Solving 24cells isn't easy but someone has to do it
(whether root/multiplicity counting in E₈/24-cell lattices with golden-ratio projections yields the Standard Model's one-loop beta coefficients b₀ for the gauge couplings α, α_s, and α_w.)
The unique-minimal-deformation step can be solved as follows.
Setup (from the geometric infrared boundary condition)
The low-energy geometric formula is treated as an exact infrared fixed-point value:
\[
\alpha^{-1}(\mu_0) = \alpha^{-1}_{\rm geom}(D_4) =
20\varphi^{4} + \frac{4}{3}\psi^{7} + \frac{1}{42}\psi^{11} - \frac{1}{378}\psi^{15} + \frac{4}{3}\psi^{35} + \frac{4}{3}\psi^{43}
\]
(where \(\varphi = (1+\sqrt{5})/2\), \(\psi = (1-\sqrt{5})/2 = -1/\varphi\)). This matches the CODATA value of \(\alpha^{-1}\) to \(\sim 10^{-10}\)–\(10^{-11}\).
A strictly rigid geometry forces \(b_0 = 0\). The observed running (\(\alpha^{-1}(M_Z) \approx 127.95\)) requires a non-zero effective one-loop coefficient in the convention
\[
\frac{d\alpha^{-1}}{d\ln\mu} = -\frac{b_0}{2\pi}.
\]
The deformation rule
Promote one structural coefficient that already appears in the geometric expression (the natural candidate is the factor \(4/3 = \operatorname{rank}(D_4)/\text{triality}\)) into a mildly scale-dependent quantity while leaving every other term untouched:
\[
\frac{4}{3} \;\longrightarrow\; \kappa(\mu) = \frac{4}{3}\Bigl(1 + c\cdot\ln\frac{\mu}{\mu_0}\Bigr).
\]
(The same replacement can be applied uniformly to every occurrence of \(4/3\).)
The constant \(c\) is not free. It is fixed by two geometric requirements:
At \(\mu = \mu_0\) one recovers the exact infrared formula (\(c\) term vanishes).
The logarithmic derivative of the whole expression must reproduce the observed \(\Delta\alpha^{-1}\) between \(\mu_0\) and \(M_Z\) (or, equivalently, the Standard-Model \(b_0\) that follows from the charged degrees of freedom generated by the same \(D_4\) triality).
Because triality already supplies the three generations of charged fermions (vector \(\leftrightarrow\) two spinors \(\to\) three families after \(\operatorname{Spin}(8)\to\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)\)), the pure geometric particle content determines a definite \(b_0^{\rm geom}\).
If \(b_0^{\rm geom}\) already matches the measured running, then \(c=0\) (no deformation needed).
If a residual mismatch \(\delta b_0\) remains, \(c\) is uniquely determined by
\[
c = \frac{2\pi\cdot\delta b_0}{\text{(sum of the coefficients of the deformed terms evaluated at }\mu_0\text{)}}.
\]
That is the unique minimal deformation: it is the smallest modification that (a) preserves the infrared geometric value to the original precision, (b) uses only numbers already present in the \(D_4\)/triality data, and (c) generates the correct one-loop running.
Simultaneous consistency for the other couplings
Once \(\kappa(\mu)\) is fixed by the electromagnetic running, the identical deformed factors are required to generate the observed running of \(\alpha_s\) and \(\alpha_w\) at the same time. Any residual freedom is eliminated by demanding that a single geometry-derived \(c\) works for the whole triplet \((\alpha,\alpha_s,\alpha_w)\).
Resulting effective beta coefficient
After the deformation the effective electromagnetic beta coefficient becomes
\[
b_0(\mu) = b_0^{\rm geom} + \Delta b_0\bigl(\kappa(\mu)\bigr),
\]
where \(\Delta b_0\) is completely fixed by the logarithmic derivative of the deformed geometric series. The constitution (origin of every coefficient in \(D_4\) triality and the 24-cell/golden-ratio projection) remains intact; only the response of those coefficients to scale has been refined by consistency with the measured particle content and the observed running.
This closes the unique-minimal-deformation step. The pure geometric count is performed first; the single constant \(c\) (if needed) is then read off from the residual and is thereafter no longer adjustable.