Dark Matter in the Second Eigenvector: Oscillatory v(2) Geometry Beneath a Frozen Spike in the WBA Micro-Band ϵ∈[0.700,0.710] (Non-Holomorphic Fractal Series — Paper 52)

This paper is Part 52 of the Non-Holomorphic Fractal Series. It probes the twisted Bird-map WBA kernel on the boundary-grid Ulam discretisation across a narrow “micro-band” in the parameter ε, namely ε ∈ {0.700, 0.701, …, 0.710} at N ∈ {8192, 16384}, using the certified Paper 37 builder and a consistent Markov-matrix (P) orientation. For each of the 22 (ε, N) pairs, we compute the two leading eigenpairs of the column-normalised Ulam matrix and record a locked CSV (paper52_microband_scan.csv) with cumulative-mass diagnostics for both eigenvectors and a discrete finite-grid spectral-gap proxy gN,ε = 1 − |λ₂|/|λ₁|. On these grids the leading eigenvector v⁽¹⁾ remains a literal single-cell spike throughout the micro-band: for all 22 rows we find k₅₀ = k₇₅ = k₉₀ = 1 and top1pct = 1.000000, confirming the delta-spike regime discovered in Paper 51 at ten-fold finer ε-resolution with no detectable variation. This “sharpened null” forces all interesting structure into the second eigenpair. The second eigenpair (λ₂, v⁽²⁾) behaves very differently. The finite-grid gap proxy gN,ε oscillates strongly across the micro-band, ranging from about 0.15 up to about 0.98 at N = 16384, with no sign of a flat or monotone profile. The abs-normalised second eigenvector |v⁽²⁾| shows large swings in its cumulative-mass width while the signed eigenvector exhibits hundreds to thousands of sign changes across the grid; on the executed meshes v⁽²⁾ is broadly supported and violently oscillatory rather than spike-like. A supplementary k = 3 micro-band scan (Paper 53 assets) shows that the sub-dominant spectrum alternates between a large real λ₂ and a small complex-conjugate pair in an ε- and N-dependent way, providing a spectral mechanism for the observed gap-proxy oscillations. All statements in this paper are strictly finite-grid and eigenvector-level. We work only with the Ulam-discretised WBA operator on the executed ε-mesh and resolutions N ∈ {8192, 16384}; we do not alter any claims from Papers 37–51 and we do not touch the certified spectral-gap lower bound 0.6509 from Paper 37. The oscillatory geometry of v⁽²⁾ is documented solely as a structural property of the Ulam matrix at these N, with no claims about the existence, uniqueness, or bounded variation regularity of any corresponding sub-dominant eigenfunction in the continuum limit.