Binary Prefix Closures and Local Corrections in Composite Detection: A Large-Scale Experimental Note

We study an experimental arithmetic structure built from the proper binary prefixes of an odd integer n. From these prefixes, we generate first-layer candidates using the bitwise operations OR, XOR, AND, and absolute difference. We then examine a shallow second layer based on pairwise differences of first-layer values, together with a local correction mechanism of the form |x − c|, where x is a generated value and c is a bounded positive integer. The central empirical finding is that nontrivial divisors of odd composite numbers often emerge within this low-depth closure. In lightweight large-scale experiments on odd integers up to 10⁶, the candidate architecture based on P (n) ∪ L1 (n) ∪ L2^ (M) (n) ∪ CC (L1 (n) ) achieved detection of 421, 441 out of 421, 502 odd composites, with zero odd primes incorrectly flagged, corresponding to 99. 9855% coverage. All odd composites with at least three prime factors were detected, and semiprimes were detected with 99. 9667% coverage. The residual failures are not random: they are concentrated in a narrow family of rigid semiprimes and a few prime squares with medium or moderately large prime factors. This note does not claim a proof of a primality criterion. Its purpose is to document a coherent large-scale experimental phenomenon suggesting that divisors of odd composites are often encoded in a structured and unexpectedly shallow closure generated by the binary prefixes of the number.