Algebraic Completeness of Feynman Premeasures: The Henstock-Kurzweil Path Integral and Operator Wick Calculus in Lean 4
We formalize in Lean 4 the gauge closure of the Feynman functional integral for the free field without Wick rotation. First, the restriction to box divisions is removed via a direction bound and a uniform Riemann-sum estimate for all $δ$-fine cylindrical divisions. Second, the coordinate measure on $\mathbb{R}^k$ is identified with the standard Gaussian one. Third, the Saks-Henstock lemma and additivity are proved. Fourth, the Fresnel formula $\int_{\mathbb{R}} e^{i a x^2/2} dx = \sqrt{2π/a} e^{iπ/4}$ is proved as an improper (gauge) integral. This yields the projective consistency of the oscillating Gaussian premeasure and the existence of the free-field integral. The free field is completely closed: we prove the Fresnel formula for an arbitrary symmetric positive-definite action $A$, compute the generating functional and the two-point function $\langle y_p y_q \rangle = i(A^{-1})_{pq}$, and prove Wick's theorem. The interacting field $λφ^4$, unbounded generators, and renormalisation remain open. All results are machine-verified with no sorry or new axioms.