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

Overview We propose W_γ, a new diagnostic metric for measuring formal-cause-induced phase mismatch in the cosmic microwave background (CMB) temperature power spectrum. The metric quantifies how a change in the effective number of neutrino species (N_eff) — treated as a formal-cause parameter — reshapes the photon acoustic phase geometry in the TT power spectrum relative to a neutrino-free reference. Definition: W_γ(N_eff, 0) = [ ∫ from ℓ=100 to 2000 ( φ_γ^(N_eff)(ℓ) − φ_γ^(0)(ℓ) )² dℓ ]^(1/2) where φ_γ^(N)(ℓ) is the instantaneous phase of D_ℓ^TT after Savitzky–Golay baseline subtraction, extracted via Hilbert transform. Origin: from acoustic engineering to cosmology The definition originates from the W-metric in Spatial Phase Alignment (SPA) theory (doi:10.5281/zenodo.19334407), which measures inter-driver geometric mismatch — acoustic chromatic aberration — in finite-aperture sound sources. The translation to cosmology is grounded in the Webster–Friedmann structural isomorphism (doi:10.5281/zenodo.19334612), which establishes a complete equation-level mapping between the acoustic horn equation and the cosmological perturbation equation. This isomorphism is why a mismatch metric of SPA type can be defined in cosmology at all. Key methodological insight The correct cosmological analogue of SPA's inter-driver mismatch is not a species-to-species comparison (photon vs neutrino). Neutrino perturbations are non-oscillatory at recombination, so Hilbert-transform phase extraction fails on them. The correct translation is: same photon acoustic oscillation under different formal-cause conditions (N_eff) — the analogue of "same instrument, changed boundary conditions." This revision, documented in the implementation memo, was essential for obtaining stable results. Validated results (TT channel) Property Result Status Null test W_γ(0, 0) = 0 exactly ✅ Monotonicity Strict, all N_eff ∈ [0.5, 5.0] ✅ Physical shape Concave (diminishing marginal slope) ✅ Follin cross-validation Spearman ρ = −0.939, p < 10⁻⁴ ✅ SG-window robustness CV < 2%, all windows monotonic ✅ Fiducial value: W_γ(3.044, 0) = 124.8 The strong rank correlation with the Follin et al. (2015, PRL 115, 091301) template-based phase-shift estimator confirms that W_γ captures the same physical information, reformulated as a formal-cause mismatch. Contents File Description concept_note_v2.pdf 1-page concept note: definition, translation principle, formal-cause reading impl_memo_v02.pdf 2-page implementation memo: operational procedure, rejected methods, results validation_note.pdf 2-page numerical validation: five tests documented above fig_w_gamma_main.png Main result figure (5 panels) fig_validation.png Validation figure: Follin comparison + SG-window robustness w_cmb_final.py Main computation script (Python / CLASS / SciPy) w_cmb_validation.py Validation suite: Follin + SG robustness README.md File manifest and execution instructions Scope and limitations This is a TT-channel-only validated release (v1.0). Extension to TE and EE polarization spectra is planned as a subsequent version. Computed with the CLASS Boltzmann code; Y_He = 0.2454 fixed. The metric is a diagnostic quantity defined on the Webster–Friedmann isomorphism, not a replacement for it. Theoretical context The proposition validated here — "formal cause carries mode selectivity" — holds across scales: Acoustics: plate thickness, enclosure geometry, and driver mounting select frequency-dependent radiation geometry. Cosmology: N_eff, sound speed c_s, and recombination epoch t_rec select ℓ-dependent photon phase response. W_γ is the first quantitative realization of this cross-scale structural invariant. Related work T. Takagi, "Acoustic Chromatic Aberration," 10.5281/zenodo.19334407 (2026) — SPA / W-metric origin T. Takagi, "The Exponential Horn as Formal Cause," 10.5281/zenodo.19334612 (2026) — Webster–Friedmann isomorphism T. Takagi, "FCPP," 10.5281/zenodo.18786296 (2026) — Form-Cause Processing Principle