Emergent Discreteness from Stability Boundaries in Reversible Dynamics

Discrete spectra are traditionally postulated as axiomatic in quantum theory. Here we demonstrate that discreteness can emerge generically as a consequence of dynamical stability in reversible, non-integrable systems under weak perturbations. We identify robust parameter plateaus (locking shelves) across quantum Floquet chains, classical area-preserving maps, and wave-transfer matrices. Crucially, every tested system follows the same boundary logic: shelves occur only within elliptic stability islands and collapse under irreversibility, strong noise, or hyperbolic instability. We do not derive quantum theory; we identify a cross-domain stability mechanism that generically selects discrete responses. This stability-selection principle provides a concrete bridge between classical mode-locking, Floquet many-body scars, discrete time crystals, and dynamical localization—suggesting that discreteness is a selected dynamical outcome rather than a fundamental axiom.