Cell-Wise Oscillation Anatomy of the Invariant-Density Proxy Near the Mixing Boundary for Twisted Bird–Map Operators (Paper 43 in the Non-Holomorphic Fractal Series)

Paper 43 in the Non-Holomorphic Fractal Series. Paper 41 and Paper 42 reported global oscillation saturation osc(φ_h, ε) = 1.0 for all three kernels on both the coarse grid E_wide and the refined boundary subgrid E_bdy. This note decomposes the oscillation spatially on the fixed N = 1024 Ulam partition using the frozen diagnostic script paper43_cellwise_osc_probe.py. For every h ∈ K and every ε ∈ E_bdy the invariant-density proxy φ_h(ε) is numerically equivalent to a single-cell Dirac delta in the central half of the partition: osc_left = osc_right = 0 exactly, osc_centre = osc_global ≈ 1.0, and the cell-wise variance σ²_cell equals the exact single-cell value (N−1)/N² = 1023/1048576 ≈ 9.756 × 10⁻⁴ to within 2 × 10⁻¹³ of the exact value. All claims are purely numerical and diagnostic on the existing certified Ulam infrastructure; no new computations, kernels, or analytic statements are introduced.