Truncation Levels and Spacetime B: The Necessary Termination of Hierarchical Chains under the Two Weakest Constraints
Year: 2026
Type: preprint
Abstract: This paper studies the termination behavior of hierarchical chains under thetwo weakest constraints of information conservation and computability. The twoconstraints presuppose no physical laws: information conservation requires that noevolution or composite map send two distinguishable states into indistinguishableoutputs; computability requires that the dynamics of the system can be approximated by a finite rule set over a finite symbol table in finitely many steps. Withinthis framework, truncation is defined as the first finite event on a hierarchical chainat which “the next level is not a lawful extension.” The argument proceeds inreductio form: assuming that the hierarchical chain admits lawful extension everywhere, three mutually independent channels respectively yield descriptive collapse,information runaway, and structural dissolution. The descriptive channel relies onthe growth bridge and the global single finite-description reading; under the condition that each level introduces at least one independent degree of freedom, thepigeonhole principle forces any description system of finite capacity to lose levelby-level tracking after finitely many levels. The information channel relies on thecapacity bridge and the condition that carrier capacity is sublinearly bounded withscale; in the face of a finite upper bound, the strictly monotone growth of capacitydemand is either nowhere to be placed or forces the loading map to be non-injective,thereby violating the algebraic reading of information conservation. The structuralchannel relies on the uniformity bridge and the framework convention that theminimal structure of causal closure takes alternativity; a coupling spectrum that iseverywhere equivalent simultaneously deprives scale separation, stable informationboundaries, and well-defined evolution operators of their footing. The premisesof the three channels do not overlap at the registration granularity; whenever itsexplicit conditions hold, any one channel already suffices to be incompatible withthe no-truncation hypothesis. The main theorem is therefore stated in conditionalform: when at least one of the growth-bridge premise, the capacity-bridge premise,and the uniformity-bridge premise holds, and the auxiliary conditions of the respective channel (global single-description reading, carrier-capacity upper bound,alternativity convention) are satisfied, the truncation set of the hierarchical chain isnonempty. Three corollaries respectively give the terminological convention for successful systems, the contrapositive reading that scale separation takes truncation asa necessary condition, and a registration-based classification of failure modes. Fivemathematical facts—Hurwitz termination, Besicovitch sets, G¨odel incompleteness,Lawvere closure, and percolation criticality—are registered as post hoc corroboration and do not enter the proof burden. The argument constructs no physicalpredictions and introduces no tunable parameters; its conditional depth is explicitlymarked in the bridge lemmas and the main theorem, and uncompressed parts areregistered as open problems.
Authors: zhou, changzheng, zhou, ziqing
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225316901