Binary Compression and Interscale Operators on Primorial Wheels

Description This publication investigates the single scale transition (from modulus M to qM) in wheel sieves from a discrete dynamical systems perspective [1, 3]. Rather than focusing on prime-generation algorithms, the study analyzes the exact information content preserved or discarded when lifting and deleting reduced residues [1]. Key Findings and Contributions Exact Sufficient Binary Encoding: It is proved that retaining only a cyclic binary word indicator—recording whether each metric gap is divisible by the new prime q—is an exact sufficient encoding for the complete family of closure fields and the compressed interscale successor operator C_q [3, 4]. Unrolled Coordinate Resolution and Monodromy: By extending ordered residues to an unrolled sequence, boundary coordinates are rigorously resolved via a monodromy law for deletion indices, showing that binary 1-runs correspond to consecutive deleted fibers [4-6]. Complete Abstract Aliasing Classification: In a purely combinatorial model with an abstract shift operator S_d, binary words with two or more zeros are shown to be faithful [4, 7]. Words with exactly one zero form the unique nontrivial aliasing class, all collapsing to the matrix of S_d [4, 8]. Arithmetic Faithfulness: Elementary unit geometry forces every nondegenerate reduced-residue wheel (M > 3) to produce gap words with at least two zeros [9, 10]. Consequently, the abstract aliasing class is arithmetically unrealizable, establishing exact operator faithfulness (B^(q) <--> C_q) for all standard wheels [9, 11]. Information Loss Localization: Genuine information loss occurs strictly in the metric reduction step (converting exact gap lengths to binary divisibility indicators), rather than in the subsequent operator representation [9, 12, 13]. Open Generator-Compression Problem: The paper formulates an open problem regarding whether full first-hit dynamics admits a dynamically closed, lower-complexity state quotient [9, 14]. Reproducibility and Artifacts The deposit includes the complete LaTeX source alongside the Python verification script verify_binary_compression_v5.py [15, 16]. The script independently validates the binary formulas and arithmetic bounds across 7,132 gap configurations and extensive modulus scans [17, 18].