TOTU Progress: The Quantized Proton, CODATA, and a Golden Ratio CMB Correlation

By Mark Rohrbaugh · October 13, 2026

The Theory Of The Universe, or TOTU, investigates whether particles can be understood as stable wave structures in a superfluid vacuum. Its most concrete numerical result is a proton length obtained from a circulation condition with winding $Q=4$: it agrees with the CODATA 2022 proton charge radius to about 0.058%. A separate golden ratio scaling connects a characteristic CMB photon wavelength to the cosmological sound horizon with a difference of about 0.104%.

These results deserve careful examination. The proton radius comparison is a numerical correspondence based on an explicit hypothesis. Several other very precise agreements follow from established identities. The CMB scaling is an exploratory correlation. Keeping those distinctions clear makes the next physical tests more useful.

The quantized proton starts with a circulation condition. For a phase winding $Q$, an ideal circular flow obeys

$$ \oint \mathbf v\cdot d\boldsymbol\ell=\frac{2\pi Q\hbar}{m}. $$

If the speed is constant around a circle of radius $r$, this becomes

$$ mvr=Q\hbar. $$

The proposed proton case takes $Q=4$, $m=m_p$, and an internal wave speed $v=c$. This is a hypothesis about the internal structure; it does not mean the proton as a massive object travels at the speed of light. Using the measured proton mass and the SI values of $h$ and $c$ gives

$$ r_4=\frac{4\hbar}{m_pc}=0.8412356402\ \mathrm{fm}. $$

CODATA 2022 recommends the proton root-mean-square charge radius $r_p=0.84075\pm0.00064\ \mathrm{fm}$. The comparison is:

Quantity	Value
$Q=4$ circulation radius	$0.8412356402\ \mathrm{fm}$
CODATA 2022 charge radius	$0.84075\pm0.00064\ \mathrm{fm}$
Relative difference	$+0.05776\%$
Difference / quoted radius uncertainty	$0.759$

Equivalently, using the CODATA charge radius in the proposed relation gives

$$ Q_{\mathrm{obs}}=\frac{m_pc\,r_p}{\hbar}=3.99769\pm0.00304. $$

The value four lies within the quoted radius uncertainty. These are calculated comparisons using the NIST CODATA 2022 constants. The measured proton mass is an input, so this is not a calculation of the mass from first principles.

There is also a specific physical identification to establish: a circulation radius and a root-mean-square charge radius are different definitions. A complete model must calculate the charge distribution and its electromagnetic form factor to show why the measured charge radius should equal $r_4$. The numerical agreement motivates that calculation; it does not replace it.

The electron mass and proton-to-electron ratio provide a precise consistency check. The standard Rydberg identity is

$$ R_\infty=\frac{\alpha^2m_ec}{2h},\qquad m_e=\frac{2hR_\infty}{\alpha^2c}. $$

Using CODATA 2022 values $R_\infty=10973731.568157\ \mathrm{m}^{-1}$ and $\alpha=0.0072973525643$, the rearranged equation gives $m_e\approx9.109383713998\times10^{-31}\ \mathrm{kg}$. The recommended value is $9.1093837139\times10^{-31}\ \mathrm{kg}$. The resulting proton-to-electron ratio is $1836.152673402$, compared with the CODATA value $1836.152673426$.

For clarity, let $\mathcal M=m_p/m_e$ denote the proton-to-electron ratio; the electron-to-proton ratio is its reciprocal. Combining the Rydberg identity with the proposed proton radius gives

$$ \frac{\alpha^2}{\pi r_4R_\infty}=\mathcal M\approx1836.152673402. $$

The close agreement follows algebraically. For a general winding radius $r_Q=Q\hbar/(m_pc)$, the same expression is $4m_p/(Qm_e)$; at $Q=4$ it becomes $m_p/m_e$. The constants are also jointly adjusted and correlated by CODATA. This check confirms consistent algebra and units, rather than supplying a second independent measurement supporting a proton vortex. A simultaneous two-body hydrogen calculation remains a separate dynamical task.

The circular wavelength reproduces the proton mass-energy.If the circumference contains $Q$ wavelengths, then

$$ \lambda=\frac{2\pi r}{Q}. $$

Substituting $r=Q\hbar/(m_pc)$ gives

$$ \lambda=\frac{h}{m_pc}=1.3214098536\times10^{-15}\ \mathrm m, \qquad f=\frac{c}{\lambda}=\frac{m_pc^2}{h}=2.2687318184\times10^{23}\ \mathrm{Hz}. $$ $$ hf=m_pc^2=938.27208943\ \mathrm{MeV}. $$

This is the proton Compton wavelength and its mass-equivalent frequency. It is exactly consistent with the mass input. The winding number cancels, so this equality does not independently select $Q=4$. Whether the proton has a physical response mode at that frequency requires a calculation of its fluctuation spectrum.

The Planck connection can be written without arbitrary units. From the standard definitions $m_{\mathrm P}=\sqrt{\hbar c/G}$ and $\ell_{\mathrm P}=\sqrt{\hbar G/c^3}$,

$$ m_{\mathrm P}\ell_{\mathrm P}=\frac{\hbar}{c},\qquad \frac{m_pr_4}{m_{\mathrm P}\ell_{\mathrm P}}=4. $$

Consequently, the proposed radius can also be written $r_4=4m_{\mathrm P}\ell_{\mathrm P}/m_p$. This is an exact rewriting of the circulation relation. Because $G$ cancels between the Planck mass and length, it does not derive gravity.

There is an exact golden ratio identity involving four. With $\phi=(1+\sqrt5)/2$,

$$ \phi^3-\phi^{-3}=4. $$

The identity is mathematically exact. It provides a possible target for a transfer or response calculation, but does not establish that a proton selects fourfold winding. Similarly, a cascade satisfying $u_{n+2}=u_{n+1}+u_n$ has a constant positive ratio obeying $q^2=q+1$, whose solution is $q=\phi$. The physical challenge is to derive that recurrence and its stability from the interactions.

Our closest golden ratio CMB correspondence links two very different length scales. Fixsen's CMB temperature review gives $T=2.72548\pm0.00057\ \mathrm K$. For a blackbody spectrum expressed as brightness per unit frequency, the maximum occurs at $h\nu/(k_BT)=2.821439372\ldots$. It gives

$$ \nu_{\mathrm{pk}}=160.22895\ \mathrm{GHz},\qquad \lambda_{\nu,\mathrm{pk}}=\frac{c}{\nu_{\mathrm{pk}}}=1.87102559\ \mathrm{mm}. $$

This is the wavelength corresponding to the frequency-spectrum maximum. It is not the maximum of the spectrum per unit wavelength, which occurs near $1.063\ \mathrm{mm}$.

The exploratory scaling with integer exponent 131 gives

$$ \lambda_{\nu,\mathrm{pk}}\phi^{131}=144.58061\ \mathrm{Mpc}. $$

Planck 2018 results VI, Table 2 gives a comoving sound horizon at photon last scattering of $r_*=144.43\pm0.26\ \mathrm{Mpc}$ for the base-$\Lambda$CDM TT, TE, EE + lowE + lensing analysis. The difference is $0.15061\ \mathrm{Mpc}$, or $0.10428\%$. Propagating the quoted temperature uncertainty into the scaled length gives approximately $0.03024\ \mathrm{Mpc}$.

The closeness is worth recording, with its scope attached. A thermal photon wavelength and a collective acoustic horizon are different physical quantities. The exponent and spectral representation were selected during exploration, and the sound horizon is inferred within a cosmological model. This was not a prediction fixed before comparing the measurements. It supplies no detection probability for golden scaling.

The proton length also cancels if the comparison is written as the difference between exponents in $r_4\phi^n$. Thus this particular correspondence does not test the proton radius independently. A direct proton scaling illustrates the limitation: $r_4\phi^{190}=139.0653\ \mathrm{Mpc}$, which is about 3.714% below the same sound horizon.

A complete CMB explanation must account for temperature and polarization together. We tested one explicitly imposed primordial pattern periodic in $\ln k$, with period $\ln\phi$, through CAMB. One shared pair of modulation coefficients was fitted to 215 public Planck TT, TE and EE bins. The diagonal residual statistic changed from $\chi^2=201.185$ to $200.733$: an improvement of only $0.452$ for two extra coefficients.

That diagnostic supplied no preference for the particular golden pattern tested. It used a fixed Planck background, assumed coherent adiabatic initial conditions, and omitted the full covariance and official likelihood. It neither establishes a proton-to-CMB mechanism nor excludes every nonlinear superfluid completion. The chosen minimal GP–KG condensate also showed no distinguished golden ratio in its local linear mode spectrum. A constant vacuum-energy term alone did not select it.

An exact oscillator calculation quantified the coherence requirement: Gaussian temporal-phase scatter $\sigma_\theta$ suppresses acoustic contrast by $D=\exp(-2\sigma_\theta^2)$. A spread of one radian leaves only 13.5% of the ideal contrast. This is a conditional calculation, not a measured Planck bound. A formation mechanism must generate the appropriate time correlations to preserve the observed pattern.

A recent Geometric Harmonic Physics announcement by David Sereda and Jenefer Om quotes $0.61845$ from a normalized Planck mass-energy divided by $\alpha$. It is 0.0673% above $1/\phi$. The physical significance depends on the independently defined reference energy and the normalization, including the image's “Harmonic 144” prescription. Until those are supplied, that claim is not an additional unit-independent constraint on TOTU.

Related exploratory comparisons include Nassim Haramein's 2013 holographic mass paper, Val Baker, Haramein and Alirol's 2019 electron paper, and the cascade and phase-conjugation ideas discussed in Dan Winter's work. Such comparisons can suggest mathematical hypotheses. Sharing an algebraic relation does not provide an independent experimental confirmation of it.

The next decisive calculation is a common action that yields a stable proton solution, its charge form factor and response modes, and the cosmological perturbations generated by its formation. Those predictions must then face independent data. The $Q=4$ radius correspondence is the strongest numerical motivation developed here; the identities provide consistency checks, and the CMB scaling remains a defined correlation awaiting a physical explanation.

Measurement references: NIST CODATA 2022 constants; Fixsen 2009 CMB temperature; Planck 2018 VI; Planck PR3 spectra. Calculated comparisons and conditional model results are from our TOTU analysis. The quoted constants are inputs where stated; they are not claimed as independently derived by TOTU.