We re-analyze the Cepheid data used to infer the value of $H_0$ by calibrating SnIa. We do not enforce a universal value of the empirical Cepheid calibration parameters $R_W$ (Cepheid Wesenheit color-luminosity parameter) and $M_H^{W}$ (Cepheid Wesenheit H-band absolute magnitude). Instead, we allow for variation of either of these parameters for each individual galaxy. We also consider the case where these parameters have two universal values: one for low galactic distances $D<D_c$ and one for high galactic distances $D>D_c$ where $D_c$ is a critical transition distance. We find hints for a $3sigma$ level mismatch between the low and high galactic distance parameter values. We then use AIC and BIC criteria to compare and rank the following types of models: Base models: Universal values for $R_W$ and $M_H^{W}$ (no parameter variation), I Individual fitted galactic $R_W$ with a universal fitted $M_H^{W}$, II Universal fixed $R_W$ with individual fitted galactic $M_H^{W}$, III Universal fitted $R_W$ with individual fitted galactic $M_H^{W}$, IV Two universal fitted $R_W$ (near and far) with one universal fitted $M_H^{W}$, V Universal fitted $R_W$ with two universal fitted $M_H^{W}$ (near and far), VI Two universal fitted $R_W$ with two universal fitted $M_H^{W}$ (near and far). We find that the AIC and BIC criteria consistently favor model IV instead of the commonly used Base model where no variation is allowed for the Cepheid empirical parameters. The best fit value of the SnIa absolute magnitude $M_B$ and of $H_0$ implied by the favored model IV is consistent with the inverse distance ladder calibration based on the CMB sound horizon $H_0=67.4pm 0.5,km,s^{-1},Mpc^{-1}$. Thus in the context of the favored model IV the Hubble crisis is not present. This model may imply the presence of a fundamental physics transition taking place at a time more recent than $100,Myrs$ ago.
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