The calculation produces a mass–radius factor of four in an explicit two-photon/core model. Physical circulation Q=4 remains unproved. I also calculated the photon correlations and tested Wenki’s geometry numerically.
The model assumes a perfectly reflecting core with positive line tension T. Deriving that confinement from the vacuum remains an open requirement.
The useful new result is an energy-balance mechanism for four. Put one photon in each fundamental traveling direction around a ring. Each has wavelength $2\pi R$, giving
$$E(R)=\underbrace{\frac{2\hbar c}{R}}_{\text{two photons}}+\underbrace{2\pi TR}_{\text{confining core}}.$$
Minimizing over radius gives
$$\frac{dE}{dR}=0\quad\Longrightarrow\quad E_{\rm photons}=E_{\rm core}.$$
Consequently,
$$\boxed{Mc^2=\frac{4\hbar c}{R}},\qquad\boxed{R=\frac{4\hbar}{Mc}}.$$
Here, four arises from two fundamental photon quanta plus the equally large core energy.
Calibrating the total mass to the CODATA proton mass gives these central values.
|
Quantity |
Calculated value |
|
Geometric ring radius |
$0.8412356402040203\ \mathrm{fm}$ |
|
Energy per photon |
$234.5680223575\ \mathrm{MeV}$ |
|
Core energy |
$469.136044715\ \mathrm{MeV}$ |
|
Required line tension |
$88.7567251441\ \mathrm{MeV/fm}$ |
This calibration fixes the tension. Identifying the geometric radius with the proton’s measured electric charge radius still requires a charge distribution and form factor.
The conserved currents distinguish this result from circulation. For equal counterpropagating occupations,
$$I_N=\frac{c(N_+-N_-)}{2\pi R}=0,\qquad J_z^{\rm orbital}=\hbar(N_+-N_-)=0.$$
The photons have opposing mode windings +1,-1. No condensate phase winding of four emerges.
A separate quantum calculation does produce a fourfold pattern. The two-photon superposition
$$|\Psi\rangle=\frac{|2,0\rangle+|0,2\rangle}{\sqrt2}$$
has coincident detection probability
$$\boxed{G^{(2)}(\vartheta)\propto1+\cos(4\vartheta)}.$$
That supplies a concrete explanation for fourfold correlations without establishing fourfold circulation.
The geometry calculation does not select Wenki’s golden separation. I varied both radius and separation in a two-lobe electromagnetic cavity, checking both cavity polarizations.
|
Calculated sector |
Result |
|
TM polarization; both photons allowed in the lowest mode |
Circular cavity: d/R=0 |
|
TM; one photon forced into each of two distinct modes |
Restricted minimum near d/R=0.380 |
|
TE; both photons allowed in the lowest mode |
Energy decreases as the neck narrows |
The image requires $d/R=\varphi\approx1.618$. In the distinct-mode TM calculation, that geometry costs approximately 9.3% more energy than the optimized shape. Mesh refinement confirmed the separation result; the near-contact TE calculation supports an energetic instability direction, rather than a lifetime prediction.
Retaining a finite vacuum contribution also matters. Writing $E_{\rm vac}=\zeta\hbar c/R$ changes the ring result to
$$\boxed{\frac{McR}{\hbar}=4+2\zeta}.$$
Thus, the exact coefficient depends on the vacuum and core dynamics. The neutral model also lacks proton charge, spin 1/2, and baryon number one.
The executed calculation therefore gives TOTU a conditional mechanism for its mass–radius correspondence, while identifying the missing work: a vacuum-derived core with the required current, topology, and stability.
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