We point out that a modified temperature–redshift relation (

*T*-

*z* relation) of the cosmic microwave background (CMB) cannot be deduced by any observational method that appeals to an a priori thermalisation to the CMB temperature

*T* of the excited states in a probe environment of independently determined redshift

*z*. For example, this applies to quasar-light absorption by a damped Lyman-alpha system due to atomic as well as ionic fine-splitting transitions or molecular rotational bands. Similarly, the thermal Sunyaev-Zel’dovich (thSZ) effect cannot be used to extract the CMB’s

*T*-

*z* relation. This is because the relative line strengths between ground and excited states in the former and the CMB spectral distortion in the latter case both depend, apart from environment-specific normalisations, solely on the dimensionless spectral variable

$x=\frac{h\nu}{{k}_{B}T}$. Since the literature on extractions of the CMB’s

*T*-

*z* relation always assumes (i)

$\nu \left(z\right)=(1+z)\nu (z=0)$, where

$\nu (z=0)$ is the observed frequency in the heliocentric rest frame, the finding (ii)

$T\left(z\right)=(1+z)T(z=0)$ just confirms the expected blackbody nature of the interacting CMB at

$z>0$. In contrast to the emission of isolated, directed radiation, whose frequency–redshift relation (

$\nu $-

*z* relation) is subject to (i), a non-conventional

$\nu $-

*z* relation

$\nu \left(z\right)=f\left(z\right)\nu (z=0)$ of pure, isotropic blackbody radiation, subject to adiabatically slow cosmic expansion, necessarily has to follow that of the

*T*-

*z* relation

$T\left(z\right)=f\left(z\right)T(z=0)$ and vice versa. In general, the function

$f\left(z\right)$ is determined by the energy conservation of the CMB fluid in a Friedmann–Lemaitre–Robertson–Walker universe. If the pure CMB is subject to an SU(2) rather than a U(1) gauge principle, then

$f\left(z\right)={\left(1/4\right)}^{1/3}(1+z)$ for

$z\gg 1$, and

$f\left(z\right)$ is non-linear for

$z\sim 1$.

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