Emergence of the Proton Mass from a Closed Quartic Variational Functional via Domain-Induced Scale Breaking

This work demonstrates that the proton mass emerges as a necessary consequence of a closed quartic variational functional, without the introduction of external parameters, dimensional constants, fitting procedures, or phenomenological inputs. Starting from the stationary condition of the functional, the admissible configuration space is constructed through explicit constraints of finiteness, normalization, internal closure, and stability. The second variation defines a Hessian operator whose spectrum exhibits a uniquely determined structure with eigenvalue ratio 1, L, L, where L = 0. 25 emerges as an internal invariant fixed by the quartic interaction and Z3 cyclic closure. A modulated instability of the stationary configuration generates a finite characteristic wave number k*, which defines an intrinsic length scale. The admissible domain enforces discretization and compatibility conditions that break scale invariance internally, leading to the identification E* = k* and to the selection of a unique numerical value for the fundamental scale. The proton is obtained as the minimal stable Z3 configuration, with energy Eₚ = E* (1 + 2L). Substitution of the internally selected scale yields Eₚ ≈ 938 MeV. All steps are derived explicitly from the variational structure. No external calibration, unit fixing, or empirical input is used. The result shows that particle masses are not free parameters but consequences of a closed, self-consistent selection process acting on the space of admissible configurations.