{"id":8352,"date":"2026-05-08T08:18:00","date_gmt":"2026-05-08T00:18:00","guid":{"rendered":"https:\/\/meta-quantum.today\/?p=8352"},"modified":"2026-05-07T16:22:41","modified_gmt":"2026-05-07T08:22:41","slug":"axiom-math-at-the-montgomery-summit-fireside-chat","status":"publish","type":"post","link":"https:\/\/meta-quantum.today\/?p=8352","title":{"rendered":"Axiom Math at The Montgomery Summit | Fireside Chat"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">Introduction<\/h2>\n\n\n\n<p>This fireside chat features C<strong>arina Hong<\/strong>, founder and CEO of <strong>Axiom<\/strong>, in conversation with <strong>Matt Mcllwain<\/strong>, Managing Director at Madrona Venture Group (an investor in Axiom since day one), at The Montgomery Summit. Carina shares her journey from southern China \u2014 where she taught herself English to read the <em>Graduate Texts in Mathematics<\/em> (GTM) \u2014 through MIT (math and physics), Oxford as a Rhodes Scholar, and Stanford, where she dropped out of her concurrent PhD and law degree to start Axiom. Inspired by mathematical geniuses like Galois (who died at 21) and Ramanujan (who died around 30 from malnutrition and ill health), Carina is motivated by a haunting question: <em>what fundamental breakthroughs has humanity missed because we are bounded by the scarcity of outlier reasoning minds?<\/em> Axiom&#8217;s mission is to build superhuman AI reasoners \u2014 starting with the world&#8217;s first <strong>AI Mathematician<\/strong>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">About Axiom and Core Concept<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">The AI Mathematician<\/h3>\n\n\n\n<p>Axiom is building a general AI mathematician \u2014 not just a model that solves a narrow class of math problems, but a system that reasons across mathematics broadly. The historical trajectory is striking:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>2016<\/strong>: Google began applying deep neural networks to mathematics (pre-Transformer, pre-ChatGPT).<\/li>\n\n\n\n<li><strong>2019<\/strong>: Fran\u00e7ois Charton and Guillaume Lample (Mistral co-founder) showed Transformers could outperform Mathematica on symbolic integration. <em>Charton is now full-time technical staff at Axiom.<\/em><\/li>\n\n\n\n<li><strong>December 2024 (Putnam Exam)<\/strong>: Axiom Prover scored <strong>a perfect 120\/120<\/strong> on the Putnam Competition \u2014 an exam where the median score among 150,000 math majors is <strong>zero<\/strong>, and only <strong>five humans have scored a perfect 120 in 98 years<\/strong>. The top human this year (an MIT undergraduate) scored 110. The best LLM (DeepSeek by Mass Arena) scored 103.<\/li>\n\n\n\n<li><strong>Late January 2025<\/strong>: Axiom Prover <strong>independently solved four open research conjectures<\/strong> sent by professors from Tacoma Israeli University, Boston College, and Williams College \u2014 problems the professors themselves could not solve. Reported by Will Knight in WIRED.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Verified AI \u2014 The Real Product Insight<\/h3>\n\n\n\n<p>The crucial differentiator is not just answer-generation but <strong>verifiable correctness<\/strong>. Axiom Prover outputs <strong>computer programs for mathematical proofs<\/strong> (in Lean 4). If the program compiles and you see a check mark, the logic is <strong>100% verified<\/strong> \u2014 you don&#8217;t need to be a math professor to trust the output. This positions Axiom not just as a math tool but as the foundation for <strong>verification across the entire generative-AI economy<\/strong> (code verification, chip verification, agent verification).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Axiom Math \u2014 Complete Guide: Products, Installation, Setup, Configuration &amp; Prover Use Cases<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">1. Overview \u2014 What is Axiom Math?<\/h3>\n\n\n\n<p><strong>Axiom Math<\/strong> (<a href=\"http:\/\/axiommath.ai\">axiommath.ai<\/a>) is the Silicon Valley company founded by Karina Hong (featured in the Montgomery Summit fireside chat) building AI mathematicians and verified-AI infrastructure. They ship two distinct products developers should know about:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Product<\/th><th>What it is<\/th><th>Access<\/th><\/tr><\/thead><tbody><tr><td><strong>AxiomProver<\/strong><\/td><td>Their flagship autonomous multi-agent ensemble theorem prover for Lean 4 \u2014 the system that scored <strong>12\/12 on Putnam 2025<\/strong><\/td><td>Currently <strong>internal \/ partnership-based<\/strong> (not yet a public API). Solutions are open-sourced on GitHub.<\/td><\/tr><tr><td><strong>AXLE (Axiom Lean Engine)<\/strong><\/td><td>Public API + Python SDK + CLI exposing the proof verification and manipulation primitives Axiom uses internally to train AxiomProver<\/td><td>Public release \u2014 sign up at the console for API key<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>AXLE is what you can install and use today. AxiomProver is what <em>uses<\/em> AXLE under the hood.<\/p>\n<\/blockquote>\n\n\n\n<h3 class=\"wp-block-heading\">2. AXLE \u2014 The Axiom Lean Engine<\/h3>\n\n\n\n<p>AXLE provides interactive tools for exploring, validating, and manipulating Lean 4 mathematical proofs. According to the AXLE homepage, <em>&#8220;AXLE provides proof verification and manipulation primitives we&#8217;ve used across all of our research efforts, including training AI models and AxiomProver&#8217;s 12\/12 on Putnam 2025.&#8221;<\/em><\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Core toolset (all available via Python, CLI, and HTTP API)<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong><code>verify_proof<\/code><\/strong> \u2014 validate a candidate Lean theorem against a formal statement<\/li>\n\n\n\n<li><strong><code>check<\/code><\/strong> \u2014 check if Lean code is valid<\/li>\n\n\n\n<li><strong><code>extract_decls<\/code> \/ <code>extract_theorems<\/code><\/strong> \u2014 extract declarations from Lean source<\/li>\n\n\n\n<li><strong><code>rename<\/code><\/strong> \u2014 rename declarations<\/li>\n\n\n\n<li><strong><code>theorem2lemma<\/code> \/ <code>theorem2sorry<\/code> \/ <code>sorry2lemma<\/code><\/strong> \u2014 proof-structure transformations<\/li>\n\n\n\n<li><strong><code>merge<\/code><\/strong> \u2014 merge proof files<\/li>\n\n\n\n<li><strong><code>simplify_theorems<\/code><\/strong> \u2014 simplify theorem statements<\/li>\n\n\n\n<li><strong><code>repair_proofs<\/code><\/strong> \u2014 auto-repair broken proofs<\/li>\n\n\n\n<li><strong><code>have2lemma<\/code> \/ <code>have2sorry<\/code><\/strong> \u2014 convert local <code>have<\/code> statements<\/li>\n\n\n\n<li><strong><code>disprove<\/code><\/strong> \u2014 counter-example tooling<\/li>\n\n\n\n<li><strong><code>normalize<\/code><\/strong> \u2014 canonicalization<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">3. Installation<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">Requirements<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Python 3.11+<\/strong><\/li>\n\n\n\n<li>Internet connection (to reach the AXLE API \u2014 the Lean compilation runs server-side)<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Option A \u2014 Install from PyPI (recommended)<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code>pip install axiom-axle\n<\/code><\/pre>\n\n\n\n<h4 class=\"wp-block-heading\">Option B \u2014 Install from Source<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code>git clone &lt;https:\/\/github.com\/AxiomMath\/axiom-lean-engine.git&gt;\ncd axiom-lean-engine\npip install -e .\n<\/code><\/pre>\n\n\n\n<p>Or directly via SSH:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>pip install git+ssh:\/\/git@github.com\/AxiomMath\/axiom-lean-engine\n<\/code><\/pre>\n\n\n\n<h4 class=\"wp-block-heading\">Option C \u2014 Development Installation (with dev tools + pre-commit hooks)<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code>git clone &lt;https:\/\/github.com\/AxiomMath\/axiom-lean-engine.git&gt;\ncd axiom-lean-engine\nmake setup-env\n<\/code><\/pre>\n\n\n\n<h4 class=\"wp-block-heading\">Verify the install<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code># CLI check\naxle --version\n\n# Python import check\npython -c \"from axle import AxleClient; print('OK')\"\n<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">4. Setup \u2014 API Key &amp; Authentication<\/h3>\n\n\n\n<p>AXLE uses API key auth for rate limiting. Without a key, you get <strong>10 concurrent active requests max<\/strong>.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Step 1: Get an API key<\/h4>\n\n\n\n<p>Visit the console: **<a href=\"https:\/\/axle.axiommath.ai\/app\/console**\">https:\/\/axle.axiommath.ai\/app\/console**<\/a><\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Step 2: Set it as an environment variable (recommended)<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code>export AXLE_API_KEY=your-api-key\n<\/code><\/pre>\n\n\n\n<p>Add to <code>~\/.bashrc<\/code>, <code>~\/.zshrc<\/code>, or your project&#8217;s <code>.env<\/code> for persistence.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Step 3 (optional): Pass directly in code<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code>from axle import AxleClient\nclient = AxleClient(api_key=\"your-api-key\")\n<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">5. Configuration<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">Environment variables<\/h4>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Variable<\/th><th>Default<\/th><th>Description<\/th><\/tr><\/thead><tbody><tr><td><code>AXLE_API_KEY<\/code><\/td><td>\u2014<\/td><td>API key for authentication<\/td><\/tr><tr><td><code>AXLE_API_URL<\/code><\/td><td><code>https:\/\/axle.axiommath.ai<\/code><\/td><td>API server URL<\/td><\/tr><tr><td><code>AXLE_TIMEOUT_SECONDS<\/code><\/td><td><code>1800<\/code><\/td><td>Base retry-window timeout<\/td><\/tr><tr><td><code>AXLE_MAX_CONCURRENCY<\/code><\/td><td><code>20<\/code><\/td><td>Max concurrent requests<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<pre class=\"wp-block-code\"><code>export AXLE_API_KEY=your-api-key\nexport AXLE_API_URL=https:\/\/axle.axiommath.ai\nexport AXLE_TIMEOUT_SECONDS=600\nexport AXLE_MAX_CONCURRENCY=50\n<\/code><\/pre>\n\n\n\n<h4 class=\"wp-block-heading\">Python client configuration<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code>from axle import AxleClient\n\n# Custom URL, timeout, and concurrency\nclient = AxleClient(\n    api_key=\"your-api-key\",\n    url=AxleClient.DEFAULT_URL,\n    base_timeout_seconds=600,\n    max_concurrency=50,\n)\n<\/code><\/pre>\n\n\n\n<h4 class=\"wp-block-heading\">CLI configuration (global options)<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code># Custom API URL\naxle --url &lt;https:\/\/axle.axiommath.ai&gt; check file.lean --environment lean-4.28.0\n\n# JSON output\naxle --json check file.lean --environment lean-4.28.0\n\n# Output to file\naxle theorem2sorry input.lean --environment lean-4.28.0 -o output.lean\n<\/code><\/pre>\n\n\n\n<h4 class=\"wp-block-heading\">Lean environments<\/h4>\n\n\n\n<p>Every API request <strong>requires<\/strong> an <code>environment<\/code> parameter. AXLE supports multiple Lean versions with pre-built dependencies (typically Mathlib). At time of writing, the recommended starter is <code>lean-4.28.0<\/code>.<\/p>\n\n\n\n<p><strong>Discover available environments:<\/strong><\/p>\n\n\n\n<pre class=\"wp-block-code\"><code># CLI\naxle environments\n\n# Python\nasync with AxleClient() as client:\n    envs = await client.environments()\n    for env in envs:\n        print(f\"{env.name}: {env.description}\")\n\n# HTTP\ncurl -s &lt;https:\/\/axle.axiommath.ai\/v1\/environments&gt; | jq\n<\/code><\/pre>\n\n\n\n<p><strong>Example environment definitions:<\/strong><\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>&#91;\n  {\n    \"name\": \"lean-4.21.0\",\n    \"lean_toolchain\": \"leanprover\/lean4:v4.21.0\",\n    \"imports\": \"import Mathlib\",\n    \"description\": \"Lean 4.21.0 with Mathlib\"\n  },\n  {\n    \"name\": \"pnt-4.26.0\",\n    \"lean_toolchain\": \"leanprover\/lean4:v4.26.0\",\n    \"repo_url\": \"&lt;https:\/\/github.com\/AlexKontorovich\/PrimeNumberTheoremAnd&gt;\",\n    \"imports\": \"import Mathlib\\\\nimport PrimeNumberTheoremAnd\",\n    \"description\": \"Lean + Mathlib 4.26.0 with Terence Tao's Prime Number Theorem Project\"\n  }\n]\n<\/code><\/pre>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>Custom environments can pin a specific GitHub repo + revision + subdir \u2014 useful for working with particular formalization projects.<\/p>\n<\/blockquote>\n\n\n\n<h3 class=\"wp-block-heading\">6. Quick Start \u2014 First &#8220;Hello, Theorem&#8221; Check<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">Python<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code>import asyncio\nfrom axle import AxleClient\n\nasync def main():\n    async with AxleClient() as client:\n        result = await client.check(\n            content=\"import Mathlib\\\\ntheorem citation_needed : 1 + 1 = 2 := by decide\",\n            environment=\"lean-4.28.0\",\n        )\n        print(f\"Valid: {result.okay}\")\n        if result.lean_messages.errors:\n            print(\"Errors:\", result.lean_messages.errors)\n\nasyncio.run(main())\n<\/code><\/pre>\n\n\n\n<h4 class=\"wp-block-heading\">CLI<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code># From file\naxle check mytheorem.lean --environment lean-4.28.0\n\n# From stdin\necho \"def meaning_of_life := 42\\\\n#print meaning_of_life\" \\\\\n  | axle check - --environment lean-4.28.0\n<\/code><\/pre>\n\n\n\n<h4 class=\"wp-block-heading\">HTTP<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code>curl -s -X POST &lt;https:\/\/axle.axiommath.ai\/api\/v1\/check&gt; \\\\\n    -H \"Authorization: Bearer $AXLE_API_KEY\" \\\\\n    -H \"Content-Type: application\/json\" \\\\\n    -d '{\n      \"content\": \"import Mathlib\\\\ntheorem citation_needed : 1 + 1 = 2 := by decide\",\n      \"environment\": \"lean-4.28.0\"\n    }' | jq\n<\/code><\/pre>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Try it in-browser first<\/strong>: an interactive Colab demo notebook is linked from the AXLE Quick Start page \u2014 handy for kicking the tires before touching pip.<\/p>\n<\/blockquote>\n\n\n\n<h3 class=\"wp-block-heading\">7. The Core Use Case \u2014 <code>verify_proof<\/code> (the Prover Verification Workflow)<\/h3>\n\n\n\n<p>This is the heart of the verified-AI loop Karina described in the fireside chat. The flow:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>You (or your AI agent) write a <strong>formal statement<\/strong> with <code>sorry<\/code> placeholders \u2014 defining <em>what must be proved<\/em>.<\/li>\n\n\n\n<li>You produce a <strong>candidate proof<\/strong> \u2014 the actual Lean code attempting the proof.<\/li>\n\n\n\n<li>AXLE compiles both, checks signatures match, runs the kernel, and tells you whether the proof is <strong>100% valid<\/strong>.<\/li>\n<\/ol>\n\n\n\n<h4 class=\"wp-block-heading\">Conceptual example<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code>-- formal_statement: defines the theorem signature\nimport Mathlib\ntheorem add_comm (a b : Nat) : a + b = b + a := by sorry\n<\/code><\/pre>\n\n\n\n<pre class=\"wp-block-code\"><code>-- content: provides the actual proof\nimport Mathlib\ntheorem add_comm (a b : Nat) : a + b = b + a := Nat.add_comm a b\n<\/code><\/pre>\n\n\n\n<h4 class=\"wp-block-heading\">Python API<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code>result = await axle.verify_proof(\n    formal_statement=\"import Mathlib\\\\ntheorem citation_needed : 1 = 1 := by sorry\",\n    content=\"import Mathlib\\\\ntheorem citation_needed : 1 = 1 := rfl\",\n    environment=\"lean-4.28.0\",\n    permitted_sorries=&#91;\"helper\"],  # Optional: helper lemmas allowed to remain unproven\n    mathlib_options=False,          # Optional: enable conventional Mathlib options\n    ignore_imports=False,           # Optional: handle import mismatches\n    timeout_seconds=120,            # Optional: max 900s (15 min) for non-admin\n)\n\nprint(result.okay)         # True if proof is valid\nprint(result.content)      # The processed Lean code\n<\/code><\/pre>\n\n\n\n<h4 class=\"wp-block-heading\">CLI<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code># Basic\naxle verify-proof statement.lean proof.lean --environment lean-4.28.0\n\n# With permitted sorries\naxle verify-proof statement.lean proof.lean \\\\\n  --permitted-sorries helper1,helper2 \\\\\n  --environment lean-4.28.0\n\n# Pipeline-friendly (proof from stdin)\ncat proof.lean | axle verify-proof statement.lean - --environment lean-4.28.0\n\n# Strict mode (non-zero exit on invalid) \u2014 perfect for CI\nif axle verify-proof statement.lean proof.lean --strict --environment lean-4.28.0 &gt; \/dev\/null; then\n    echo \"Proof valid \u2705\"\nelse\n    echo \"Proof invalid \u274c\"\nfi\n<\/code><\/pre>\n\n\n\n<h4 class=\"wp-block-heading\">HTTP<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code>curl -s -X POST &lt;https:\/\/axle.axiommath.ai\/api\/v1\/verify_proof&gt; \\\\\n    -H \"Authorization: Bearer $AXLE_API_KEY\" \\\\\n    -d '{\n      \"content\": \"import Mathlib\\\\ntheorem citation_needed : 1 = 1 := rfl\",\n      \"formal_statement\": \"import Mathlib\\\\ntheorem citation_needed : 1 = 1 := by sorry\",\n      \"environment\": \"lean-4.28.0\"\n    }' | jq\n<\/code><\/pre>\n\n\n\n<h4 class=\"wp-block-heading\">Output structure<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code>{\n  \"okay\": false,\n  \"content\": \"import Mathlib\\\\n\\\\ntheorem foo : 1 = 1 := rfl\\\\n\",\n  \"lean_messages\": { \"errors\": &#91;], \"warnings\": &#91;], \"infos\": &#91;] },\n  \"tool_messages\": {\n    \"errors\": &#91;\"Theorem 'foo' does not match expected signature: expected type 2 = 2, got 1 = 1\"],\n    \"warnings\": &#91;],\n    \"infos\": &#91;]\n  },\n  \"failed_declarations\": &#91;\"foo\"],\n  \"timings\": { \"total_ms\": 160, \"formal_statement_ms\": 3, \"declarations_ms\": 0, \"candidate_ms\": 28 }\n}\n<\/code><\/pre>\n\n\n\n<h4 class=\"wp-block-heading\">Common verification error patterns<\/h4>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Pattern<\/th><th>Meaning<\/th><\/tr><\/thead><tbody><tr><td><code>Missing required declaration '{name}'<\/code><\/td><td>A symbol in <code>formal_statement<\/code> is missing from <code>content<\/code><\/td><\/tr><tr><td><code>Kind mismatch for '{name}': candidate has {X} but expected {Y}<\/code><\/td><td>E.g., <code>theorem<\/code> vs <code>def<\/code><\/td><\/tr><tr><td><code>Theorem '{name}' does not match expected signature<\/code><\/td><td>Theorem type changed<\/td><\/tr><tr><td><code>Unsafe\/partial function '{name}' detected<\/code><\/td><td>Use of a disallowed function<\/td><\/tr><tr><td><code>Axiom '{axiom}' is not in the allowed set of standard axioms<\/code><\/td><td>Disallowed axiom used<\/td><\/tr><tr><td><code>Declaration '{name}' uses 'sorry' which is not allowed<\/code><\/td><td>Theorem not actually proven<\/td><\/tr><tr><td><code>Candidate uses banned 'open private' command<\/code><\/td><td>Disallowed <code>open private<\/code><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h4 class=\"wp-block-heading\">Important security note<\/h4>\n\n\n\n<p><code>verify_proof<\/code> trusts the Lean environment for speed. A creative adversary could exploit Lean metaprogramming to make invalid proofs appear valid. For <strong>untrusted proofs (e.g., from competitors, untrusted AI outputs)<\/strong>, additionally run:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>lean4checker<\/strong> \u2014 Lean FRO&#8217;s <code>.olean<\/code> verifier<\/li>\n\n\n\n<li><strong>Comparator<\/strong> \u2014 Lean FRO&#8217;s gold-standard proof judge<\/li>\n\n\n\n<li><strong>SafeVerify<\/strong> \u2014 battle-tested public proof checker (used by AxiomProver in the Putnam 2025 verification pipeline)<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">8. Real-World Case Study \u2014 AxiomProver on Putnam 2025<\/h3>\n\n\n\n<p>Their public Putnam 2025 repo is the cleanest case study of the prover-plus-verification stack working end-to-end. Headline numbers from the GitHub README:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Final score: 12\/12<\/strong> (8 solved during the competition window, the remaining 4 in the days after)<\/li>\n\n\n\n<li>AxiomProver is described as <em>&#8220;an autonomous multi-agent ensemble theorem prover for Lean 4.21.0&#8221;<\/em><\/li>\n\n\n\n<li>Each solution is a fully verified Lean 4 proof, checked by <strong>SafeVerify<\/strong><\/li>\n<\/ul>\n\n\n\n<p><strong>Sample proof complexity<\/strong> (from their published metrics):<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Problem<\/th><th>Prover time<\/th><th>Tokens<\/th><th>Proof size<\/th><\/tr><\/thead><tbody><tr><td>2025 A1<\/td><td>110 min<\/td><td>7M<\/td><td>652 lines, 23 theorems, 561 tactics<\/td><\/tr><tr><td>2025 A3<\/td><td>165 min<\/td><td>8M<\/td><td>1,333 lines, 78 theorems, 1,701 tactics<\/td><\/tr><tr><td>2025 B5<\/td><td>354 min<\/td><td>18M<\/td><td>1,495 lines, 66 theorems, 1,967 tactics<\/td><\/tr><tr><td>2025 B6<\/td><td>494 min<\/td><td>21M<\/td><td>1,019 lines, 30 theorems, 1,052 tactics<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><strong>Reproducing the verification yourself:<\/strong><\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>git clone &lt;https:\/\/github.com\/AxiomMath\/Putnam2025&gt;\ncd Putnam2025\nlake run verify\n# Output:\n# Verifying A1 solution ... \u2705\n# Verifying A2 solution ... \u2705\n# ... through B6\n<\/code><\/pre>\n\n\n\n<p>This is the exact pattern Karina described: AxiomProver generates the proof; a verifier independently confirms it; you trust the check mark, not the model.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">9. Practical Patterns for Your Own AI-Verification Pipeline<\/h3>\n\n\n\n<p>A reasonable architecture if you want to use AXLE in your own agent loop (mirrors AxiomProver&#8217;s design):<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510     \u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510     \u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510\n\u2502   LLM Agent     \u2502 \u2500\u2500\u25b6\u2502  Lean Code Gen   \u2502 \u2500\u2500\u25b6 \u2502  AXLE check    \u2502\n\u2502  (Claude\/etc.)  \u2502     \u2502  (formal proof)  \u2502     \u2502  (syntax valid?)\u2502\n\u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2518     \u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2518     \u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2518\n        \u25b2                                                 \u2502\n        \u2502                                                 \u25bc\n        \u2502                                         \u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510\n        \u2502       \u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510      \u2502 AXLE verify_proof\u2502\n        \u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2502  Repair \/ regenerate on  \u2502\u25c0\u2500\u2500\u2500\u2500\u2502  (semantic \u2705?)  \u2502\n                \u2502  tool_messages.errors    \u2502      \u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2518\n                \u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2518               \u2502\n                                                           \u25bc\n                                                  \u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510\n                                                  \u2502  SafeVerify       \u2502\n                                                  \u2502  (adversarial \u2705) \u2502\n                                                  \u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2518\n<\/code><\/pre>\n\n\n\n<p>Tips:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Use <code>permitted_sorries<\/code> to let your agent ship partial proofs and iterate (very useful in RL training loops with verifiable rewards \u2014 exactly the RLVR pattern Karina described).<\/li>\n\n\n\n<li>Use <code>repair_proofs<\/code> and <code>simplify_theorems<\/code> as deterministic post-processing to clean up LLM output before verification.<\/li>\n\n\n\n<li>Pin a specific environment (<code>lean-4.28.0<\/code> or your custom repo-pinned env) for reproducibility \u2014 bumping Lean versions mid-experiment will silently break things.<\/li>\n\n\n\n<li>Set realistic timeouts: hard ceiling is 900s (15 min) per request. AxiomProver&#8217;s longest Putnam proof took ~8 hours of wall-clock prover time, but each individual verification call is much shorter.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Video &#8211; Axiom Math at The Montgomery Summit<\/h2>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Axiom Math at The Montgomery Summit | Fireside Chat\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/PljrqbytxGc?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-pale-cyan-blue-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">Related Sections<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">From Deterministic to Non-Deterministic Workflows \u2014 Why Verification Matters Now<\/h3>\n\n\n\n<p>The world has shifted from deterministic, predictable workflows to <strong>non-deterministic LLM outputs<\/strong> \u2014 essentially conjectures. Just as math conjectures need a prover, LLM-generated code, designs, and agent actions need a <strong>verifier<\/strong>. Carina illustrated the risk with a recent OpenClaw anecdote: a user had OpenClaw book a Davos speaking slot for him, only to discover he owed <strong>$31,000<\/strong> he could not afford. As autonomous agents are deployed at scale into safety-critical and regulated industries, <strong>one agent should certify another agent&#8217;s actions<\/strong> before they propagate.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">The &#8220;Why Now&#8221; \u2014 Three Movements Converging in 2024<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>LLMs became genuinely good at informal reasoning<\/strong> \u2014 high-level sketching, planning, and outlining (a capability that solidified only after September 2024).<\/li>\n\n\n\n<li><strong>Lean 4 matured into industry-grade infrastructure<\/strong>. The formal-proof language Lean has been developed since 2019 (notably by undergrads at Imperial College London \u2014 many of Carina&#8217;s Math Olympiad friends, several now full-time at Axiom). Lean 4&#8217;s release in <strong>September 2023<\/strong> enabled engineering-scale work, and the broader mathematical community accepted it in 2024.<\/li>\n\n\n\n<li><strong>RLVR (Reinforcement Learning with Verifiable Rewards)<\/strong> delivered breakthrough gains in coding because code provides instant pass\/fail feedback. Math is the other natively verifiable domain \u2014 and it&#8217;s exactly where Axiom plays.<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Real-World Applications Already Emerging<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Hardware\/chip verification<\/strong>: GPUs need 100% correctness \u2014 there is no partial credit. Verilog\/RTL verification is a natural target.<\/li>\n\n\n\n<li><strong>CPU hypervisor optimization<\/strong>: AWS recently published <strong>260,000 lines of formal-language code<\/strong> that took world experts <strong>3\u20135 years<\/strong> to write by hand. Carina notes wryly: <em>&#8220;vibe coding has not changed their life.&#8221;<\/em><\/li>\n\n\n\n<li><strong>Beyond SMT\/SAT solvers<\/strong>: Traditional formal tools suffer from <strong>state-space explosion<\/strong> (a counter value of 100 unrolls 100 times). Lean&#8217;s mathematical abstractions and invariants let Axiom Prover bypass exhaustive search \u2014 a key technical insight.<\/li>\n\n\n\n<li><strong>Historical precedent<\/strong>: The Paris trade union once demanded formal verification of the subway&#8217;s automatic switching system; ESA used formal verification on the Ariane project; Boeing and Airbus have applied it for decades.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">The Team \u2014 A Magnet for Top Talent<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Fabian Gloeckle<\/strong> \u2014 formerly Director at Facebook AI Research, his prior work was deployed across all of Meta for LLM-generated software testing. He champions <strong>Donald Knuth&#8217;s literate programming<\/strong> vision: programmers reason in natural language while verified code ships directly to deployment.<\/li>\n\n\n\n<li><strong>Professor Ken Ono<\/strong> \u2014 former Vice Provost of the University of Virginia and former Vice President of the American Mathematical Society, who left academia to join Axiom after watching Axiom Prover solve number-theory conjectures he could not solve himself. For Ken, this represents <em>a new way of doing mathematics<\/em> \u2014 mathematicians operate at higher abstraction (intuitions about proof direction) while Axiom Prover fills in the low-level details in real time.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">The Abstraction Ladder<\/h3>\n\n\n\n<p>Carina draws a parallel to programming history: punch cards (1950s) \u2192 Fortran \u2192 low-level compilers \u2192 Python \u2192 <strong>natural language<\/strong>. Each lift in abstraction freed humans for higher-leverage work. AI mathematicians do the same for mathematics: instead of solving 1\u20132 Millennium Prize problems in a lifetime, a mathematician like Ken could potentially attempt <strong>a hundred<\/strong>. The Hardy\u2013Littlewood\u2013Ramanujan analogy lands beautifully here \u2014 Lean is the modern proof assistant that converts intuition into rigorous logic.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">The Code-with-Proof Insight<\/h3>\n\n\n\n<p>A key technical contribution: rather than running RL with two divergent objective functions (Python code + English math proof), Axiom <strong>converts math proofs into formal-language form<\/strong> that structurally resembles Python or C \u2014 making joint optimization tractable. This is why Axiom Prover&#8217;s capability transfers directly from math to code verification, <strong>saturating several code-verification benchmarks without modification<\/strong>.<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Conclusion and Key Takeaways<\/h2>\n\n\n\n<p>Axiom is not building &#8220;another LLM that does math.&#8221; It is building the <strong>verification substrate<\/strong> for the entire generative-AI economy \u2014 starting with the most rigorous testbed humanity has (mathematics) and generalizing outward to code, chips, and autonomous agents.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Key takeaways:<\/strong><\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Verified AI is the next frontier.<\/strong> As LLMs flood the world with non-deterministic output, the bottleneck shifts from generation to <strong>trustworthy verification<\/strong>.<\/li>\n\n\n\n<li><strong>Math is the proving ground, not the product.<\/strong> Anything reducible to <em>code and logic<\/em> is a candidate market \u2014 chip verification, regulated enterprise code, safety-critical agent operations.<\/li>\n\n\n\n<li><strong>The &#8220;why now&#8221; is a triple convergence<\/strong>: LLMs reasoning informally + Lean 4 industrial maturity + RLVR breakthroughs \u2014 all aligned in 2024.<\/li>\n\n\n\n<li><strong>Abstraction lifts unlock human leverage.<\/strong> Mathematicians stop checking nitty-gritty steps and operate on intuition; Axiom Prover handles the formal details.<\/li>\n\n\n\n<li><strong>Trust enables autonomy.<\/strong> Without verification, high-stakes industries cannot deploy AI agents fully. Axiom is building the agent that certifies other agents.<\/li>\n\n\n\n<li><strong>Recruitment is a leading indicator.<\/strong> When a former AMS Vice President leaves a Vice Provost role to join an 8-month-old company, the technical signal is strong.<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\">Related References<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Axiom Math home<\/strong>: <a href=\"https:\/\/axiommath.ai\" target=\"_blank\" rel=\"noopener\" title=\"\">https:\/\/axiommath.ai<\/a><\/li>\n\n\n\n<li><strong>AXLE site<\/strong>: <a href=\"https:\/\/axle.axiommath.ai\">https:\/\/axle.axiommath.ai<\/a><\/li>\n\n\n\n<li><strong>AXLE docs<\/strong>: <a href=\"https:\/\/axle.axiommath.ai\/v1\/docs\/\" target=\"_blank\" rel=\"noopener\" title=\"\">https:\/\/axle.axiommath.ai\/v1\/docs\/<\/a><\/li>\n\n\n\n<li><strong>AXLE console (get API key)<\/strong>: <a href=\"https:\/\/axle.axiommath.ai\/app\/console\" target=\"_blank\" rel=\"noopener\" title=\"\">https:\/\/axle.axiommath.ai\/app\/console<\/a><\/li>\n\n\n\n<li><strong>AXLE GitHub<\/strong>: <a href=\"https:\/\/github.com\/AxiomMath\/axiom-lean-engine\" target=\"_blank\" rel=\"noopener\" title=\"\">https:\/\/github.com\/AxiomMath\/axiom-lean-engine<\/a><\/li>\n\n\n\n<li><strong>Putnam 2025 solutions repo<\/strong>: <a href=\"https:\/\/github.com\/AxiomMath\/Putnam2025\" target=\"_blank\" rel=\"noopener\" title=\"\">https:\/\/github.com\/AxiomMath\/Putnam2025<\/a><\/li>\n\n\n\n<li><strong>Axiom blog (&#8220;Building The Reasoning Engine at Axiom&#8221;)<\/strong>: <a href=\"https:\/\/axiommath.ai\/blog\/\" target=\"_blank\" rel=\"noopener\" title=\"\">https:\/\/axiommath.ai\/blog\/<\/a><\/li>\n\n\n\n<li><strong>SafeVerify (third-party adversarial checker)<\/strong>: <a href=\"https:\/\/github.com\/GasStationManager\/SafeVerify\" target=\"_blank\" rel=\"noopener\" title=\"\">https:\/\/github.com\/GasStationManager\/SafeVerify<\/a><\/li>\n\n\n\n<li><strong>lean4checker (Lean FRO official verifier)<\/strong>: <a href=\"https:\/\/github.com\/leanprover\/lean4checker\" target=\"_blank\" rel=\"noopener\" title=\"\">https:\/\/github.com\/leanprover\/lean4checker<\/a><\/li>\n\n\n\n<li><strong>Lean 4 reference on validating proofs<\/strong>: <a href=\"https:\/\/lean-lang.org\/doc\/reference\/latest\/ValidatingProofs\/\" target=\"_blank\" rel=\"noopener\" title=\"\">https:\/\/lean-lang.org\/doc\/reference\/latest\/ValidatingProofs\/<\/a><\/li>\n\n\n\n<li><strong>Wall Street Journal<\/strong> <a href=\"https:\/\/www.wsj.com\/tech\/ai\/math-ken-ono-carina-hong-axiom-startup-649bc417\" target=\"_blank\" rel=\"noopener\" title=\"\">coverage of Ken Ono leaving UVA for Axiom<\/a><\/li>\n\n\n\n<li><strong>Putnam Competition<\/strong> \u2014 <a href=\"https:\/\/en.wikipedia.org\/wiki\/Putnam_Competition\" target=\"_blank\" rel=\"noopener\" title=\"\">annual undergraduate math exam (median score: 0)<\/a><\/li>\n\n\n\n<li><strong>Lean 4<\/strong> \u2014 formal proof assistant: <a href=\"https:\/\/lean-lang.org\/\"><\/a><a href=\"https:\/\/lean-lang.org\">https:\/\/lean-lang.org<\/a><\/li>\n\n\n\n<li><strong>Madrona Venture Group<\/strong> \u2014 <a href=\"https:\/\/www.venturecapitaljournal.com\/ai-focused-madrona-venture-labs-collects-11m-for-fifth-fund\/\" target=\"_blank\" rel=\"noopener\" title=\"\">Axiom&#8217;s early investor<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Axiom Math is the Silicon Valley startup behind AxiomProver \u2014 the AI mathematician that scored a perfect 12\/12 on Putnam 2025 \u2014 and AXLE, its public Lean 4 verification engine. This guide walks through installing the Python SDK and CLI, configuring API keys and Lean environments, and using `verify_proof` to build verified-AI pipelines where every output carries a machine-checkable proof.<\/p>\n","protected":false},"author":1,"featured_media":8353,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15,18,13,7],"tags":[],"class_list":["post-8352","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-ai","category-education","category-quantum-and-u","category-quantum-mindset-programme"],"aioseo_notices":[],"featured_image_src":"https:\/\/meta-quantum.today\/wp-content\/uploads\/2026\/05\/Axiom-Math-Verified-AI-Reasoning-Engine-Overview-scaled.jpg","featured_image_src_square":"https:\/\/meta-quantum.today\/wp-content\/uploads\/2026\/05\/Axiom-Math-Verified-AI-Reasoning-Engine-Overview-scaled.jpg","author_info":{"display_name":"coffee","author_link":"https:\/\/meta-quantum.today\/?author=1"},"rbea_author_info":{"display_name":"coffee","author_link":"https:\/\/meta-quantum.today\/?author=1"},"rbea_excerpt_info":"Axiom Math is the Silicon Valley startup behind AxiomProver \u2014 the AI mathematician that scored a perfect 12\/12 on Putnam 2025 \u2014 and AXLE, its public Lean 4 verification engine. This guide walks through installing the Python SDK and CLI, configuring API keys and Lean environments, and using `verify_proof` to build verified-AI pipelines where every output carries a machine-checkable proof.","category_list":"<a href=\"https:\/\/meta-quantum.today\/?cat=15\" rel=\"category\">AI<\/a>, <a href=\"https:\/\/meta-quantum.today\/?cat=18\" rel=\"category\">Education<\/a>, <a href=\"https:\/\/meta-quantum.today\/?cat=13\" rel=\"category\">Quantum and U<\/a>, <a href=\"https:\/\/meta-quantum.today\/?cat=7\" rel=\"category\">Quantum Mindset Programme<\/a>","comments_num":"0 comments","_links":{"self":[{"href":"https:\/\/meta-quantum.today\/index.php?rest_route=\/wp\/v2\/posts\/8352","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/meta-quantum.today\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/meta-quantum.today\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/meta-quantum.today\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/meta-quantum.today\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=8352"}],"version-history":[{"count":2,"href":"https:\/\/meta-quantum.today\/index.php?rest_route=\/wp\/v2\/posts\/8352\/revisions"}],"predecessor-version":[{"id":8355,"href":"https:\/\/meta-quantum.today\/index.php?rest_route=\/wp\/v2\/posts\/8352\/revisions\/8355"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/meta-quantum.today\/index.php?rest_route=\/wp\/v2\/media\/8353"}],"wp:attachment":[{"href":"https:\/\/meta-quantum.today\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8352"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/meta-quantum.today\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8352"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/meta-quantum.today\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8352"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}