Foundations of Rational Distance Geometry: The DIC Method and a Unified Framework over Q, Q_p, and F_q

This monograph develops a systematic and rigorous methodology for studying rational-distance problems—questions of whether a point exists at rational distances from all vertices of a given finite configuration in Euclidean space. Starting from the classical unit‑square problem, we introduce the Decomposition–Intersection–Control (DIC) method, which reduces any such problem to a finite, verifiable algebraic certificate. The method consists of three stages: Decomposition: cover the vertex set by simpler subconfigurations (e.g., triangles); Intersection: the original rational‑distance locus equals the intersection of the subconfiguration loci; Control: equip each subconfiguration with faithful algebraic charts (e.g., elliptic fibrations), record all exceptional loci, and enforce compatibility equations, culminating in a direct substitution check of the original distance equations. The power of the DIC method lies in its uniform applicability across three intimately connected arithmetic worlds: Over ℚ, we obtain a directional fibration of triangle loci into elliptic curves, leading to a precise compatibility condition that reduces the unit‑square conjecture to an explicit polynomial system. Over ℚₚ (p‑adic fields), we develop square‑class tests, Hensel lifting, formal groups, and good/bad reduction analysis, providing local obstructions and residue‑disk control. Over 𝔽_q (finite fields), the method becomes fully constructive: every chart is finite, and we give exact enumeration algorithms, character‑sum formulas, and connections to coding theory, Paley graphs, and association schemes. A central achievement is the rigorous reduction of the unit‑square rational‑distance conjecture to the verification that a specific biquadratic system has only the trivial rational solutions m=0,±1,∞m=0,±1,∞ (the symmetry axes). This reduction is unconditional and algorithmic; the remaining computation is a finite algebraic task that can be executed with computer algebra. While the unit‑square conjecture is not settled here, the DIC framework provides a complete, auditable proof blueprint. More importantly, it transforms rational‑distance geometry from a collection of isolated techniques into a unified mathematical discipline with its own language, certificates, and bridges to arithmetic geometry, p‑adic analysis, finite‑field combinatorics, and coding theory. Keywords Rational distance geometry; DIC method (Decomposition–Intersection–Control); elliptic fibrations; unit square problem; p‑adic distance geometry; finite field distance geometry; Hensel lifting; quadratic characters; Groebner bases; resolvent elimination; Paley graphs; association schemes; distance codes; compatibility condition