Stability-Induced Discreteness from the Second Variation of Action

A common route to discreteness in physics is to postulate a Hilbert-space operator and then solve
its eigenvalue problem. Here a different, stability-based route is formulated. The starting point
is a variational stability principle: a physically realized stationary state is required not only to
satisfy the stationarity condition δS = 0, but also to be stable with respect to the second variation,
δ2S ≥ 0, understood as a positive stability form. Under standard assumptions on the second
variation–symmetry, closedness, lower semiboundedness, and coercivity after a shift–this form defines
a self-adjoint stability operator. If the physical boundary conditions make the relevant embedding
compact, the stability operator has compact resolvent and therefore a discrete spectrum. In this
sense, discreteness is not introduced as an independent quantum postulate; it arises as a spectral
consequence of the second variation together with stability and boundary conditions. The result is
stated as a theorem and proved using the representation theorem for closed semibounded forms and
compactness of the Sobolev embedding. The physical meaning is clarified by distinguishing three
levels of discreteness: discrete stability modes, discreteness of action variables, and quantum-type
energy quantization. Periodic classical systems, including bounded orbital motion, naturally give
discrete stability modes, whereas quantization of energies requires an additional minimal action scale
and a single-valued phase condition.