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 W_CMB 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: N_ur (free-streaming radiation density) and ω_cdm (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 (E_N, E_ω, A, P), the signed direction separation is recovered:

All four subfamilies achieve S > 100. The signed axis was not weak; Family E was internally heterogeneous.

Sign inversion

The critical finding is that E_N and E_ω have opposite pos_frac polarity: increasing N_ur lowers pos_frac (more negative-phase deformation), while increasing ω_cdm raises it. This sign inversion is physically interpretable: free-streaming damping (N_ur) and expansion-rate modification (ω_cdm) 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:

1. Mechanism axis (unsigned): identifies perturbation family (S ≈ 99 for E/A/P; S ≈ 3–5 for E_N/E_ω sub-separation).
2. Direction axis (signed): encodes perturbation polarity (S > 100 in all parameter-coherent subfamilies). Cross-parameter aggregation destroys the signal through sign inversion.
3. 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: 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: 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)