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+Lyα full-shape vector. At fixed dimensionless clock q=H₀t₀=0.951245, with a free ruler scale and positive normalization throughout the finite future, the three families improve χ² over their constant-normalization limit by $(0.842)$, $(4.944)$ and $(0.842)$, respectively. The exponential improvement corresponds to =-2.944 and a normalization decreasing with time; requiring a nondecreasing normalization reduces its improvement to $(1.171)$. The calibrated values remain H₀73.3--$(73.7)$, with conditional profile widths of approximately $(1.0)$. Profiling q/q₀∈[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-q₀ histories admit canonical scalar reconstructions. Independent forward evolution of their tabulated potentials recovers the histories to relative errors below 1.4×10⁻¹⁰ on 0≤ z≤2.33. 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.