The SATOR Matrix - Integer-Information Parry Measures and Golden Extremality: Exact Arithmetic Coding, Galois Attractors, and the Triple Extremality of the Golden Ratio

This paper establishes the information-theoretic and number-theoretic foundation for the pentagonal gauge geometry programme. We isolate a rigid class of symbolic dynamical systems — integer-information Parry presentations — in which all transition probabilities are exact powers of the Perron eigenvalue. This quantisation condition is equivalent to the existence of an integer-valued height function satisfying local Kraft equalities, and it makes two constructions exact rather than numerical: Markov partitions with algebraic-integer endpoints, and arithmetic coding with no rounding required. The height function defines natural Galois-channel coding maps. For each contracting embedding of the number field, one obtains a compact graph-directed self-similar attractor carrying the pushforward of the Parry measure. Demanding simultaneous compactness across all nontrivial embeddings singles out the Pisot numbers, providing a clean geometric separation between Pisot and non-Pisot systems. Specialising to the golden ratio, we show that the golden-mean shift occupies a uniquely economical position: its Kraft structure is the minimal polynomial itself; its coding is Fibonacci-synchronous via the Binet decomposition; its Zeckendorf representation provides a canonical greedy code; and the three-distance theorem ensures maximal hierarchical uniformity. A triple-extremality theorem proves that the golden ratio is the unique algebraic integer simultaneously most irrational (Hurwitz), most compact in the Galois channel (Pisot), and most economical algebraically (minimal field discriminant) — three independent criteria from Diophantine approximation, symbolic dynamics, and algebraic number theory that intersect in a singleton, linked by a single arithmetic identity.