Existence-Determining Configuration Theory

Abstract

Description This paper introduces Existence-Determining Configuration Theory, a framework for representing and manipulating lattice configurations through sequences of directional arrow operations. Rather than relying on matrices or coordinate transformations to describe a configuration, the framework treats the configuration itself—its nodes, edges, and spatial relationships—as the primary mathematical object. The paper develops the theory in several stages. First, it formalizes eight directional operations on the two-dimensional integer lattice, distinguishing reversible cursor movements from edge-drawing operations that permanently add edges to a configuration. It then introduces an equivalence relation under which different sequences of operations are regarded as equivalent whenever they produce the same configuration. Next, the paper defines a folding operation, called a telescope, that combines a collection of nodes into a single folded node while preserving the total node count, or mass. Folded nodes generate shadows in neighboring lattice cells. These shadows are treated as a separate, non-interfering layer, allowing information about folded structures to be represented and accessed from the surrounding configuration. A central construction is the negative luminance, a quantity obtained by sorting overlapping shadow masses in decreasing order and combining them with successively halved weights. This quantity provides a mechanism for converting information about a configuration into new arrow operations. The main mathematical result is an integrality theorem that gives necessary and sufficient conditions for the negative luminance to be an integer. The conditions are expressed through a congruence modulo a power of two and an equivalent recursive characterization. Several corollaries describe the arithmetic constraints imposed by the number and masses of overlapping shadows. The framework establishes a preliminary mathematical foundation for configuration-based geometric representation, folding, information readout, and the generation of configurations through operations. The paper concludes with open problems concerning the completeness of the equivalence rules, automatic selection of folding ports, motion of folded nodes, and extensions to three-dimensional configurations and generalized weighting ratios.