Application of the γ-Equation to the Proton: A Parameter-Free Hadronic Mass Scale

[61 Application] 

This paper takes a relation that the 0-Sphere model first established for the electron and applies it, without any change, to the proton. Using only two well-measured properties of the proton — its mass and its anomalous magnetic moment — the construction yields an internal mass scale of about 336 MeV that falls squarely within the range long associated with the constituent quark mass in QCD, offered here as a new point of empirical contact for the framework at the scale of the strong interaction.

— Core Equations —

The bridge equation, taken from the electron treatment and applied here unchanged, in its mass-determining form:

\[ \gamma = 1 + |a| \]

Direct substitution of the empirical proton inputs \(m_p \approx 938.27\ \mathrm{MeV}/c^2\) and \(a_p \approx 1.793\) gives the mass-determining Lorentz factor and the kernel rest mass:

\[ \gamma_p = 1 + |a_p| = 2.793, \qquad m_0^{\,p} = \frac{m_p}{\gamma_p} \approx 335.94\ \mathrm{MeV}/c^2 \]

The velocity-determining form fixes the internal Zitterbewegung velocity, which for the proton is highly relativistic:

\[ \gamma_{v,p} = 1 + \frac{|a_p|}{\sqrt{2}} = 2.268, \qquad \frac{v_{\mathrm{ZB}}^{\,p}}{c} = \sqrt{1 - \frac{1}{\gamma_{v,p}^{2}}} \approx 0.898 \]

The entire derivation chain \(\{m_p, a_p\} \to \gamma_p \to m_0^{\,p}\) contains no free parameter at any stage.

— What This Paper Establishes —

• Parameter-free hadronic mass scale: a single framework, applied without modification, places the electron in a near-non-relativistic internal regime and the proton in a strongly relativistic one, with the empirical \(|a|\) the sole quantity driving the transition.
• Coincidence with the QCD constituent quark mass band: the inferred \(336\ \mathrm{MeV}/c^2\) lands within the model-dependent constituent quark mass band (about \(300\)–\(350\ \mathrm{MeV}/c^2\)) near its frequently cited benchmark.
• Structural disjointness of the two routes: the 0-Sphere route (inputs \(\{m_p, a_p\}\), bridge equation) and the QCD route (chiral condensate, dynamical mass generation, spectroscopic fits) share no input measurement, no intermediate machinery, and no interpretive mechanism, yet agree on the output value.
• Internal consistency check: the kernel Compton wavelength \(\bar\lambda_C^{\mathrm{kernel}} \approx 0.59\ \mathrm{fm}\) is comparable to the proton charge radius (about \(0.84\ \mathrm{fm}\)), where the naive value \(0.21\ \mathrm{fm}\) differs by a factor of four.
• A third empirical anchor: the result complements the existing anchors at the electron mass scale and at the gravitational-redshift scale.

— What Is and Is Not Claimed —

The paper does not claim to derive the constituent quark mass from first principles, nor that the QCD-side value is itself a first-principles QCD result; both routes produce phenomenological values from empirical data. The framework’s parameter-free character denotes the absence of fitting freedom, not numerical privilege among the model-dependent benchmarks. Because the QCD-side value was already known when the prediction was computed, the consistency is a postdiction rather than a fit — the same epistemological category as the framework’s earlier gravitational-redshift check. Extension to negative-\(a\) particles such as the neutron, the geometric origin of the constituent mass scale, and the connection to the SU(3) structure of QCD are explicitly deferred as open problems.

— Position in the 0-Sphere Model Series —

This paper continues Redefining Electron Spin and the Anomalous Magnetic Moment (10.5281/zenodo.17764997), the source of the bridge equation, extending it to the hadronic scale. The construction rests on the two-kernel Hamiltonian identity of the foundational electron model (10.5281/zenodo.16759284).

The complete series is available on Zenodo: Hanamura, Satoshi — 0-Sphere Model.