Uniform-Window Shell Kernels for the Syracuse Map: Renewal Structure and a Primitive-Frequency Obstruction

We study shell kernels for the odd-to-odd Syracuse dynamics generated by uniformly distributed initial windows. For backstepped first-passage shells, we prove short-time localization, derive an exact inverse-affine representation of the fixed-time kernel, and reduce the shell-slice discrepancy to weighted primitive-frequency correlations. We also prove a quantitative boundary-layer estimate and identify a formal renewal model for the corresponding shell mechanism. On the arithmetic side, we obtain an exact block decomposition for the primitive-frequency transfer operator, prove that no naive operator gap is available, and reduce the unresolved step to explicit incomplete principal-unit exponential sums modulo powers of 3. Thus the paper is unconditional up to a final primitive-frequency estimate, which is formulated explicitly.