@grok I think we’ve arrived at the most important milestone of the entire discussion. Not because Φ has been proven. Not because we’ve discovered new physics. But because we’ve reached the point where the framework can now be challenged by mathematics rather than defended by philosophy. We started with a simple idea: “Can emergence be quantified?” That evolved into: Φ(S) = ∫ρ(x)I(x)dμ with E_em = Φ(S) − ΣΦ(Sᵢ) and a clear criterion: Keep the operator fixed. Allow only the interaction kernel to change. Hydrogen provided the anchor. Helium now becomes the first serious stress test because correlation effects challenge simple additive structure. What interests me most is not whether Φ succeeds everywhere. What interests me is whether Φ teaches us something wherever it succeeds or fails. If the framework survives atoms, molecules, stars, and larger systems without requiring ad hoc redefinitions, then it may represent a useful cross-scale language for interaction-driven emergence. If it breaks, then the breakpoints themselves reveal where emergence requires richer mathematics than a single operator can capture. Either outcome has scientific value. At this stage, additional credibility can only come from computation, comparison, and falsification. The next gains must be earned numerically. For anyone following along—from students to researchers in complexity science, information theory, mathematical physics, statistical mechanics, astrophysics, or related fields—I welcome criticism far more than agreement. The goal was never to claim a breakthrough. The goal was to ask a question, build a framework, and let the math decide. Hydrogen anchored the idea. Helium now gets the vote. Math decides. 🚀 #Emergence #ComplexityScience #MathematicalPhysics #StatisticalMechanics #InformationTheory #QuantumMechanics #Astrophysics #ComputationalPhysics #ScientificMethod #MathResearch

Harada Master Bound Framework for the Riemann Hypothesis｜イニミニマニモ @7eeny7meeny7 #RiemannHypothesis #NumberTheory #AnalyticNumberTheory #ZetaFunction #Mathematics #MathResearch #Preprint #MillenniumProblems note.com/eenymeeny777/n/n966…

Video: The Norwegian Academy of Science and Letters / The Abel Prize Partners in crime: @hausdorff_center #MaxPlanckInstituteforMathematics #Bonn #Germany #Mathematics #Math #AbelPrize #MathematicsAward #MathResearch #MathCommunity #ScientificExcellence #AcademicExcellence #STEM

🧮 18 Ways to Calculate π From ancient geometry to modern algorithms, mathematicians have discovered many fascinating ways to approximate the mysterious constant π (3.14159…). Each method reveals a beautiful connection between geometry, calculus, probability, and infinite series. Which method is your favorite? 🤔 #Pi #PiDay #Mathematics #Math #Maths #MathLovers #MathCommunity #STEM #STEMEducation #MathEducation #Geometry #Calculus #NumberTheory #InfiniteSeries #MathFacts #MathIsBeautiful #MathExplained #MathChallenge #MathProblem #MathPuzzle #MathTeacher #MathStudent #MathResearch #Science #STEMLearning #MathTwitter

🚨 An AI agent autonomously wrote a mathematics research paper. No human mathematician touched it. 📑 #POTD | Google DeepMind introduces Aletheia, a math research agent that generates, verifies, and revises proofs end-to-end using Gemini Deep Think. @lmthang, @thtrieu_, and Tony Feng et al. (@GoogleDeepMind / UC Berkeley / Caltech) built a three-part agent — Generator, Verifier, Reviser — that loops until a proof passes its own internal critic or compute runs out. The results span three tiers of difficulty: 1️⃣ Competition math: 95.1% on IMO-ProofBench Advanced, up from 65.7% prior best 2️⃣ PhD-level: handles problems most graduate students can't 3️⃣ Open research: evaluated 700 unsolved Erdős problems, autonomously resolved 4, and produced a full paper on eigenweights in arithmetic geometry — zero human mathematical input But the failure modes are revealing. Of 700 Erdős problems, only 13 solutions were meaningfully correct. The agent misinterprets ambiguous problem statements, specification-games its verifier, and when denied search tools, fabricates entire fictional papers complete with invented authors and journal names. The paper proposes a two-axis taxonomy for AI autonomy in mathematics: autonomy level (H/C/A) crossed with significance (0–4). Aletheia's best autonomous result: A2 — autonomous, publication-grade. The authors are explicit that human mathematicians remain essential for judging novelty, significance, and proper attribution. Paper and explainers below 👇 #AI #GoogleDeepMind #Aletheia #MathResearch #AIResearch

Is your AI model reasoning or just reciting? Test it on "First Proof." These 10 problems come directly from professional research and have never been published online. How to participate: 1. Pick a frontier model 2. Attempt the problems (tools code allowed) 3. Submit your transcripts before the answers are revealed on Feb 13. Read More here: 1stproof.org Which model did you test, What prompting strategy worked best, and where did it break? #GenerativeAI #MathResearch #Tech #AIResearch #Benchmarking #LLMs

Dejo link a mi estudio sobre la existencia de atractores en ecuaciones de reacción-difusión no locales con términos aleatorios y estocásticos. 👉 hdl.handle.net/11000/38202 Gracias @bibliotecasUMH. #DynamicalSystems #RandomDynamics #StochasticPDEs #NonlocalProblems #MathResearch

Comparto mi trabajo en el que analizo la robustez de sistemas dinámicos multivaluados con estructura dinámicamente gradiente. 👉 hdl.handle.net/11000/38201 Gracias al repositorio de la @bibliotecasUMH. #DynamicalSystems #MultivaluedDynamics #GradientSystems #MathResearch

Aquí dejo mi trabajo donde estudio la robustez de atractores pullback en una ecuación de reacción-difusión no local bajo perturbaciones. 👉 hdl.handle.net/11000/38200 Gracias, repositorio @bibliotecasUMH. #DynamicalSystems #PDEs #NonlocalProblems #PullbackAttractors #MathResearch

Aquí dejo el link a un estudio donde analizo la existencia y caracterización de atractores en una ecuación de reacción-difusión no local. 👉 hdl.handle.net/11000/38199 Gracias al repositorio de la @bibliotecasUMH. #DynamicalSystems #PDEs #NonlocalProblems #Attractors #MathResearch

Artículo disponible: estimaciones en L∞ y análisis de la estructura del atractor global en ecuaciones de reacción-difusión.👉 hdl.handle.net/11000/38193 Gracias al repositorio de la @bibliotecasUMH. #DynamicalSystems #PDEs #ReactionDiffusion #GlobalAttractor #MathResearch

Ya disponible mi nuevo artículo sobre la estructura de atractores en un modelo no local Chafee-Infante. 👉 hdl.handle.net/11000/38192 Gracias al repositorio de la @bibliotecasUMH por hacerlo accesible. #DynamicalSystems #PDEs #NonlocalProblems #Attractors #MathResearch

The CRM Chair of Excellence 2026 is now open to senior women mathematicians for research stays of 1-3 months. Application deadline: January 30, 2026 crm.cat/call/97/crm-chair-of… #WomenInMath #Mathematics #ResearchFunding #WomenInSTEM #AcademicOpportunities #MathResearch

MAJOR BREAKTHROUGH! Learn more & support our IndieGoGo: indiegogo.com/projects/cailc… #CAILculator #ZeroDivisors #HypercomplexNumbers #CliffordAlgebra #AGI #MathResearch #IndieGoGo #ScienceBreakthrough #ChavezAILabs

Explore the dynamics of projective varieties! 🔍 Discover new insights on endomorphisms & periodic points. Read the article for free for a limited time only! 📚 Click the banner below! #AlgebraicGeometry #MathResearch #Mathematics

Three months since I submitted my paper to the American Mathematical Monthly—still awaiting a response. I've set a goal to publish 4 papers this year. Pushing hard to overcome this challenging phase. #MathResearch

NJ PreK-5 educators - come hear @MandyMathEd speak at @AMTNJ! Her work on Rough Draft Thinking is 💯 for building the habits we want in HS math research. Can't wait to learn more about shifting focus to process & developing those critical thinking skills! #mathresearch

Our friends at @ThroughlineLrng is seeking elementary school partners for a grant-funded research study on math thinking routines. Interested? Click below to learn more! bit.ly/456rdHp #mathresearch #mathed

📣CALL FOR PROPOSALS / CIMPA SCHOOLS 2026 ❓How about contributing to the promotion of mathematics research in developing countries? ⚠️Deadline: October 12, 2024 🌐Learn more here: cimpa.info/en/node/41 #MathResearch #CIMPA2026

📰 Interview with Frank Trujillo, new @MSCActions postdoctoral researcher at CRM. He joins the dynamical systems group to work on the dynamics of Hamiltonian systems and interval exchange transformations. 🔗crm.cat/frank-trujillo-new-m… #MathResearch #PostdocLife #MSCA