Computer-Verified Physics from One Equation: Formal Verification of the Hopf Soliton Programme in Lean 4

Title: Computer-Verified Physics from One Equation: Formal Verification of the Hopf Soliton Programme in Lean 4

Author: Alexander Novickis (alex.novickis@gmail.com)

We present the first comprehensive formal verification of a physics framework in the Lean 4 proof assistant. Three theorems from the Hopf soliton programme --- spanning soliton existence with stability, constructive quantum field theory with mass gap, and topological charge conservation --- are formalised with full logical chains from axioms to conclusions. The formalisation comprises approximately 3,400 lines of Lean 4 code across thirteen modules, approximately 120 axioms encoding physics and mathematics inputs, and approximately 40 proved theorems and lemmas. We describe the methodology, classify axioms by epistemic status (mathematical, physical, computer-assisted), and discuss what formal verification means for theoretical physics. The key methodological contribution is the axiom classification: while the axioms themselves are not machine-checked, the logical chain from the axioms to every claimed result is verified by the Lean kernel. We argue that this "conditional correctness" --- if the axioms hold, then the theorems follow --- is the appropriate standard for formal verification of physics, analogous to the role of axioms in mathematics.

Key results include:

• Three core theorems of the Hopf soliton programme have been formalised in Lean 4 + Mathlib: soliton existence (Paper CIV), constructive QFT with mass gap (Paper CI), and topological charge conservation (Paper CV).
• The combined formalisation comprises ~3,400 lines of Lean 4 code, ~120 axioms encoding physical and mathematical inputs, and ~40 proved lemmas and theorems.
• Every axiom is classified by epistemic status: mathematical (not yet in Mathlib), physical (proved in referenced papers), or computer-assisted (interval arithmetic). The logical chain from axioms to conclusions is machine-checked.
• This is the first formal verification of a soliton existence proof, a constructive quantum field theory, and a topological charge conservation theorem in any proof assistant.

Keywords: formal verification, lean4, mathlib, proof assistant, hopf soliton, mass gap, constructive QFT, topological soliton, computer proof

Series: Paper CXXIII in the Hopf Soliton Programme