The Planck Constant as an Unavoidable Spectral Unit of a Closed Quartic Variational Functional

This work demonstrates that the Planck constant can arise as a structural consequence of a closed quartic variational functional. The framework is defined entirely by a global integral functional, its stationary Euler–Lagrange equation, the corresponding dynamical equation obtained from the same functional derivative, and a global decision operator selecting the physically admissible configuration. No external constants, scales, couplings, boundary conditions, or phenomenological inputs are introduced at any stage. Within this closed structure, the static second variation defines a Hessian operator whose spectrum determines a discrete set of energy eigenvalues. Independently, the linearized dynamical equation generated by the same functional produces a spectrum of eigenfrequencies. Because both spectra originate from the same quartic variational structure and are evaluated on the unique configuration selected by the global decision functional, their ratio defines a structural invariant of the framework. The smallest non-zero ratio between the lowest static eigenvalue and the corresponding dynamical eigenfrequency establishes a unique action scale that cannot be tuned, adjusted, or modified by any parameter. This quantity emerges solely from the internal variational structure and therefore constitutes an unavoidable numerical invariant of the theory. The analysis shows that the Planck constant is not a fundamental input but the numerical expression of this invariant action scale generated by the closed quartic variational functional. Quantization and the existence of an elementary unit of action therefore follow from structural consistency rather than from postulates. The result indicates that the Planck constant can be interpreted as a necessary consequence of variational closure and spectral compatibility within an internally self-consistent functional framework.