An Inverse-Squared-Logarithm Hubble Normalization: Conditional Distance Fits and Scalar-Field Reconstructions

Motivated by our numerical study of a non-autonomous spectral construction, we test inverse-squared-logarithm time dependence of an effective Hubble normalization, with exponential and power-law forms as comparisons. The extension from an iteration-dependent parameter to cosmic time is an additional physical hypothesis. We solve the time and distance equations self-consistently and fit public DESI distance products and full-covariance Pantheon+ likelihoods, first without absolute calibration and subsequently using the archived Cepheid-calibrated product. The latter contains 1657 light-curve rows, including 77 calibrators, and is combined with an author-released 14-entry BAO+Ly$\alpha$ full-shape vector. At fixed dimensionless clock $q=H_0t_0=0.951245$, with a free ruler scale and positive normalization throughout the finite future, the three families improve $\chi^2$ over their constant-normalization limit by $(0.842)$, $(4.944)$ and $(0.842)$, respectively. The exponential improvement corresponds to $\Delta\mathrm{AIC}=-2.944$ and a normalization decreasing with time; requiring a nondecreasing normalization reduces its improvement to $(1.171)$. The calibrated values remain $H_0\simeq73.3$--$(73.7)$\kmsmpc, with conditional profile widths of approximately $(1.0)$\kmsmpc. Profiling $q/q_0\in[0.85,1.50]$ yields no closed clock interval at the reported likelihood levels; the preferred exponential and power-law solutions reach clock or slope bounds. Under flat general relativity and explicit physical-matter assignments, four fitted fixed-$q_0$ histories admit canonical scalar reconstructions. Independent forward evolution of their tabulated potentials recovers the histories to relative errors below $1.4\times10^{-10}$ on $0\leq z\leq2.33$. These checks establish numerical background consistency, not a detection of evolving normalization or scalar fields. The clock boundary is not a measured cosmic age, the fitted ruler is not an early-universe prediction, and no resolution of the Hubble tension is established.