Integral Eulerian magnitude homology of paths

For a unit-edge path, each fixed-frame Eulerian magnitude chain complex is identified with a signed, degree-reindexed augmented join of discrete occurrence sets, one set for each vertex absent from the frame. Its integral homology is free and concentrated in the degree equal to the width of its interval hull. Translated inclusions of minimal subpaths consequently induce a direct-sum decomposition of every bigraded homology group. This yields the path-rank recurrence of Martin II and Sazdanović, a finite frame formula and explicit integral bases. A four-vertex example corrects a previously stated maximum-length value. The results are presented as candidate theorems, supported by a complete integral argument and reproducible finite checks; established-status verification is not claimed.