A Two-Channel Cosmological W-Metric for Formal-Cause Phase Mismatch: Concept, Implementation, and Numerical Validation

Overview W_γ is a diagnostic metric for measuring formal-cause-induced phase mismatch in the CMB power spectra. It quantifies how a change in the effective number of neutrino species (N_eff) reshapes the photon acoustic phase geometry relative to a neutrino-free reference. v2.0 advance: The metric, first established in TT alone (v1.0, doi:10.5281/zenodo.19350571), is now independently reproduced in TE with near-identical magnitude and perfect rank-order agreement. This yields a robust two-channel joint metric, while the EE extension is deferred due to channel-specific phase instabilities. Two-channel results Channel W_γ(3.044, 0) Monotonic Notes TT 124.8 ✅ Density oscillation TE 125.1 ✅ Density–velocity cross-correlation Joint 176.7 ✅ sqrt(W_TT² + W_TE²) EE — ❌ Excluded: instability at N_eff ≈ 2 Cross-channel agreement Measure Value Spearman ρ (TT vs TE) 1.000000 Pearson r 0.999996 Mean W_TE / W_TT 1.001 ± 0.002 The two channels lie on the identity line to within 0.1%. This demonstrates that the formal cause (N_eff) reshapes the entire photon acoustic phase geometry uniformly — density oscillations and density–velocity cross-correlation carry the same mismatch signal. Origin The definition originates from the W-metric in Spatial Phase Alignment (SPA) theory (doi:10.5281/zenodo.19334407), translated to cosmology via the Webster–Friedmann structural isomorphism (doi:10.5281/zenodo.19334612). The correct cosmological analogue is not species-to-species comparison (photon vs neutrino — fails because δ_ν is non-oscillatory), but same photon acoustic oscillation under different formal-cause conditions (N_eff). EE exclusion An ℓ_max sweep (800, 1000, 1200, 1500, 1800, 2000) showed that EE develops a phase instability at N_eff ≈ 2.0–2.5 that persists at all tested ℓ_max values, ruling out a high-ℓ damping artifact. The instability is intrinsic to the analytic-signal method applied to D_ℓ^EE. The EE extension requires separate methodological development. Full validation summary Property Result Status TT null test W(0,0) = 0 ✅ TE null test W(0,0) = 0 ✅ TT monotonicity Strict, N_eff ∈ [0.5, 5.0] ✅ TE monotonicity Strict, N_eff ∈ [0.5, 5.0] ✅ Joint monotonicity Strict ✅ TT–TE correlation r = 0.999996 ✅ TT–TE ratio 1.001 ± 0.002 ✅ Follin cross-validation ρ = −0.939 ✅ SG-window robustness CV < 2%, all monotonic ✅ EE systematic test Instability at all ℓ_max Excluded Contents File Description concept_note_v2.pdf 1-page concept note: two-channel definition impl_memo_v02.pdf 2-page implementation memo: procedure and rejected methods validation_note_v2.pdf 2-page validation: TT+TE results, EE exclusion fig_tt_te_main.png Main 6-panel figure fig_tt_validation.png TT validation (Follin + SG robustness) fig_ee_exclusion.png EE ℓ_max sweep evidence w_cmb_v2_tt_te.py TT+TE computation script w_cmb_validation.py TT validation suite w_cmb_ee_lmax_sweep.py EE exclusion test Related work T. Takagi, "Acoustic Chromatic Aberration," 10.5281/zenodo.19334407 — SPA / W-metric origin T. Takagi, "The Exponential Horn as Formal Cause," 10.5281/zenodo.19334612 — Webster–Friedmann isomorphism T. Takagi, "W-Metric v1.0 (TT only)," 10.5281/zenodo.19350571 — Superseded by this record T. Takagi, "FCPP," 10.5281/zenodo.18786296 — Form-Cause Processing Principle