Abstract This work investigates the structural form of the infrared linear response of stable homogeneous phases of relational fundamental dynamics. It builds on the operational identification of emergent spacetime phases and on the determination of macroscopic dynamics as the linear response of stabilized collective degrees of freedom. The starting point is neither the construction of a concrete microscopic theory nor the introduction of an additional dynamical postulate. Rather, the central question is what form the infrared response operator must already assume once a stable, homogeneous, and collectively readable phase with a well-defined linear response is present. The analysis is carried out within a clearly delimited structural class. The assumptions are relationality of the fundamental degrees of freedom, relabeling invariance, the existence of a stable homogeneous phase, a self-adjoint positive linear response on the physical fluctuation space, a symmetric Markov structure in the collective infrared sector, and locality of the associated Dirichlet form in the sense of the absence of nonlocal jump contributions. These assumptions introduce neither a geometric spatial structure nor a continuous manifold, a fundamental time variable, or prescribed fields. They determine only the operator-theoretic framework within which infrared collective dynamics can be meaningfully analyzed. Under these conditions, the leading infrared generator in the scalar collective sector is not freely selectable. Within the regularity class under consideration, its structural form is fixed up to a positive global rescaling. The generator has a Laplace-like normal form in the precise sense of a self-adjoint positive Markov generator with a local Dirichlet form. Fractional, Lévy-type, or heavy-tail-dominated infrared generators therefore do not belong to the same structural class, but instead indicate the violation of at least one of the assumed regularity, locality, or Markov conditions. The result is model-independent in a limited but precise sense. It does not depend on the microscopic details of a specific relational dynamics, yet it does not classify all conceivable linear operators. What is determined is the space of possible infrared response operators of stable homogeneous phases under the stated structural conditions. Within this framework, the infrared linear response appears as the generator of a diffusion-like contraction semigroup. In the further course of the work, this structural identification is referred to as the Gültekin–Liebchen identity. The work thereby determines the mathematical status of the infrared operator constraint of emergent spacetime. Once a stable homogeneous phase in the collective infrared sector carries a local symmetric Markov structure, its leading macroscopic response is not arbitrary, but is structurally fixed by the underlying stability, symmetry, and locality conditions.
Jan Ercan Gültekin (Wed,) studied this question.