Shellsort's best general lower and classical upper bounds differ by an iterated-logarithmic factor. Lower bounds use signed, order-free cancellation, whereas upper bounds require positive representations respecting pass order. We develop a common certificate framework for these two geometries. A prefix Fourier phase defect lower-bounds worst-case exchanges and hence comparisons. At pass $j$, the chronological ray quotient records multiples of the current gap already eliminated by earlier passes. Its truncated, pair-weighted hole mass bounds exchanges in that pass, and adding the $np$ overhead bounds comparisons. In a sufficiently long active window, $g_j^{(n)}$ ray holes imply signed transfer length at most $4g_j^{(n)}-1$, so approximate prefix characters propagate to the current gap. Signed transfer length two can coexist with arbitrarily large ray genus; scale-local Apéry representatives control both parameters. The framework recovers Pratt's $O(n\log^2 n)$ scale and the sparse two-parent scale, and yields the following structural results. Uniformly bounded full ray genus after a fixed prefix forces $W_n=Ω(n^{1+1/j_0-o(1)})$ when $p_n-j_0=o(\log n)$. An affine family has ordinary global semigroup genus and conductor $Θ(n)$ but relevant ray genus two. Balanced three-generator full grids have terminal truncated ray genus at least $\exp(Ω((\log n)^{2/3}))$. The last result is a certificate barrier.

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