Eigenphysics from Persistence - Spectral Structure as a Consequence of Discernibility

A structural derivation of spectral structure and quantization is obtained from minimal requirements on discernibility and persistence under variation. Rather than postulating quantum degrees of freedom, operators, or Hilbert space, the analysis identifies which assumptions are required for eigenstructure to arise as a consequence of stability under irreversible loss. Building on prior results establishing accumulated loss, path dependence, and curvature as necessary consequences of persistence, the paper introduces a clear admissibility ledger. Under explicit conditions—ordered variation, local loss density, and a quadratic stability regime—the second variation of accumulated loss defines a curvature operator that classifies admissible deviations by their susceptibility to persistence loss. Eigenmodes emerge as stability classes selected by persistence filtering, not as fundamental postulates. Discrete spectra appear only when admissibility conditions enforce effective compactness. Additional restrictions concerning closed continuations and phase-valued transport lead to a closure condition under repeated traversal, reproducing Bohr-type quantization as a conditional consequence rather than an axiom. Global spectral weights and regularization are introduced solely as technical bookkeeping devices when persistence comparison is extended to families of modes. The framework deliberately separates structural derivation from physical interpretation. Hilbert space structure, probabilistic interpretation, and standard quantum dynamics arise only under further assumptions and are not required for spectral stability itself. Standard quantum mechanics is recovered as a specialization of the general framework under additional linearity, normalization, and interpretive assumptions. The paper isolates the minimal structural content underlying eigenstructure and quantization, making explicit which assumptions are responsible for each step. It stops short of system-specific applications, but the entropy-weighted variational object derived here serves as the foundation for predictive extensions developed in companion works, including corrections to tunneling behavior, interference suppression, and collapse-like phenomena emerging from resolution instability.