Methodological Audit of Large-Scale Signals in Cosmicflows-4: Shuffle-Null Persistent Homology and the Bulk Flow by Distance Indicator

We audit two large-scale signals in the Cosmicflows-4 (CF4) compilation: (i) an excess of H1 persistent-entropy ordering relative to an angularly shuffled null, and (ii) the cosmic bulk flow at 90–130 Mpc. (i) In nested spheres the contrast reaches z_shuf ≈ 8σ, and a core-excised volume (30–120 Mpc) retains ΔS_shuf = +0.91% (+8.63σ), while the 60–90 and 90–120 Mpc shells alone are consistent with the null. Because the shuffle null removes all spatial correlations, any clustered field lies below it; this statistic detects ordinary clustering of the cosmic web and is not a test of statistical homogeneity. We make no homogeneity claim from it. (ii) A joint monopole-dipole fit of the full sample gives U0 = -114 ± 21 km/s and V_bulk = 482 ± 42 km/s. Split by indicator, Tully-Fisher / Fundamental Plane (TF/FP) groups give 549 ± 47 km/s with U0 = -143 ± 24 km/s, and Type Ia supernova (SN Ia) groups (N_grp = 193) give 363 ± 99 km/s with U0 = -35 ± 47 km/s. The indicators differ by 2.0σ in the monopole and 1.7σ in amplitude. The significance relative to ΛCDM depends strongly on the adopted linear-theory V_rms and on the convention for the component error: for V_rms = 185.3 km/s the full sample deviates by 3.7σ and the SN Ia sample by 2.2σ (p = 0.03) or 1.6σ (p = 0.10); for V_rms = 254 km/s these fall to 2.4σ and 1.4–1.2σ. Both indicators point within 14°–27° of the Great Attractor / Shapley region.