Conditional Identifiability of Multiscale Mechanistic Roles in Finite-Observation Dynamical Systems: From Information Budgets and Canonical Mechanism Quotients to Stable Role Structures and Structured Information Bottlenecks

Problem. Finite observation can provide only partial information about the true state, but can information deficit by itself determine a hierarchy of roles in a complex system? We show that observational information deficit alone is neither sufficient nor necessary for a hierarchical structure. An identifiable structure must instead be constrained jointly by dynamical scales, the conditional effect kernel, and representational minimality. Accordingly, we study the conditional identifiability of multiscale mechanistic roles in finite discrete dynamical observation systems, treating the familiar rule–execution dichotomy as a two-scale special case. Method. We first establish a general information-cascade budget theorem, expressing the total state-information deficit from observation, factorization, and final representation as a sum of stagewise conditional mutual informations; the existing three-level deficit decomposition is thereby obtained as a special case. Next, we define an information persistence scale through lagged mutual information and introduce a scale-separation margin, explicitly distinguishing the dynamical attributes of “slow/fast” from the mechanistic attributes of “regulatory/executive.” We then use the conditional effect kernel K_f(y) = P(Y = y | F = f) to define a block-local mechanism equivalence relation on the support, construct the canonical coarsest mechanism quotient, and prove its essential uniqueness within blocks under the coarsening partial order. We further define mechanism activity for each scale block and the minimal number of active roles k_act, thereby avoiding the circular identification of “number of scale clusters” with “number of roles.” Finally, we introduce mechanism- and scale-separation margins, derive stable-identifiability conditions under finite estimation error, and generalize the hierarchical information bottleneck to a general structurally constrained information bottleneck, defining the cost gap of structural constraints relative to the unconstrained information bottleneck. Main results. We obtain eight core results: (i) every finite Markov representation cascade satisfies an additive information-budget identity; (ii) systems with the same observational information deficit can be constructed with different numbers of roles and different scale structures, so information deficit cannot select the number of levels; (iii) the support-relative block mechanism quotient is the canonical coarsest object among mechanism-preserving block-local representations and is essentially unique on positive-probability values; (iv) an active mechanism block cannot be deleted under the block-local constraint, so the minimum number of roles equals the number of active mechanism blocks; (v) a conditional regulatory–execution dichotomy arises when there are two scales, both blocks are active, and a nonzero second-order interaction is present; (vi) mechanism equivalence classes are stable when kernel estimation error is sufficiently smaller than the mechanism-separation margin, while the corresponding scale-error condition stabilizes the slow/fast partition; (vii) a purely synergistic XOR mechanism shows that a zero marginal main effect does not imply that the mechanism disappears, but should instead be treated as a higher-order interaction; (viii) for any class of structurally constrained encoders, the optimum of the structured information bottleneck is no lower than that of the unconstrained information bottleneck, and the resulting gap rigorously quantifies the information-theoretic cost of structural preference. Positioning. We do not claim “multiscale structure,” “hierarchical information bottlenecks,” or “minimal predictive representations” themselves as sources of novelty; these directions have substantial prior foundations. The core objects of this paper are the block-local conditional mechanism quotient, active-role minimality, and their stable-identifiability interface; on this basis, the rule–execution dichotomy is reformulated as a special structure that is computable, provable, falsifiable, and extensible to k levels. Research Paradigm Statement: The core methodology, research direction, and final decisions were independently determined by the author. Multiple AI tools assisted with code implementation, data presentation, and text drafting. The author bears full academic responsibility for all research content.