Sunyaev–Zel’dovich, Fast Radio Burst, and Metal-Absorption Tests of the Predicted CMB Temperature and the Haug–Tatum Redshift Relation in RH = ct Cosmology

<jats:p>We test the predicted CMB temperature of the Haug--Tatum $R_H=ct$ framework against Sunyaev--Zel'dovich (SZ), supernova, fast-radio-burst (FRB), and metal-absorption observations. The temperature relation, first proposed heuristically in 2015 and later derived from the Stefan--Boltzmann law \cite{TatumSeshavatharamLakshminarayana2015,HaugWojnow2024}, is $T_{\rm CMB}=T_{\rm P}(8\pi)^{-1}\sqrt{2\ell_{\rm P}/R_H}$, implying $T_{\rm CMB}\propto\sqrt{H}$. Combined with the Haug--Tatum redshift relation $1+z=\sqrt{H(z)/H_0}$, it yields the closure $T_{\rm CMB}(z)/T_0=\sqrt{H(z)/H_0}=1+z$. This reproduces the observed linear CMB temperature--redshift law and preserves the standard redshift invariance of the thermodynamic kSZ normalization and non-relativistic tSZ spectral scale. The same redshift function has separately been reported to fit all 2,287 PantheonPlusSH0ES Type Ia supernova redshifts with one fitted cosmological parameter, $H_0$. We further examine the low-redshift baryon census. For Haug--Tatum, the absorption coordinate reduces to $dX/dz=1$, and a 51-sightline O\,VII/O\,VIII WHIM catalogue receives a geometry/critical-density reweighting of about $1.25$, shifting its published baseline central values from $0.83$ to $1.04$ and from $0.79$ to $0.99$ of the standard benchmark baryon inventory. Independently, re-kerneling the localized-FRB dispersion relation gives a first model-internal estimate $\Omega_{b,0}^{\rm HT,FRB}\simeq0.0289^{+0.0119}_{-0.0141}$ (95\% confidence, first order), without inserting the standard CMB/BBN value of $\Omega_b$. These results are not yet full spectral or FRB likelihood re-fits, but together they provide four observationally distinct tests---SN Ia, SZ thermometry, FRB dispersion, and metal absorption---of a common temperature--Hubble--redshift structure and establish a concrete route toward a model-internal resolution of the low-redshift missing-baryon census.</jats:p>