Projection-Limited Coherence: Exact Criteria for Effective Dynamics and a Cross-Domain Research Program

Abstract

Many sciences replace a detailed state by a smaller set of effective variables. This paper develops a substrate-neutral framework for asking what such a projection does and does not imply. For a measurable source space, dynamics, observation map, admissible domain, and declared margin, we separate five questions: whether deterministic dynamics descends to the effective space, whether the descended map is invertible, whether a projected stochastic process is Markov, whether source information can be recovered, and whether the realizable image obeys compatibility constraints. Each question has its own criterion. We prove the exact fiber criterion for deterministic descent and the corresponding lumpability criterion for stochastic kernels. We show by counterexample that a many-to-one projection, or even the absence of a continuous section, need not make the effective dynamics irreversible. Conversely, failure of the fiber criterion produces history dependence but does not by itself prove computational irreducibility. Information loss is expressed by conditional entropy and data processing; monotone physical entropy requires an additional dynamics and entropy-production law. Likewise, a capacity is a control margin until a constitutive equation, conservation law, or source term is supplied. The resulting framework is useful precisely because it is selective. It organizes current Modal Triplet Theory constructions and provides a template for models in physics, biology, cognition, artificial intelligence, and collective systems. The mathematical template transfers across domains; physical conservation laws, entropy laws, selection principles, and claims about consciousness or civilization do not. Such applications are stated as hypotheses with explicit validation obligations.