Constant Decomposition and Directional State Limits: Arithmetic Restoration and N-Block Digit Laws

This paper presents a foundational study of constant decomposition and directional state limits. The central expression is A = limA + lim0, where limA is not the standard limit operator, but a state-number symbol representing a value that approaches A without being identified with A itself. Likewise, lim0 is not zero itself, but a difference state number representing the remaining gap. The study introduces the N-th digit difference state number lim0_N, directional signs for approaches from below and above, and shows that the full binomial expansion of the state number and the difference state number restores A^m for natural-number exponents m. It also develops N-block digit laws describing how digit blocks appear in upper and lower approaches, with direct verifications for the square, cube, and fourth power. This work proposes a state-number framework for recording how constant decomposition is preserved and restored through arithmetic expansion and radix digit structures.