Parameter-Coherent Direction Recovery in CMB Acoustic Phase Deformation — W_CMB v10.1 Package

This deposit resolves the remaining limitation of the v10.0 mechanism–direction decomposition. The v10.0 result established that the WCMB response decomposes into an unsigned mechanism axis (S ≈ 99) and a signed direction axis (r ≈ −0.008, orthogonal). However, the direction axis failed within Family E (pos_frac gap ≈ 0.001). This v10.1 note identifies the cause and recovers the signal. The problem Family E aggregates two physically distinct parameter types: Nur (free-streaming radiation density) and ωcdm (cold dark matter density). These produce opposite-sign phase deformations at corresponding multipoles. When combined into a single family, their signed indicators cancel, creating an apparent failure of the direction axis. The result After parameter-coherent splitting into four subfamilies (EN, Eω, A, P), the signed direction separation is recovered: Subfamily pos_frac (+δ) pos_frac (−δ) Gap S EN (Nur) 0.460 0.541 0.081 400 Eω (ωcdm) 0.540 0.461 0.078 774 A (ωb) 0.414 0.581 0.166 164 P (ns) 0.592 0.408 0.184 908 All four subfamilies achieve S > 100. The signed axis was not weak; Family E was internally heterogeneous. Sign inversion The critical finding is that EN and Eω have opposite pos_frac polarity: increasing Nur lowers pos_frac (more negative-phase deformation), while increasing ωcdm raises it. This sign inversion is physically interpretable: free-streaming damping (Nur) and expansion-rate modification (ωcdm) deform the acoustic phase in opposite senses, even though their unsigned kernel shapes place them in the same mechanism family. Updated decomposition principle The mechanism–direction decomposition is now complete: Mechanism axis (unsigned): identifies perturbation family (S ≈ 99 for E/A/P; S ≈ 3–5 for EN/Eω sub-separation). Direction axis (signed): encodes perturbation polarity (S > 100 in all parameter-coherent subfamilies). Cross-parameter aggregation destroys the signal through sign inversion. Orthogonality: r = −0.008, preserved after splitting. Spine (v3.0 → v10.1) 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: Σmν 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: Sign encodes direction, not mechanism v10.0: Mechanism–direction decomposition (S ≈ 99, r ≈ −0.008) ★ v10.1: E-family sign inversion resolved — direction recovery (S > 100, all subfamilies) ★ Contents Research notes: v10.1 direction recovery + v10.0 decomposition (Markdown) Two computation scripts (Python): v9.0 signed-phase, v10.0 decomposition Numerical results (JSON) and diagnostic figures (PNG)