#lean4 — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #lean4, aggregated by home.social.
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🙈 New preprint, Part V: *What the Higgs Potential Cannot See*.
If you build gauge–Higgs unification models: there is a discrete choice you may be scanning over for nothing. On \(T^2/\mathbb{Z}_2\) each bulk multiplet carries boundary signs \(\eta_0,\eta_1\). For a whole class of bulk matter, the product \(\eta=\eta_0\eta_1\) has no observable consequence in the Higgs sector at one loop. Not suppressed. Identically zero.
Why. The one-loop Wilson-line potential is one operator traced twice. Even windings give a graded dimension, odd windings an index:
\[\Sigma_\lambda=s_\lambda(1,1,t,t^{-1}),\qquad D_\lambda=s_\lambda(1,-1,t,t^{-1}),\]
and AHMN's \(\{A+B(-1)^{k_2}\}\) is exactly \((\Sigma\pm D)/2\). \(\Sigma\) has non-negative coefficients and can never cancel; \(D\) can. η multiplies \(D\) and nothing else — so η is invisible exactly where \(D_\lambda\equiv 0\).Which matter is blind is a parity condition you read off the Young diagram: \(\lambda_1\not\equiv\lambda_2\not\equiv\lambda_3\not\equiv\lambda_4\), or \(\lambda_i+\lambda_{5-i}=c\) odd. Counted in closed form — \(\lceil (k+1)^2/2\rceil\) at \(\lambda_1=2k+1\), none for \(\lambda_1\) even — and machine-checked in Lean 4, sorry-free. A second, disjoint cause: only \(\lfloor (N+1)^2/2\rfloor\) of the \((N+1)^2\) boundary-condition classes of \(SU(N)\) have a coset sector at all, for every \(N\).
Anchored, not fitted: twelve printed coefficients of arXiv:2312.08608 come out exactly, and two further published potentials follow from the same mode counts.
📄 https://doi.org/10.5281/zenodo.21727094
💻 https://github.com/karlesmarin/higgs-blind-class#Physics #ParticlePhysics #HEP #BSM #Higgs #ExtraDimensions #Lean4
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🙈 New preprint, Part V: *What the Higgs Potential Cannot See*.
If you build gauge–Higgs unification models: there is a discrete choice you may be scanning over for nothing. On \(T^2/\mathbb{Z}_2\) each bulk multiplet carries boundary signs \(\eta_0,\eta_1\). For a whole class of bulk matter, the product \(\eta=\eta_0\eta_1\) has no observable consequence in the Higgs sector at one loop. Not suppressed. Identically zero.
Why. The one-loop Wilson-line potential is one operator traced twice. Even windings give a graded dimension, odd windings an index:
\[\Sigma_\lambda=s_\lambda(1,1,t,t^{-1}),\qquad D_\lambda=s_\lambda(1,-1,t,t^{-1}),\]
and AHMN's \(\{A+B(-1)^{k_2}\}\) is exactly \((\Sigma\pm D)/2\). \(\Sigma\) has non-negative coefficients and can never cancel; \(D\) can. η multiplies \(D\) and nothing else — so η is invisible exactly where \(D_\lambda\equiv 0\).Which matter is blind is a parity condition you read off the Young diagram: \(\lambda_1\not\equiv\lambda_2\not\equiv\lambda_3\not\equiv\lambda_4\), or \(\lambda_i+\lambda_{5-i}=c\) odd. Counted in closed form — \(\lceil (k+1)^2/2\rceil\) at \(\lambda_1=2k+1\), none for \(\lambda_1\) even — and machine-checked in Lean 4, sorry-free. A second, disjoint cause: only \(\lfloor (N+1)^2/2\rfloor\) of the \((N+1)^2\) boundary-condition classes of \(SU(N)\) have a coset sector at all, for every \(N\).
Anchored, not fitted: twelve printed coefficients of arXiv:2312.08608 come out exactly, and two further published potentials follow from the same mode counts.
📄 https://doi.org/10.5281/zenodo.21727094
💻 https://github.com/karlesmarin/higgs-blind-class#Physics #ParticlePhysics #HEP #BSM #Higgs #ExtraDimensions #Lean4
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A #lean4 paper cut that I hate: it absolutely kills me that they mixed the fields `val` and `property` on Subtype, so every time i try to use it i go down the dead ends of trying BOTH `value` and `prop` and being wrong.
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A #lean4 paper cut that I hate: it absolutely kills me that they mixed the fields `val` and `property` on Subtype, so every time i try to use it i go down the dead ends of trying BOTH `value` and `prop` and being wrong.
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Formalized in Lean4:
The map F : K³ → K³ has constant Jacobian −2 yet identifies three points. This is no accident of algebra: F is, definitionally, the proof is rfl, the map "binary cubic with a marked simple root ↦ forget the marking", for the family cT³ − 2T²U + bTU² − 2aU³.
So every fiber is the set of simple roots of one cubic. The Keller condition says the root you stand on is simple, a local fact. It cannot control how many simple roots the cubic has: 3 off the discriminant hypersurface, 1 on it, 0 on its singular curve (the triple roots), which is exactly the locus the image misses. All machine-checked now, including disc = −4W and "missed curve = singular locus of the discriminant".
My favorite part is arithmetic: over any field of char ≠ 2, a fiber can never have exactly two rational points, two roots force a third. Vieta, made constructive.
The Jacobian Conjecture failed because pointwise simplicity of roots doesn't count roots.
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Com començar a desenvolupar en Lean4? – Blog of Joaquim Puig
https://web.mat.upc.edu/joaquim.puig/posts/apunts_lean4_1/ -
A computer found one solution to a hard model-building problem. Then we proved you will *always* find one — and machine-checked the proof in Lean 4. 🧮
The setting: which fermions complete a quark block in a 6D \(SU(4)\) gauge–Higgs model on an orbifold, so that every local consistency condition (anomalies + "tadpoles") cancels? These are exact integer / representation-theory conditions over \(SU(4)\) weights.
One witness is easy to distrust — a fluke? The structural answer is no. Writing the anomaly map \(A\) and the tadpole map \(\Theta\) as linear functionals of the added matter, two exact facts settle it:
• \(\operatorname{rank}[A;\Theta]=8+2=10\): the tadpole is *independent* of the anomalies — no conserved invariant traps it;
• anomaly-neutral additions realise *every* tadpole direction (Farkas certificates), so their cone is all of \(\mathbb{R}^2\).Hence *every* anomaly-free completion is tadpole-compatible: an Existence theorem, not luck. The certificate is checked by the Lean 4 kernel, depending only on propext. The same rank+cone test ships as a reusable tool for any orbifold model. Honest scope: one infrared step stays open.
📄 https://zenodo.org/records/21432626
💻 https://github.com/karlesmarin/ghu-su4-completion#Lean4 #FormalMath #ProofAssistant #RepresentationTheory #Maths #Physics
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A computer found one solution to a hard model-building problem. Then we proved you will *always* find one — and machine-checked the proof in Lean 4. 🧮
The setting: which fermions complete a quark block in a 6D \(SU(4)\) gauge–Higgs model on an orbifold, so that every local consistency condition (anomalies + "tadpoles") cancels? These are exact integer / representation-theory conditions over \(SU(4)\) weights.
One witness is easy to distrust — a fluke? The structural answer is no. Writing the anomaly map \(A\) and the tadpole map \(\Theta\) as linear functionals of the added matter, two exact facts settle it:
• \(\operatorname{rank}[A;\Theta]=8+2=10\): the tadpole is *independent* of the anomalies — no conserved invariant traps it;
• anomaly-neutral additions realise *every* tadpole direction (Farkas certificates), so their cone is all of \(\mathbb{R}^2\).Hence *every* anomaly-free completion is tadpole-compatible: an Existence theorem, not luck. The certificate is checked by the Lean 4 kernel, depending only on propext. The same rank+cone test ships as a reusable tool for any orbifold model. Honest scope: one infrared step stays open.
📄 https://zenodo.org/records/21432626
💻 https://github.com/karlesmarin/ghu-su4-completion#Lean4 #FormalMath #ProofAssistant #RepresentationTheory #Maths #Physics
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👨🚀🚀 Ah, yes, because nothing says "I'm serious about math" like naming your #AI system 'Star Fleet' and using Lean 4 to conquer #math problems like it's the final frontier. 🖖 Just a casual desktop app with a mere *2,000 vCPUs* per starship, as if everyone's got a MacBook Pro strapped to an IBM mainframe. 😂 Who knew solving Erdős problems was as easy as running a script from a Star Trek fan's garage?
https://www.starfleetmath.com/ #StarFleet #Lean4 #ErdősProblems #TechHumor #HackerNews #ngated -
👨🚀🚀 Ah, yes, because nothing says "I'm serious about math" like naming your #AI system 'Star Fleet' and using Lean 4 to conquer #math problems like it's the final frontier. 🖖 Just a casual desktop app with a mere *2,000 vCPUs* per starship, as if everyone's got a MacBook Pro strapped to an IBM mainframe. 😂 Who knew solving Erdős problems was as easy as running a script from a Star Trek fan's garage?
https://www.starfleetmath.com/ #StarFleet #Lean4 #ErdősProblems #TechHumor #HackerNews #ngated -
The Proof in the Code: How a Truth Machine Is Transforming Math and AI
by Kevin Hartnett
#LeanLang #Lean4 #LeanProver
https://amzn.to/4xnAIj8 -
The Proof in the Code: How a Truth Machine Is Transforming Math and AI
by Kevin Hartnett
#LeanLang #Lean4 #LeanProver
https://amzn.to/4xnAIj8 -
От augmentation к symbiosis: новая парадигма программирования
Использование средств генеративного искусственного интеллекта (ИИ) в разработке программного обеспечения радикально ускоряет создание кода...
#искусственныйинтеллект #программирование #код #разработка #SemanticCore #KnowledgeGraphs #нейросимволическиеагенты #DOLPHIN #SYNVER #Imandra #Lean4 #symbiosis #NeuroSymbolicAI #LOGOSκ #NIGC #FAIRCARE #AUniversum #SemanticDB #Python #Λоператоры #Logos #код
Источник: https://dstglobal.ru/club/1179-ot-augmentation-k-symbiosis-novaja-paradigma-programmirovanija
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От augmentation к symbiosis: новая парадигма программирования
Использование средств генеративного искусственного интеллекта (ИИ) в разработке программного обеспечения радикально ускоряет создание кода...
#искусственныйинтеллект #программирование #код #разработка #SemanticCore #KnowledgeGraphs #нейросимволическиеагенты #DOLPHIN #SYNVER #Imandra #Lean4 #symbiosis #NeuroSymbolicAI #LOGOSκ #NIGC #FAIRCARE #AUniversum #SemanticDB #Python #Λоператоры #Logos #код
Источник: https://dstglobal.ru/club/1179-ot-augmentation-k-symbiosis-novaja-paradigma-programmirovanija
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SOLIDやLayered Architectureは何を守っているのか？――アーキテクチャ零曲率定理から見る設計原則と不変量
https://qiita.com/iroha1203/items/52d2186f0d510f820e6f?utm_campaign=popular_items&utm_medium=feed&utm_source=popular_items -
Say, I have an inductively defined type in #lean4 like
inductive foo
| quux
| bar : foo -> foo -> fooand I do a proof or simply something like
#check foo.bar .quux .quux
Why do I get an output like
foo.quux.bar foo.quux
instead of something like
foo.bar foo.quux foo.quux
which to me would be more readable? I'm pretty sure that I can get what I want and that this is "stupid" question. But how do I get it?
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Say, I have an inductively defined type in #lean4 like
inductive foo
| quux
| bar : foo -> foo -> fooand I do a proof or simply something like
#check foo.bar .quux .quux
Why do I get an output like
foo.quux.bar foo.quux
instead of something like
foo.bar foo.quux foo.quux
which to me would be more readable? I'm pretty sure that I can get what I want and that this is "stupid" question. But how do I get it?
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So, just double-checking how #Lean4 and #Mathlib work:
* Lean takes 3GiB of RAM and a minute to open Mathlib
* Lean requires about 10min to build itself in CI, only verifying required theorems
* Verifying all of Mathlib is measured in hours
* Lean's kernel is untrustworthy due to junk theoremsAnd yet I'm a clown for using #Metamath? At some point we ought to reconsider the type-theory fetish.
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🚀 Oh, look! Another attempt to make Lean 4 relevant by shoehorning it into socket state machines. 😂 Apparently, the best runtime check is one that doesn't bother existing at all—because why solve real problems when you can just redefine them as types? 🙄
https://ngrislain.github.io/blog/2026-3-25-zerocost-posix-compliance-encoding-the-socket-state-machine-in-lean-4s-type-system/ #Lean4 #Lean4StateMachines #RuntimeChecks #SoftwareDevelopment #TypeSystems #HackerNews #ngated -
🚀 Oh, look! Another attempt to make Lean 4 relevant by shoehorning it into socket state machines. 😂 Apparently, the best runtime check is one that doesn't bother existing at all—because why solve real problems when you can just redefine them as types? 🙄
https://ngrislain.github.io/blog/2026-3-25-zerocost-posix-compliance-encoding-the-socket-state-machine-in-lean-4s-type-system/ #Lean4 #Lean4StateMachines #RuntimeChecks #SoftwareDevelopment #TypeSystems #HackerNews #ngated -
Zero-Cost POSIX Compliance: Encoding the Socket State Machine in Lean's Types
#HackerNews #ZeroCostPOSIX #Compliance #SocketStateMachine #Lean4 #Types
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Zero-Cost POSIX Compliance: Encoding the Socket State Machine in Lean's Types
#HackerNews #ZeroCostPOSIX #Compliance #SocketStateMachine #Lean4 #Types
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Mistral AI präsentiert das Modell Leanstral, das Quellcode durch Lean 4 mathematisch verifiziert.
Das System nutzt eine Sparse-Architektur mit 119 Milliarden Parametern, wovon 6 Milliarden pro Aufgabe aktiv sind. Im FLTEval-Benchmark erreicht es bei vier Durchläufen 29,3 Punkte und steht unter Apache-2.0-Lizenz bereit.
#Leanstral #MistralAI #Lean4 #OpenSource #News
https://www.all-ai.de/news/news26top/mistral-leanstral-neu -
Lean 4 is apparently the new secret sauce of #AI dominance, because who knew that theorem proving could be so *riveting*? 🤔✨ But don't worry, before you can learn how to take over the world with math, you'll need to pass the Vercel Security Checkpoint IQ test, where only the chosen ones with #JavaScript enabled may proceed. 🛂🔒
https://venturebeat.com/ai/lean4-how-the-theorem-prover-works-and-why-its-the-new-competitive-edge-in #Lean4 #TheoremProving #VercelSecurity #HackerNews #ngated -
Lean 4 is apparently the new secret sauce of #AI dominance, because who knew that theorem proving could be so *riveting*? 🤔✨ But don't worry, before you can learn how to take over the world with math, you'll need to pass the Vercel Security Checkpoint IQ test, where only the chosen ones with #JavaScript enabled may proceed. 🛂🔒
https://venturebeat.com/ai/lean4-how-the-theorem-prover-works-and-why-its-the-new-competitive-edge-in #Lean4 #TheoremProving #VercelSecurity #HackerNews #ngated -
Lean 4: How the theorem prover works and why it's the new competitive edge in AI
#HackerNews #Lean4 #AI #TheoremProver #CompetitiveEdge #MachineLearning
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Lean 4: How the theorem prover works and why it's the new competitive edge in AI
#HackerNews #Lean4 #AI #TheoremProver #CompetitiveEdge #MachineLearning
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This is also not too different to how #Lean4 implements mutability and loops in its do-notation.
(Though I think Lean does not use a key-value-store, but rather just nests multiple State Monad Transformers inside of each other.)
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This is also not too different to how #Lean4 implements mutability and loops in its do-notation.
(Though I think Lean does not use a key-value-store, but rather just nests multiple State Monad Transformers inside of each other.)
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[New Blog Post] Calling Lean Functions As Python Functions https://www.philipzucker.com/leancall/ #lean4 #python
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[New Blog Post] Calling Lean Functions As Python Functions https://www.philipzucker.com/leancall/ #lean4 #python
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Lean4 proof that SSOT requires definition-time hooks and introspection
https://zenodo.org/records/18177320
#HackerNews #Lean4 #SSOT #proof #definition-time #hooks #introspection #Zenodo
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Lean4 proof that SSOT requires definition-time hooks and introspection
https://zenodo.org/records/18177320
#HackerNews #Lean4 #SSOT #proof #definition-time #hooks #introspection #Zenodo
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🎉 Behold! A groundbreaking compilation of "junk theorems" in #Lean4, because why not waste your time with math that doesn't matter? 😂 Dive into this #GitHub treasure trove and make your brain cells work overtime for zero reward! 🧠💥
https://github.com/James-Hanson/junk-theorems-in-lean #junktheorems #mathhumor #brainwork #wasteoftime #HackerNews #ngated -
In the thrilling new world of #Lean4, we've finally proven bounds for the Randomized MaxCut #Approximation algorithm that nobody asked for. 📈🔍 Now you can confidently cut those graphs like a pro while impressing... well, nobody. Because who doesn't love an NP-Complete problem just chilling with some approximation algorithms? 🙄🎉
https://abhamra.com/blog/randomized-maxcut/ #RandomizedMaxCut #NPComplete #Algorithms #GraphTheory #HackerNews #ngated -
Proving Bounds for the Randomized MaxCut Approximation Algorithm in Lean4
https://abhamra.com/blog/randomized-maxcut/
#HackerNews #ProvingBounds #RandomizedMaxCut #ApproximationAlgorithm #Lean4 #HackerNews
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🎉 Welcome to the mind-bending world of Lean 4, where scattered resources and typos unite to create the most "informal" #introduction to a formal language ever conceived. 🤪 Just remember, your keyboard shortcuts will be more useful than the article itself, as it takes you on a thrilling #journey from QED to ZZZ. 😴🚀
https://sdiehl.github.io/zero-to-qed/01_introduction.html #Lean4 #Informal #Learning #Programming #HackerNews #ngated -
From Zero to QED: An informal introduction to formality with Lean 4
https://sdiehl.github.io/zero-to-qed/01_introduction.html
#HackerNews #ZeroToQED #Lean4 #Introduction #Formality #Programming
