N-Stability of the Single-Cell Spike: Cell-Wise Oscillation at N = 2048 for Twisted Bird–Map Operators (Paper 44 in the Non-Holomorphic Fractal Series)

Paper 44 in the Non-Holomorphic Fractal Series. Paper 43 established that, on the N = 1024 Ulam grid, the invariant-density proxy for twisted Bird–map operators is numerically a single-cell Dirac-like spike in the central half of the partition. This note reruns the same frozen diagnostic script at N = 2048. Every qualitative signature persists unchanged: all 30 rows satisfy osc_left = osc_right = 0 exactly, osc_centre ≈ 1.0, maxratio = ∞, and σ²_cell = (2047/2048²) ≈ 4.880 × 10⁻⁴ to within 8.4 × 10⁻¹⁴. The peak-cell index scales by a factor of 2 relative to N = 1024, preserving the kernel ordering Alien ≥ WBA ≫ Lorentzian. We conclude that the single-cell Dirac artefact is N-stable: it tracks the grid resolution and is not an accident of the N = 1024 discretisation.