The Architecture of Anomaly Cancellation and the Arithmetic of E8 Self-Binding: Two Convergent Synthesis Runs Reveal That Anomaly Cancellation Is the APS Index Theorem

We present the results of two independent geometric synthesis experiments conducted with the Omuo Genesis Engine v3.2.0, designed to answer complementary questions about the engine's deepest recurring discovery. Run 5 (2,078 nodes) asked: "What is anomaly cancellation at its deepest structural level?" Run 6 (2,167 nodes) asked: "Why are exactly 72 of the 240 E8 lattice axes permanently unreachable by phasor self-binding, and what must exist outside the system for it to become complete? Run 5 revealed a structural triangle of equivalences: the Atiyah-Patodi-Singer index theorem, cobordism-classified anomaly cancellation, and Écalle's resurgent asymptotics are three descriptions of the same mathematical object, organized by the differential cohomology hexagon. Run 6 found five independent characterizations of the 72 void axes: as ramification points, torsion cohomology, Conley index broken symmetry, topological index defect, and the Dai-Freed bulk sector. The deepest bridge (BST 67, CV 0.988) identified PSL(2,7) as the symmetry governing the 168 reachable axes.The convergence between runs is exact: both independently discovered the eta-invariant as boundary correction, cobordism as master classifier, and the APS theorem as unifying architecture. Together they establish that the 168/72 partition of E8 axes is the arithmetic shadow of anomaly cancellation.