From E-Adjacency to Four-Family Resolution: Normalization Compression and Raw-Spread Recovery in CMB Acoustic Phase Deformation — W_CMB v6.0 Package

W_CMB v6.0 — From E-Adjacency to Four-Family Resolution Version: 6.0 Date: 2026-04-04 Author: Takayuki Takagi DOI (v5.0): 10.5281/zenodo.19401308 DOI (v4.0): 10.5281/zenodo.19387247 Principle Statement We identify CDF normalization — not sign-structure loss — as the primary compression mechanism preventing the single-kernel Wγ from separating neutrino mass (Σmν) from free-streaming radiation (N_eff), and demonstrate that reintroducing unnormalized deformation spread (raw_std) achieves four-family separation with S(M↔E) = 8.41, resolving the E-adjacency limitation established in v5.0. Discovery Narrative v9.0: Signed cumulative fails The signed cumulative cell was identical to magnitude after normalization (Spearman ρ = 1.000). Multi-cell clustering degraded from ARI = 1.000 to 0.448. The architecture was valid but Cell 2 design was wrong. v9.1: No sign structure exists Cancellation ratio S(ℓ) = cumΔφ / cum|Δφ| ≈ ±1 everywhere. Phase deformation accumulates monotonically — no sign reversals. The v8.0-lite diagnosis ("|Δφ| drops sign") was incorrect: there is no sign to drop. The compression source lies in normalization itself. v9.2: Raw spread breaks E-adjacency ★ Removing CDF normalization and extracting raw indicators: S(M↔E) comparison — the breakthrough: norm_ell50: S = 0.89 → COLLAPSED norm_IQR: S = 1.64 → PARTIAL raw_std: S = 8.41 → SEPARATION ★ raw_std achieves full 4-family separation: E↔A: S = 16.68 E↔P: S = 38.73 A↔P: S = 27.20 M↔E: S = 8.41 First single indicator to separate all four families with S > 3. Physical interpretation: raw_std measures the ℓ-spread of deformation weighted by absolute amplitude. N_eff (std ≈ 449) produces more concentrated deformation; Σm_ν (std ≈ 464) spreads more broadly — reflecting focused free-streaming vs broader lensing-mediated effects. CDF normalization erases this spread difference. Key Result: Second-Generation W_γ W_γ matures from a single observable into a two-layer measurement principle: Layer 1 (Geometry): Normalized kernel K(ℓ) — identifies family structure (E/A/P) Layer 2 (Spread): Raw deformation-weighted std — resolves family-internal compression, recovers the fourth family (M) Spine v3.0: W_CMB is established v4.0: Internal geometry is family-structured v5.0: That geometry is discriminative (ARI = 1.000, 3 families) v5.0 v6.0/v6.1: Σm_ν is E-adjacent in normalized kernel v5.0 v8.0-lite: E-adjacency is compression, not information absence v9.0: Signed cumulative ≡ magnitude after normalization v9.1: No sign structure — S(ℓ) ≈ ±1 everywhere v9.2: raw_std breaks E-adjacency → 4-family separation (S = 8.41) ★ Conclusion Normalized kernel geometry identifies perturbation families, but unnormalized deformation spread resolves family-internal compression; together they recover four-family structure in CMB acoustic phase geometry. Files W_CMB_v6_README.md — Full documentation w_cmb_v9_multicell.py — v9.0 multi-cell signed cumulative test w_cmb_v9_1_cancellation.py — v9.1 cancellation ratio test w_cmb_v9_2_raw.py — v9.2 raw spread breakthrough w_cmb_v9_multicell.png — v9.0 figure w_cmb_v9_1_cancellation.png — v9.1 figure w_cmb_v9_2_raw.png — v9.2 breakthrough figure Requirements Python 3.8+, CLASS/classy, numpy, scipy, matplotlib, scikit-learn