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: \ = 1 + |a| \ Direct substitution of the empirical proton inputs \ (mₚ 938. 27\ MeV/c²\) and \ (aₚ 1. 793\) gives the mass-determining Lorentz factor and the kernel rest mass: \ ₚ = 1 + |aₚ| = 2. 793, m₀^\, p = mₚₚ 335. 94\ MeV/c² \ The velocity-determining form fixes the internal Zitterbewegung velocity, which for the proton is highly relativistic: \ ₕ, ₏ = 1 + |aₚ|2 = 2. 268, vₙ₁^\, ₏c = 1 - 1{ₕ, ₏^{2}} 0. 898 \ The entire derivation chain \ (\mₚ, aₚ\ ₚ m₀^\, 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\ MeV/c²\) lands within the model-dependent constituent quark mass band (about \ (300\) –\ (350\ MeV/c²\) ) near its frequently cited benchmark. Structural disjointness of the two routes: the 0-Sphere route (inputs \ (\mₚ, aₚ\\), 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 \ (C^kernel 0. 59\ fm\) is comparable to the proton charge radius (about \ (0. 84\ fm\) ), where the naive value \ (0. 21\ 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.