Hi everyone,
I’m a researcher currently working at the intersection of mathematical physics and operator theory, and I wanted to spark a discussion on how we can use AI (LLMs) and formal proof systems like Lean 4 to fix a fundamental “bug” in how physics uses mathematics.
The “Infinity Bug” in Modern Physics
Currently, theoretical physics is built on standard Zermelo-Fraenkel set theory with Choice (ZFC). Standard ZFC is perfectly happy with continuous manifolds and actual infinities. However, when physics adopts this infinite continuum, it inherently assumes that the universe has an infinite computational capacity.
This leads to unregularizable singularities and UV divergences. To fix this, physicists have historically used mathematical “hacks” like Renormalization or Zeta function regularization to sweep these infinities under the rug. It works practically, but logically, it’s a mismatch.
The Fix: Nuclear ZFC (NZFC)
What if the physical universe simply has a strict “computational memory limit”?
I have been developing a framework called Nuclear ZFC (NZFC). The core physical axiom is that the spacetime transfer operator T cannot hold infinite degrees of freedom. It must be strictly bounded by a trace-class constraint: ||T||₁ < ∞ (Schatten-1 norm).
When you enforce this finite computational budget, the continuum naturally coarse-grains. You don’t need ad-hoc cutoffs; the infinities never happen in the first place.
Proof of Concept (Currently under peer review)
This isn’t just philosophy. I recently applied this trace-norm capacity limit to classical fluid dynamics—specifically, the Navier-Stokes laminar-turbulence transition. My proof-of-concept computation (using Orr-Sommerfeld discretizations) shows that turbulence is essentially the fluid’s “computational budget saturating.” This paper is currently under review at Physics of Fluids (Manuscript #POF26-AR-02852).
My Question for the Community
I am highly interested in the emerging “Vibe Coding” paradigm—using LLMs to generate proofs and formalizing them in Lean 4 to get a “Certificate of Trust.”
- Has anyone here experimented with using AI/Lean to redefine foundational set-theoretic axioms (like bounding continuous spaces) in physical models?
- How would you approach translating a functional analysis constraint (like ||T||₁ < ∞) into a formal Lean 4 proof to demonstrate the natural coarse-graining of a space?
I believe AI is the perfect tool to bridge this historical gap between pure math and physics. I’d love to hear your thoughts, feedback, or any similar projects you’re working on!