Independent Researcher

We prove that every unitary-invariant observable of a positive trace-one operator K is completely determined by the pairwise spectral differences μij=(λi−λj)2, where {λi} is the spectrum of K.We further establish that admissible renormalization group flows are unique up to monotone conjugacy, the induced spectral metric is unique up to rescaling, and all admissible formulations belong to a single equivalence class under monotone spectral isomorphism.No additional geometric, algebraic, or group-theoretic structures are required.This work provides a minimal, closed, and self-sufficient mathematical foundation intended to serve as a base for multiple follow‑up developments, each addressing different aspects of emergent physical structures.