMax H3 Series: from open weights to 35x throughput","MiniMax H3 自 2026 年 8 月开源以来节奏密集：官方把生成、参考与编辑收回一个模型；ComfyUI 当天压进 RTX 3060；摩尔线程 3 小时完成国产 GPU 适配；fal 后训练版把吞吐拉到 35 倍；FastH3 蒸馏再砍推理成本。本专题持续追踪 H3 的发布—开源—蒸馏—部署全链路。","Since MiniMax open-sourced H3 in August 2026 the pace has been relentless: one unified omni-modal model, same-day ComfyUI support down to an RTX 3060, a 3-hour Day-0 port to Moore Threads GPUs, fal's post-trained H3 Max at 35x throughput, and FastH3 distillation cutting inference cost further. This topic tracks the full H3 chain — release, open weights, distillation, deployment.","83ef0daa-3c31-4cb3-86ed-e5ee58654d5f","2026-09-08T07:33:19.942193Z","2026-09-08T07:33:19.942209Z",{"items":58},[59,64,69,74,79,84],{"id":60,"title":61,"news_slug":62,"published_at":63},"5480e0ca-7b17-481c-8a24-8498f1a0aecd","8B模型OST涨13.41分:空间推理的接口锁定","chronoreplay-spatial-interface-lock-in","2026-10-28T05:09:43+00:00",{"id":65,"title":66,"news_slug":67,"published_at":68},"4b693fb4-541f-47ed-8923-6e280cec965f","大模型的“记忆”还没过视觉这一关：MEMLENS 把长上下文的短板测出来了","memlens-multimodal-long-term-memory","2026-08-03T02:00:00+00:00",{"id":70,"title":71,"news_slug":72,"published_at":73},"c3a956f8-dd42-46df-a1fd-1322dd38c15c","MentalThink 把 SVG 当作「心智草稿纸」:让多模态大模型学会用代码画心像做空间推理","mentalthink-svg-spatial-reasoning","2026-07-10T22:30:00+00:00",{"id":75,"title":76,"news_slug":77,"published_at":78},"39e6e64f-a3de-4dce-bdd8-c8643f9413a1","Orca：把\"世界状态\"焊进潜空间——BAAI 推出通用世界基础模型新范式","baai-orca-world-foundation","2026-07-03T02:00:00+00:00",{"id":80,"title":81,"news_slug":82,"published_at":83},"b0a2cefc-7a4e-4f2c-83a0-f1e4911f04e5","RNG-Bench：GPT-5.4\u002FGemini 3.1 Pro 闭环记忆现形","rng-bench-shanghai-ai-lab-non-markov-memory","2026-06-24T18:15:00+00:00",{"id":85,"title":86,"news_slug":87,"published_at":88},"94de69e8-d5b3-4199-ac19-aa7ca4e2299f","视频RL的奖励自己调权重:EmoReward公开代码","emoreward-video-rl-self-recalibrating-rewards","2026-11-06T03:15:00+00:00"]