You will find the mathematical solution published Jan. 25, 2023 in the Intl. J. Geom. Methods Mod. Phys (doi: 10.1142/S021988782350069X) (Appx. BF-BH).

Article Measurement Quantization

Not only is the CMB temperature resolved to five significant digits 2.7255K, but its age, quantity and present-day density are also resolved. The calculation compares with the Fixsen study - a survey of CMB measurements over a 10-year period - with no difference, (doi: 10.1088/0004-637X/707/2/916), that is digit-for-digit correspondence (Appx. BH).

Article The Temperature of the Cosmic Microwave Background

Notably, the total mass/energy of the CMB has never before been derived from first principles, resolved in the summary review paper - Measurement Quantization - as a combination of the fixed rate of universal mass accretion (Appx. BE) and the fixed radial rate of universal expansion (Appx. AZ-BA), the latter a metric description of expansion not to be confused with observations of expansion between galaxies.

The derivation is carried out using only Planck Units, more precisely fundamental units which are a geometry describing the relation between the system and internal frames of the universe. This is to say, a derivation from first principles using only a quantum description of observed phenomena may be found in the noted references. Your approach is a wonderful affirmation of the significance of this important quality of our universe.

For historical purposes it should be noted that these calculations were first published in the J. High Energy Phys, Gravit. Cosmol. on Mar. 31, 2020 (doi: 10.4236/jhepgc.2020.62015) (Sec. 3.13, Eq. 144).

Article Measurement Quantization Describes History of Universe—Quant...

Reviewing the cited MQ paper we find that the dependencies start with a determination of the size of the universe - the square root of three Planck lengths - this being the size of the universe at which the quantum epoch ends and the expansionary epoch begins. The quantum epoch is denoted by the discrete geometry, a period by which external referencing has no discrete mathematical solution. Without external referencing, there is no solution to an internal expansion at the speed of light. Specifically, the expansion velocity at the time when the quantum epoch ends is as resolved in Eq. BG.5 and as this is a geometry, we are limited in precision only by our measure of the radial rate of expansion, a function of the measure of theta which is usually derived from a physical measure of the fine structure constant (Appx. AA).

We do not do that here. Rather, we use a quantum measure of theta which in turn allows a solution to the fundamental measures. This calculation is limited to 6 digits of precision, a function of the measure of half of the Planck momentum - which we show equals the polarization angle of entangled photons at their degenerate frequency (Appx. S) as carried out by Shwartz and Harris in their 2011 paper (theta=3.26239 rad)

Article Polarization Entangled Photons at X-Ray Energies

Notably, MQ research typically reverses this calculation (Appx. AD) to provide 12 significant digits with 2 uncertain digits for theta and the fundamental measures (Appx. BM), and therein solutions with the same precision for nearly all of the physical constants. As that approach derives from the CMB temperature, we must resort to the Shwartz and Harris measures which are constrained to 6 digits.

Therein, having a calculation of the size of the universe at the end of the quantum epoch we can resolve the time elapsed associated with this period (Eq. BH.2). There is a time dilation between epochs, but fortunately this is known as a function of the quantum epoch formulation without the introduction of additional parameters (Eq. BH.3). The precision remains unchanged.

And where the fixed rate of mass accretion is known (Appx. BE) - also entirely a function of theta - then we combine the elapsed time with the rate of mass accretion to resolve the mass/energy associated with the quantum epoch. This represents the total mass/energy making up the CMB (Eq. BH.4). To be more precise, the calculations can be extended to account for the Recombination Epoch, a period where CMB formation is still occurring. That is carried out here (Eq. 142-144),

Article Measurement Quantization Describes History of Universe—Quant...

but the difference in CMB temperature due to this difference is reflected in digits that exceed the fifth digit of precision. So we may ignore the difference as physically beyond our current precision inputs.

For anyone looking to better understand theta, this is an angle with respect to certain Planck scale measurements and half of the Planck momentum in nearly all cases of measure relative to the internal frame. The term carries no units when defined against the system frame of the universe as the system frame has no external reference. In Appx. S, we show how to demonstrate mathematical equality of angle and momentum at the Planck bound relative to the internal frame. The calculation is most interesting, as it is an implicit outcome of the expression for the Planck Length, a formulation that has been around for nearly 100 years.

At this point we can then calculate the current density and temperature of the CMB, which is a function of the measured age of the universe (nTu mf) (Eq. BH.4). Present universal age is the limiting parameter which affects precision. We use a measure which has a precision of five digits. Naturally, there are several measures of universal age, some equal some with less precision, but what is important is that we directly identify the source of the measurement constraints, thus addressing the precision inquiry. With this, the remaining terms include the radiation constant (Eq. BH.6), which in turn produces the CMB temperature (Eq. BH.7). All remaining terms have more significant digits.

Precision is an important and understated quality of the calculation. Developing a method which can be identified as from 'first principles' implies also that there exists the least of inputs and fundamentally no other approach with a finer description and therein potentially greater precision.

Lastly, we draw attention to the unique qualities of this derivation. In short, using the elapsed time associated with the quantum epoch and the fixed rate of universal mass accretion, we can derive the accumulated mass/energy during this period, which in turn becomes the CMB. The approach is straight-forward, explaining where the CMB comes from, why its density is as it is. thus why its present-day temperature is as is, and what physical principles were involved that ended the quantum epoch and began the expansionary epoch. Moreover, the solution integrates the internal and system frames of the universe in such a way as to provide an consistent description of the universe across the quantum, macroscopic and cosmological domains.