High-winding first-hit geometry and singular action finite parts near a reversible Hamiltonian saddle-focus

We consider a four-dimensional reversible exact Hamiltonian flow with a real-analytic saddle-focus and a transverse homoclinic orbit. Long saddle passages produce the familiar high-winding return strips, but separate continuation of these strips does not exclude additional return or exit components. From a compact segment of the homoclinic orbit and two selected outgoing orbit germs we construct a compact observation block whose boundary faces record the first return and exits. We identify every sufficiently high-winding cell with one fixed reference cell and prove uniform C² convergence of the finite family of first-event functions. A face-preserving ambient isotopy carries the limiting normal-crossing arrangement, including its sign strata and connected components, to every sufficiently high-winding cell, while strict first-hit margins preserve the physical labels. Hence the local return–exit decomposition is exhaustive inside the block. We also treat two noncompact exit channels. At an infinite-time algebraic end, action is compared with a fixed reference orbit at the same value of the compactifying coordinate. At a finite-time regular-singular blow-up end with fixed simple positive-integer indicial roots, a finite Laurent–log counterterm is subtracted at fixed remaining physical time. Both relative finite parts are limits of finite physical orbit integrals and retain the exact concatenation law. The hypotheses are verified for the reversible folded saddle-node-II normal form by exact calculation and interval enclosures. This record contains the manuscript, a self-contained LaTeX source archive, and the M1–M4 validation evidence used by the RFSN-II realization. It excludes research history, exploratory computations, editorial records, and unrelated follow-up projects.