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 (Nₑff) 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 (WTT² + WTE²) EE — ❌ Excluded: instability at Nₑff ≈ 2 Cross-channel agreement Measure Value Spearman ρ (TT vs TE) 1. 000000 Pearson r 0. 999996 Mean WTE / WTT 1. 001 ± 0. 002 The two channels lie on the identity line to within 0. 1%. This demonstrates that the formal cause (Nₑff) 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 (Nₑff). EE exclusion An ℓₘax sweep (800, 1000, 1200, 1500, 1800, 2000) showed that EE develops a phase instability at Nₑff ≈ 2. 0–2. 5 that persists at all tested ℓₘax values, ruling out a high-ℓ damping artifact. The instability is intrinsic to the analytic-signal method applied to D_ℓEE. 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, Nₑff ∈ 0. 5, 5. 0 ✅ TE monotonicity Strict, Nₑff ∈ 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 ℓₘax Excluded Contents File Description conceptₙoteᵥ2. pdf 1-page concept note: two-channel definition implₘemoᵥ02. pdf 2-page implementation memo: procedure and rejected methods validationₙoteᵥ2. pdf 2-page validation: TT+TE results, EE exclusion figₜtₜeₘain. png Main 6-panel figure figₜtᵥalidation. png TT validation (Follin + SG robustness) figₑeₑxclusion. png EE ℓₘax sweep evidence wcmbᵥ2ₜtₜe. py TT+TE computation script wcmbᵥalidation. py TT validation suite wcmbₑeₗmaxₛweep. 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
Takayuki Takagi (Wed,) studied this question.