Deriving a Persistence-Weighted Path Integral

This paper derives a persistence-weighted path integral from first principles of coherent comparison under composed admissible variation. Rather than postulating probabilistic, geometric, or dynamical structure, the analysis asks which bookkeeping devices are unavoidable if comparison is to remain well defined when admissible variations are recomposed, refined, or reordered. Starting from minimal operational requirements—discernibility, admissible variation, and persistence—the paper shows that path dependence, accumulated loss, stationarity, and quadratic stability classification are structurally forced. An accumulated loss functional emerges as the minimal additive bookkeeping required for coherence, and persistence weighting arises as a monotone representation of the induced ordering on admissible paths. Under additional stability requirements, the exponential weighting appears as a canonical representative compatible with arbitrary recomposition. Second-order variation around persistence-stationary paths yields a quadratic stability form, which can be repackaged as a curvature operator acting on admissible perturbations. Spectral structure then appears as a classification of persistence-relevant deviations, without presupposing geometry, dynamics, or operator postulates. Familiar formalisms—path integrals, entropy-weighted amplitudes, and spectral descriptions—arise only as restricted realizations once further admissibility, locality, and representational constraints are imposed. The entropy-weighted Feynman path integral is recovered explicitly as such a downstream specialization, with all additional assumptions labeled. The result is not a constructive physical model but a boundary-forming derivation: it fixes which structures are forced by coherence of comparison and which questions are deferred to later, more constrained analyses. The paper provides a foundational anchor for subsequent applications that treat persistence-weighted amplitudes and curvature operators as fixed inputs rather than postulates.