A Weighted Pair-Symbol Encoding System with Relational States: From the 2-Bit Table to the 4-Bit Table

This paper presents a reversible system that represents each byte as a single decimal number of the form X.YYY. Each 2-bit pair carries a weight (00→2, 01→3, 10→5, 11→13). Two pairs form a 4-bit group, and two groups form a byte. Because summing weights loses ordering information, the system adds ten relational states that record where each group starts and which group is smaller, together with a rule-built repetition-offset table. The result is 256 distinct values for 256 bytes, verified by exhaustive enumeration. Four bytes are then composed into one 32-bit number across four graduated sections (10⁻³ to 10⁻¹²). The paper states the algorithmic rules, the encoding and decoding steps, and the system’s constraints: it is a complete, reversible representation, not a compression method. It also documents what changed from two earlier versions.