Truncation Levels and Spacetime A: Information Conservation, Computability, and Necessary Conditions for the Generation of Hierarchical Structure

This paper investigates the truncation problem of hierarchical chains under the twoconstraints of “information conservation” and “computability.” A hierarchical chain isformalized as an indexed system of embedded objects; a legal extension requires onlythat the previous level remains traceable and that the rules can be executed finitely,without promising that each level adds irremovable information. Under this weak definition, information conservation and program-level computability alone cannot implythat the set of legal truncations is nonempty: a consistent computable prefix chainsatisfies both constraints and legal extension, but its set of legal truncations is empty.This yields a negative boundary. On this basis, the paper introduces strong generativeextension and fixed carrier as explicit additional structures and proves a conditionalcounting theorem: if every level of a hierarchical chain is injectively packed into a finitecarrier of capacity N, and every nonterminal extension step is strongly generative, thenthe number of strong generations does not exceed N − n0, and the chain saturates ator before this upper bound. The theorem uses only strong generation, injectivity, andfinite counting; it does not rely on physical entropy bounds and is not derived frommathematical analogy samples. The paper further distinguishes program-level computability from finite-state tracking, pointing out that the algorithmic informationof a consistent computable chain has a logarithmic-order upper bound and thereforecannot sustain linear algorithmic-information growth indefinitely. Scale separation istreated as an operational concept and bears conditional structural interpretation onlyin an explicit scale-coarse-graining model. The conclusions are graded by strictnessinto five levels: negative propositions, conditional existence theorems, growth-rate obstructions, conditional structural interpretations, and open problems; the premises andapplicability boundaries of each conclusion are explicitly registered.