The Prime Radix Invariance Theorem: Closed-Form Counting Function and Bijective Generator for Odd Prime-Order Progression-Free Sets

For any odd prime p >= 3, let S_p denote the lexicographically earliest sequence of non-negative integers containing no p-term arithmetic progression initialized with s_0 = 0 (the Stanley sequence of order p). We prove that S_p is identically equal to the set A_p of non-negative integers whose base-p expansion contains no digit equal to p - 1. The proof uses the invertibility of non-zero residues in the finite field Z_p, which implies that every non-trivial progression of length p contains at least one element with maximal base-p digit p - 1. From this characterization, we obtain: (i) an exact, non-asymptotic counting formula r_p(N) = Psi_p(N) = |S_p cap [0, N-1]| valid for all N >= 1; (ii) an explicit bijection computing the n-th element a_n in O(log n) operations; and (iii) the exact Hausdorff dimension d_H = ln(p-1)/ln(p) together with the absolute convergence of sumn in S_p, n >= 1 1/n. The counting formula is verified numerically across 24,000 checkpoints up to N = 100,000 with zero discrepancies.