Prime-Layer Rigidity in Canonical Dyadic First-Return Systems

We study the discrete folding permutationBQ(x) = 2x,2x ≤(Q−1)/2,Q−2x, 2x>(Q−1)/2XQ ={1,...,(Q−1)/2},for odd integers Q ≥ 3. Every state admits a unique dyadic decomposition x = 2ku withu odd. We prove that the resulting dyadic tower decomposition is canonical and that thewhole finite system is reconstructed from the first-return datumIQ,ΦQ,hQ ,ΦQ(u) = Q−2hQ(u)u.We then attach to each odd head u > 1 the arithmetic label spf(u) and lift these labels tothe complete dyadic towers. For odd Q ≥ 7, the central result is the exact equivalencecomplete prime-layer crystallization ⇐⇒ rad(Q) =p≤Pp odd primep,P =Pmax(Q).Equivalently, every prime-labelled layer is preserved by the canonical first-return permutationif and only if every odd prime not exceeding the largest prime factor of Q divides Q. Theproof separates two mechanisms: prime labels are constant along dyadic towers, while allpossible label changes occur at the terminal first-return jump. For the converse, a singlewitness is provided by the smallest odd prime missing from the support of Q.Thus the arithmetic content of complete crystallization is precisely completeness of theprime support, while the square-free condition supplies the remaining step from primesupport to the exact primorial. The paper is related to the classical doubling map moduloodd integers and to first-return compressions of doubling dynamics; those constructionsare not claimed as novel here. The contribution is the specific rigidity theorem linking thecanonical dyadic first-return decomposition to complete prime-layer preservation.