Binary Window Detection of Smallest Prime Factors Through the Transform 3n (Experimental note)

We study an experimental factor-detection mechanism based on binary windows. Startingfrom an odd integer n coprime to 30, we first inspect the binary windows of n itself. Then,for the unresolved cases, we inspect the binary windows of the transformed integer 3n. Inlarge-scale computations up to 10^6, after removing non-squarefree composites, the transform3n reveals the smallest prime factor spf (n) for every tested composite in the working universe,while producing no nontrivial false positives on primes.More precisely, in the squarefree composite universeU(106) = {n ≤ 106 : n composite, gcd(n, 30) = 1, n squarefree},we obtain:• direct exact visibility in the windows of n for 43.581456% of the cases;• after passing to 3n, revelation of spf(n) in 100% of the tested composites;• zero nontrivial false positives in 78,495 tested primes p ≤ 106 with gcd(p, 30) = 1 and p > 5.The note is experimental: it records a verified finite-range phenomenon, formulates itprecisely, and proposes a carry-based interpretation of why multiplication by 3 may exposehidden factors through binary-window deformation.