Slit Monodromy and Commutator Moments of Iterated Integrals

This supplement to Phase Moments on the Critical Strip asks how the shape of an integration contour determines what iterated (Chen) integrals can detect about the zeros inside it. It works with keyhole and figure-eight contours. A single integral sees only winding numbers, and the two walls of a slit cancel. From length two onward, the slit carries the monodromy of the preceding loop into the next kernel, so iterated integrals detect homotopy rather than homology. Their values change only when the slit boundary crosses a singularity (a "step principle"). The paper gives an exact finite-width keyhole identity and shows that zero offsets δ can be read from the boundary only while the corridor stays narrower than δ. Integrating Littlewood's lemma over the abscissa gives a ladder of offset moments, Σ(δ−ε)₊ᵏ/k!, which all switch off at ε = δ. In a two-letter projection of the free Lie algebra, commutator moments between two figure-eight contours are divisible by δ₁δ₂ at every order, and they first become nonzero at order four. A longitudinal barrier acts by conjugation: it preserves order two and adds −4δ₁δ₂ at order three. A transverse corridor wider than δ removes all orders at once. Finally, the paper re-derives the explicit formulas of von Mangoldt and Riemann as instances of the same mechanism. Moving the Perron line leftward deposits a residue −x^ρ/ρ at each zero it crosses, or, when working with log ζ, a slit whose walls contribute −li(x^ρ). Classical criteria equivalent to the Riemann hypothesis are then placed in this framework. All numerical values are reproducible, and the author states explicitly that none of the results bears on the truth of the Riemann hypothesis