The Evolution of Sufficient Statistics in Cosmological Structure Formation

This study presents empirical evidence that challenges the Markovian assumption of the standard $\Lambda$CDM cosmological model. Statistically significant correlations are identified between primordial phase structure in the cosmic microwave background (CMB) and the late-time metallicity field reconstructed from SDSS, DESI, MaNGA, Gaia, and JWST observations. Bipolar Spherical Harmonics (BiPoSH) quantify the primordial anisotropy, while a tomographic three-dimensional metallicity field captures late-time structure. A peak cross-correlation significance of $3.9\sigma$ is detected within the multipole range $30 \le \ell \le 100$ and redshift interval $0.5 < z < 2.0$, contradicting the expectation that non-linear gravitational evolution erases primordial phase information. These findings are formalised through the Infinite Transformation Principle (ITP), which introduces a minimal extension to $\Lambda$CDM by allowing path-dependent structural information $\Psi(t)$ within an evolving state space $\Omega(t)$. The framework is fully falsifiable and predicts observable signatures for upcoming missions, including JWST, Euclid, Roman, LSST, and CMB-S4.