Kernel Discriminability and Coordinate Compression in CMB Acoustic Phase Deformation: W_CMB v5.0 Package

W_CMB v5.0 — Kernel Discriminability and Coordinate Compression in CMB Acoustic Phase Geometry Version: 5.0 Date: 2026-04-04 Author: Takayuki Takagi DOI (v4.0): 10.5281/zenodo.19387247 DOI (v3.0): 10.5281/zenodo.19365861 Principle Statement Abstract form (≤ 2 sentences) We show that unsupervised clustering on kernel location parameters perfectly recovers the three v4.0 families (ARI = 1.000), that neutrino mass Σm_ν does not create a new kernel-manifold direction, and that the resulting E-adjacency reflects coordinate compression — not information absence — in the |Δφ|-cumulative kernel construction. Body form (full) The accumulation-kernel family structure established in v4.0 (Expansion, Acoustic, Primordial) is tested for discriminative power in location-parameter space. Six indicators — ℓ₅₀, ℓ₇₅, ℓ₉₀, IQR, tail ratio, Bowley skewness — extracted from the fractional accumulation kernel K(ℓ) are used as coordinates. The analysis is then extended to massive neutrinos (Σm_ν) and connected to per-ℓ Fisher information to determine what the kernel measures, what it misses, and why. Discovery Narrative v5.0: Discriminability (ARI = 1.000) Separation ratio S = (between-family median gap) / (max within-family IQR spread) was computed for each indicator, with judgment thresholds fixed a priori: S > 3: separation 1 < S ≤ 3: partial separation S ≤ 1: collapsed Separation ratio results: ℓ₅₀: E↔A = 2.97, E↔P = 66.2, A↔P = 288.5, Min S = 2.97 → PARTIAL IQR: E↔A = 5.49, E↔P = 4.60, A↔P = 3.14, Min S = 3.14 → SEPARATION tail_ratio: E↔A = 0.39, E↔P = 88.6, A↔P = 88.7, Min S = 0.39 → COLLAPSED K-means (k=3) on standardized 6-indicator space: match rate = 100%, ARI = 1.000. The bottleneck is exclusively E↔A; all other family pairs separate with S > 50. v6.0/v6.1: Σm_ν is E-adjacent v6.0 (unlensed TT): Σm_ν (ℓ₅₀ = 1409.7) lands near Family E (ℓ₅₀ = 1422.5) with S(M↔E, ℓ₅₀) = 0.89. PCA shows no new dimension (PC1+PC2 = 99.99%). However, k=4 clustering perfectly separates all four groups (ARI = 1.000), indicating M is distinguishable but E-adjacent. v6.1 (lensed TT): Lensing does not activate a new direction. S(M↔E) decreased in 4/6 indicators. k=4 ARI dropped from 1.000 to 0.556. Lensing smoothing destroys acoustic fine structure that aided the subtle M/E discrimination. v8.0-lite: Coordinate compression, not information absence Fisher information decomposition resolved the ambiguity: Total Fisher F: N_eff = 21,071; Σm_ν = 22,130 Fisher ℓ₅₀: N_eff = 1,509; Σm_ν = 1,555 Kernel ℓ₅₀: N_eff = 1,426; Σm_ν = 1,409 Key findings: Fisher ratio Σm_ν / N_eff = 1.05 → comparable information Fisher ℓ₅₀ gap = 46 → Fisher sees the difference Kernel ℓ₅₀ gap = 17 → kernel compresses it Fisher profile overlap = 0.72 → partial degeneracy, not total The compression originates from the kernel's construction: cumulating |Δφ| discards the sign structure of the phase deformation, collapsing directional Fisher information onto the amplitude axis. Key Reinterpretation W_γ is a structured but compressive phase observable: What it measures: accumulation geometry — where phase deformation concentrates across ℓ What it resolves: perturbation families defined by distinct physical mechanisms (E/A/P) What it compresses: directional differences between free-streaming (N_eff) and mass-induced (Σm_ν) deformation Why it compresses: the |Δφ| operation discards phase sign, which carries the Fisher-visible direction difference This motivates a future multi-cell phase diagnostic architecture assigning distinct kernels to magnitude, sign, and differential modes of phase deformation. Spine v3.0: W_CMB is established v4.0: Internal geometry is family-structured v5.0: That geometry is discriminative (ARI = 1.000) v6.0/6.1: Σm_ν does not create a new direction v8.0-lite: E-adjacency is coordinate compression, not information absence Files W_CMB_v5_README.md — Full documentation w_cmb_v5_discriminability.py — v5.0 pipeline: location parameters, separation ratio, clustering w_cmb_v6_manifold.py — v6.0 Σm_ν manifold expansion test (unlensed) w_cmb_v6_1_lensed.py — v6.1 lensed vs unlensed comparison w_cmb_v8_fisher.py — v8.0-lite Fisher information decomposition w_cmb_v5_scatter.png — v5.0 scatter plots and correlation matrix w_cmb_v6_manifold.png — v6.0 PCA and manifold analysis w_cmb_v6_1_comparison.png — v6.1 unlensed vs lensed comparison w_cmb_v8_fisher.png — v8.0-lite Fisher vs kernel geometry Requirements Python 3.8+ CLASS/classy (Boltzmann solver) numpy, scipy, matplotlib, scikit-learn