Spectral Failure Geometry in Modular Arithmetic Networks: Direct Takeover vs Runner-Up Handoff Across Moduli

Neural networks trained on modular arithmetic can fail to grok by collapsing from a shared harmonic scaffold to a competing spectral mode. We characterise the structure of this collapse across two moduli, N=127 and N=257, and describe qualitatively different failure geometries in the two frozen panels examined here (a within-panel descriptive contrast, not a claim generalised to unstudied moduli — see §2.3). In N=127, the eventual failure endpoint is typically not the strongest non-scaffold competitor at the moment of visible scaffold loss — it is the spectral runner-up, rising behind a more powerful predecessor that then decays. In N=257, the first visible non-scaffold mode is already the dominant spectral competitor before scaffold loss in trajectories tested: there is no predecessor to intercept. We characterise the runner-up handoff mechanism for N=127 using trajectories from the main experiment grid (Paper12), showing that off-initial endpoints satisfy a hybrid pre-terminal selector and that multi-hop rows place the endpoint as runner-up before terminal takeover. Handoff margin growth is directionally consistent in handoff-runner-up rows. For N=257, we characterise the mode 7 attractor family using epsilon-initialized trajectories (Paper6 seed family plus main N=257 exp2, rows total) and a broader no-epsilon surrogate panel (50 seeds). The mode 7 family {7, 14, 21, 42} covers first non-scaffold events; the remaining are internal scaffold harmonics (k88, k81, k95) that appear only in no-epsilon conditions. We provide algebraic identification of these modes as the 2nd, 4th, and 8th folded harmonics of the N=257 scaffold (k44), and contrast them with the 6th-order external harmonic (mode 7) that dominates under epsilon. These findings motivate distinct intervention hypotheses for each modulus and geometry type, and define a preregistered mode-selective weight decay scout for N=257. The structural cause of the cross-modulus geometry difference remains uncharacterised at current logging resolution.