Eigenpair analysis reveals oscillatory behavior in the coarse band, suggesting complex dynamics in dark matter models.
We extend the WBA boundary-grid eigenpair scan from the micro-band (Paper 52, step 0.001) to the coarse band epsilon in {0.70, 0.71, 0.72, 0.73, 0.74} at N in {8192, 16384}. The leading eigenvector remains a literal single-cell delta spike (k50 = k75 = k90 = 1, top1pct = 1.000000) at all ten coarse-band points, confirming the Paper 51 sharpened null across the full band and both grid sizes. The second eigenvector forms a highly compressed, ultra-high-frequency wavepacket: its signed version carries sign-change counts near 3800 (approximately 0.23N at N = 16384), yet its absolute mass collapses to k90^(2) = 2 cells near the spike centre. The finite-grid gap proxy g(N, epsilon) = 1 - |lambda_2| / |lambda_1| remains strongly non-monotone across the coarse band, ranging from 0.253 at epsilon = 0.72 to 0.984 at epsilon = 0.73 at N = 16384. Combining coarse-band and micro-band data reveals a Nyquist-style aliasing trap: coarse sampling at step 0.01 can give the impression of smooth parameter dependence while interstitial micro-band sampling at step 0.001 exposes violent finite-grid oscillations in the sub-dominant spectrum. A k = 3 micro-band probe records six epsilon-N-dependent conjugate-pair configurations, confirming that the real versus complex-conjugate classification of the sub-dominant spectrum remains unstable under grid refinement near the B2 band. All statements are strictly finite-grid, scoped to the executed epsilon-mesh and N in {8192, 16384}. Data locked in paper53_coarseband_scan.csv and paper53_k3_microband.csv (both checkers passed). No new kernels, Banach spaces, certified gaps, or universal constants are introduced.
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Michael Bird (2026) studied this question.
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