Spike-Quotient Diagnostics for the Invariant-Density Proxy of Twisted Bird-Map Operators (Paper 45 in the Non-Holomorphic Fractal Series)

Paper 45 in the Non-Holomorphic Fractal Series. Papers 43–44 established that the invariant-density proxy φ_h(ε) on the boundary subgrid E_bdy ⊂ I_rig converges, on Ulam grids of size N = 1024 and N = 2048, to an N-stable single-cell Dirac-like spike in the central half of the partition. This makes the global oscillation functional osc_global saturated and unable to distinguish trivial spikes from genuinely rough densities. Paper 45 defines a battery of five spike-quotient functionals on finite probability vectors u ∈ ℝ^N: Variance-off-Peak (VOP), Entropy-after-Exclusion (EAE), BV-seminorm-of-Complement (BV_COMP), Spike-Quotient Oscillation (SQ_OSC), and Coarse-Rebinned Support Count (CR_SUPP). Each functional is defined by an explicit finite-dimensional algebraic expression and provably vanishes on exact one-hot vectors. Moreover, for the class of k-cell uniform clusters, BV_COMP and EAE are invertible: they encode k exactly via k = 1 + 2/BV_COMP(u) and k = 1 + e^{EAE(u)} (Lemma 2.2). All five functionals evaluate to the Regime A baseline on the observed N = 1024 and N = 2048 data, confirming Papers 43–44 and establishing explicit falsifiability thresholds for future higher-resolution runs.