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Universal Modular Dynamics: The Law of Renormalization
We present a formulation of renormalization within the framework of Universal Modular
Dynamics (UMD), in which the density operator ρ is taken as the fundamental object encoding
physical structure. In this approach, geometry, locality, and dynamics are not assumed a priori, but emerge from properties of ρ and its associated modular generator K = −log ρ. We demonstrate that renormalization can be consistently interpreted as a flow in the space of quantum states, parameterized by an internal ordering parameter λ. Within this formulation, the critical scale of the renormalization group (RG) flow is not determined by
a single spectral characteristic, such as a gap, but by the full statistical structure of the
modular spectrum. Using explicit numerical constructions, we show that the RG critical point λ∗(ρ) is a
stable functional of the distribution of modular energies, leading to a law of the form
λ∗(ρ) = F Spec(−log ρ) ,
where F depends on statistical properties of the spectrum, including its mean, variance, and
quantile structure. This result establishes a distributional law of renormalization, in which critical behavior is governed by global spectral features rather than isolated eigenvalues. As a consequence, renormalization becomes intrinsically state-dependent, and the notion of scale is replaced by a spectral-statistical structure defined directly at the level of the density operator. The proposed framework provides a unified perspective on renormalization, quantum
structure, and emergent geometry, and suggests a shift from scale-based to distributionbased
descriptions of physical laws.
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