N = 4096 Path-A Ulam Density Extraction and Spike-Quotient Falsification for Regime A (Paper 47 in the Non-Holomorphic Fractal Series)

Paper 47 in the Non-Holomorphic Fractal Series. Paper 46 defined a two-path falsification protocol for the N-stable single-cell Regime A hypothesis: Path A pushes the Ulam resolution from N = 2048 to N = 4096; Path B stress-tests the spike-quotient inversion formulas with planted probe kernels. Paper 47 executes the full Path A protocol: for each of the three kernel families (WBA, Lorentzian, Alien) and all ten ε ∈ E_bdy, the twisted Bird-map Ulam operator is assembled at N = 4096, the leading eigenvector is extracted via Arnoldi iteration, and the Paper 45 spike-quotient battery (VOP, EAE, BV-COMP, SQ-OSC, CR-SUPP) is evaluated. On the N = 4096 Ulam grid every boundary cell satisfies EAE > 0 and BV-COMP > 0, so Regime A is numerically falsified at every (h, ε) pair on E_bdy at N = 4096 (all thirty boundary cells) and every cell lies in Regime B in the sense of Paper 45. The inferred cluster widths k_EAE = 1 + e^{EAE} and k_BV = 1 + 2/BV-COMP remain in the single- or few-digit range for the Lorentzian corner ε ∈ {1.02, 1.05, 1.07, 1.08} and the Alien cell at ε = 0.76, while several other cells exhibit very broad k_EAE with modest k_BV.