Local Infrared Rigidity and Finite Stratification for Herglotz-Nevanlinna Functions

We establish a structural rigidity theorem for the infrared (IR) behavior of Herglotz-Nevanlinna functions under a strict locality hypothesis.Working within the category of Herglotz functions with dominant singularity at z → 0, and restricting morphisms to local IR-admissible transformations—those preserving the Herglotz property, acting analytically near the origin, and introducing no stronger singularities—we address a fundamental question: can the infrared regime support an internally generated infinite hierarchy?We prove that it cannot.First, we show a pure point collapse: any purely point spectral measure accumulating only at the essential boundary yields a meromorphic resolvent, and the fine topological complexity of the support (including arbitrary Cantor–Bendixson height) is invisible at the infrared level. The entire pure point sector reduces to a single infrared class.Second, in the continuous algebraic regime, the infrared structure is completely governed by the leading exponent γ inG(z) ~ K(-iz)^γ.The exponent determines disjoint infrared leaves [Branch_γ], and we prove a universality theorem: every local continuous infrared invariant factors uniquely through the functorγ : C_IR^local → (-1, ∞).Thus γ is the sole continuous structural parameter compatible with locality.Finally, combining pure point, continuous, and hybrid configurations, we establish a finite infrared stratification theorem: the local infrared order has finite height, and no long strictly realizable chains can be generated by local admissible morphisms. In particular, no internally generated infinite infrared tower exists under locality.We conclude with a principle of local infrared rigidity: under Herglotz structure, locality, and algebraic dominant singularities, the infrared classification is exhaustive and structurally finite. Any genuinely richer hierarchy necessarily requires leaving the local algebraic category.