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 Rᵏ is identified with the standard Gaussian one. Third, the Saks-Henstock lemma and additivity are proved. Fourth, the Fresnel formula ∫R ei a x²/2 dx = √2π/a eiπ/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 yₚ yq = i(A⁻¹)pq, and prove Wick's theorem. The interacting field λφ⁴, unbounded generators, and renormalisation remain open. All results are machine-verified with no sorry or new axioms.