Accumulation-Kernel Families in CMB Acoustic Phase Deformation: W_CMB v4.0 Package

W_CMB v4.0 — Kernel Families in CMB Phase Accumulation Version: 4.0 Date: 2026-04-02 Author: Takayuki Takagi DOI (v3.0): 10.5281/zenodo.19350570 DOI (v4.0): 10.5281/zenodo.19387247 Principle Statement Abstract form (≤ 2 sentences) We show that the fractional phase-accumulation kernel $K(\ell)$ of each cosmological parameter is invariant under perturbation strength and that kernel shape clusters parameters into three families — expansion (E), acoustic driving (A), and primordial (P) — reflecting the physical mechanism by which each parameter distorts CMB acoustic phase. Body form (full) Each cosmological parameter species possesses a strength-invariant accumulation kernel whose shape encodes the physical mechanism by which that parameter distorts acoustic phase. Parameters sharing a common perturbation mechanism cluster into kernel families. Within a family, cross-species variation is captured by a simple multipole shift $K(\ell) \approx K_{\mathcal{F}}(\ell - \Delta\ell)$ with residual RMS $\lesssim 0.02$; between families, the kernel shape itself changes (RMS $\gtrsim 0.05$), encoding a qualitative difference in the perturbation channel. Discovery Narrative v3.0 starting point v3.0 (Zenodo DOI: 10.5281/zenodo.19350570) established two facts: Kernel invariance: The fractional accumulation curve $K(\ell) = W(\ell)/W(\ell_{\max})$ is insensitive to perturbation strength within a parameter species (intra-species spread $\leq 10$). Species-dependent shift: Different parameter species produce kernels offset in $\ell{50}$ ($\Delta\ell{50} \approx 80$). The natural first hypothesis for v4.0 was: $K(\ell;\, p, \delta p) = K{\text{universal}}(\ell - \Delta\ell{\text{species}})$ That is, a single universal kernel shape shifted by a species-dependent offset. v4.0 Stage 1: Initial test (3 species) Testing with ${N{\rm ur}, \omega{\rm cdm}, \omega_b}$ at 5 perturbation strengths each, the shift-only decomposition yielded: Reference → Target Shift $\Delta\ell$ RMS $N{\rm ur}$ → $N{\rm ur}$ (intra) $\leq 7 $N{\rm ur}$ → $\omega{\rm cdm}$ $-23$ to $-43$ $0.008$–$0.012$ $N_{\rm ur}$ → $\omega_b$ $-141$ to $-149$ $\mathbf{0.054}$–$\mathbf{0.056}$ Result: The universal-kernel hypothesis fails for $\omega_b$. The kernel shape itself is different, not merely shifted. v4.0 Stage 2: Family structure (5 species) Extending to ${N{\rm ur}, \omega{\rm cdm}, \omega_b, h, n_s}$ and computing the full cross-species RMS matrix reveals three clusters: Family E (Expansion): ${N{\rm ur}, \omega{\rm cdm}, h}$ Mutual shifted-kernel RMS $\leq 0.018$ Physical basis: all modify the background expansion rate $H(\tau)$ and/or radiation-to-matter transition Shape: IQR $\approx 930$–$1004$, Bowley skew $\approx -0.07$ to $-0.12$ Family A (Acoustic driving): ${\omega_b}$ RMS from Family E: $\approx 0.05$ Physical basis: baryon loading changes the photon-baryon sound speed $c_s(\tau)$, modifying acoustic driving rather than expansion geometry Shape: IQR $\approx 1290$, $\ell_{25} \approx 455$ (low-$\ell$ accumulation — baryon loading signature) Family P (Primordial): ${n_s}$ RMS from all others: $0.076$–$0.091$ Physical basis: spectral tilt modifies initial power spectrum, not dynamical evolution Shape: IQR $\approx 1335$, Bowley skew $\approx -0.51$ (extreme high-$\ell$ concentration) Key reinterpretation The v3.0 conclusion that "shape separation does not hold" is reversed: shape separation does hold, but at the family level. Within a family, the kernel is universal up to a small shift. Between families, the kernel shape changes qualitatively, encoding the physical perturbation channel. Numerical Evidence Intra-species kernel invariance (self-reference test) Species Self-ref RMS (max) $\ell_{50}$ spread Verdict $N_{\rm ur}$ 0.0047 3.1 Invariant $\omega_{\rm cdm}$ 0.0026 26.5 Invariant (smooth drift) $\omega_b$ 0.0071 18.7 Invariant $h$ 0.0007 2.1 Invariant $n_s$ 0.0006 3.0 Invariant All five species pass. Note: $\omega{\rm cdm}$ shows a smooth monotonic drift in $\ell{50}$ with strength, consistent with a weak higher-order correction rather than kernel shape breakdown. Cross-species RMS matrix (median-strength kernels) $N_{\rm ur}$ $\omega_{\rm cdm}$ $\omega_b$ $h$ $n_s$ $N_{\rm ur}$ — 0.010 0.054 0.017 0.089 $\omega_{\rm cdm}$ 0.010 — 0.049 0.018 0.086 $\omega_b$ 0.054 0.049 — 0.053 0.076 $h$ 0.017 0.018 0.053 — 0.079 $n_s$ 0.089 0.086 0.076 0.079 — Quantile profiles (median-strength, each species) Species $\ell_{10}$ $\ell_{25}$ $\ell_{50}$ $\ell_{75}$ $\ell_{90}$ IQR Skew $N_{\rm ur}$ 451 849 1351 1784 2016 935 −0.074 $\omega_{\rm cdm}$ 436 790 1313 1765 2000 975 −0.071 $\omega_b$ 395 459 1158 1749 1984 1290 −0.084 $h$ 431 853 1410 1856 2006 1003 −0.110 $n_s$ 346 632 1642 1967 2100 1335 −0.513 Physical Interpretation The family structure maps directly onto the Boltzmann hierarchy: Family E parameters modify $H(\tau)$, changing the conformal time-to-$\ell$ mapping. This produces a uniform stretching of the phase accumulation profile — hence similar kernel shapes differing only by a translation in $\ell$. Family A ($\omega_b$) modifies the baryon-photon sound speed $c_s = c/\sqrt{3(1+R)}$ where $R = 3\rho_b/4\rho_\gamma$. This alters the acoustic oscillation frequency scale-dependently, producing a wider kernel with enhanced low-$\ell$ accumulation. Family P ($n_s$) tilts the initial power spectrum $\mathcal{P}(k) \propto k^{n_s - 1}$, creating more initial amplitude (and hence phase distortion) at high $\ell$. The kernel reflects this multiplicative tilt, yielding extreme negative skewness. Formal Decomposition $$K(\ell;\, p, \delta p) \;=\; K{\mathcal{F}(p)}!\bigl(\ell - \Delta\ell{\text{intra}}(\delta p)\bigr) + \varepsilon(\ell)$$ where: $\mathcal{F}(p) \in {E, A, P}$ is the family assignment $K_{\mathcal{F}}(\ell)$ is the family-specific kernel shape $\Delta\ell_{\text{intra}}(\delta p)$ is the small intra-species shift ($|\Delta\ell| \lesssim 10$ for weak perturbations) $\varepsilon(\ell)$ is the residual (RMS $\lesssim 0.01$ within family) File Hierarchy File Role Content w_cmb_v4_families.py Main analysis 5-species kernel family computation w_cmb_v4_families.png Main figure 4-panel: kernels, cross-RMS, quantiles, self-consistency w_cmb_v4_kernel.py Exploratory 3-species initial analysis (discovery path) w_cmb_v4_kernel.png Exploratory figure 4-panel: initial kernel analysis w_cmb_v4_summary.json Data Consolidated numerical results (5-species, family structure) w_cmb_v4_synthesis.md This file Principle statement, evidence, interpretation Relation to TSTT/SRTA The kernel family structure exemplifies the TSTT principle that formal cause is measurable: the kernel shape is a formal-causal signature of the perturbation mechanism, not a material property of the perturbation amplitude. The family decomposition $\mathcal{F}: \text{parameters} \to {E, A, P}$ is itself a formal-causal classification — it groups parameters not by their numerical values but by the structural role they play in the Boltzmann dynamics.