A Decimal–Hexadecimal Block Encoding for Prime Storage with Modular Compression via the Ternary Constraint

We describe a decimal–hexadecimal block encoding for primality over a finite stored range. Since every prime greater than 5 must lie in one of the residue classes 1, 3, 7, 9 (mod 10), each decimal block of size ten can be encoded by a 4-bit word indicating which of the candidates 10k + 1, 10k + 3, 10k + 7, and 10k + 9 are prime. This yields a nibble-based storage scheme supporting exact primality queries and exact recovery of the prime-counting function π(x) by cumulative popcount. We then establish a structural theorem arising from the congruence 10 ≡ 1 (mod 3): for k ≡ 0 (mod 3) the candidates 10k + 3 and 10k + 9 are always composite, and for k ≡ 2 (mod 3) the candidates 10k + 1 and 10k + 7 are always composite. This partitions the nibble alphabet into three classes of sizes 4, 16, and 4, reducing the Shannon entropy from 4 bits to 2.42 bits per nibble and yielding a lossless compression of 39.4% over the original encoding with O(1) decode complexity. We present data structures, Python routines, and experimental validation up to 300,000.