When Does Less Information Pay? An Exact Finite Atlas of Representation, Payoff, and Cost

Abstract A representation that retains one bit about a four-state world can solve a task perfectly, while another that retains 1.5 bits on average reaches only 75% accuracy. The difference lies in which distinctions each representation preserves. We examine this relationship in a uniform four-state world by enumerating all admissible representations and binary action rules. Constructions are compared both by replacing one component while holding the others fixed and by choosing the best action rule separately for each module. We consider three tasks, their mixtures, and four separate hypothetical cost tariffs. In some comparisons, the representation retaining less world information has the higher payoff; in others, the more informative representation does. There are also pairs with different information amounts but equal payoff—that is, equal task accuracy. If a representation preserves all existing distinctions and adds new ones, its best zero-cost payoff cannot decrease. Lower-information advantages therefore depend on which distinctions are retained. Equal counts of opposing advantages under representation replacement follow from coding symmetry. In broader comparisons, the shares depend on how repeated realizations are counted. A different task set or resource price can change which member of a fixed pair performs better. These results are exact, complete censuses of the specified finite sets of constructions and comparisons. The atlas provides a reproducible worked model for examining how action constraints, counting rules, and cost assumptions affect comparisons. It illustrates established distinctions between information quantity and decision value rather than proposing a new general law of their relation. I(W;R) measures retained world information, and payoff measures task accuracy. Contents and scope This record brings together the English preprint, a Russian translation with localized figures, the English supplement (Tables S1–S17), and an article-scoped code-and-data archive. The archive includes editable manuscript sources, exact model configurations, unchanged computational results, independent checker code, figure source tables, and verification instructions. The study enumerates a declared finite mathematical model; the data are computational results, not observations of organisms. This is a preprint, not a peer-reviewed journal article. Licenses The article, translation, original figures, and textual supplement are available under CC BY 4.0. Computational code and data retain the MIT license. LICENSES.txt and the archive manifest specify the boundaries; existing MIT grants are not revoked. AI assistance and responsibility OpenAI Codex assisted with source organization, code development and testing, figure preparation, manuscript drafting, editing, translation, and internal consistency checks. The authors are responsible for the work; the first author reviewed the scientific decisions and interpretation. AI-assisted internal checks are not external human peer review. No AI system is an author. No external funding was received. The authors declare no competing interests.