Asymptotic Sufficiency Depth, Volume IV: Integer Cycles, Absolute Periods, and Branch-Dependent Stokes Data for a Polynomial Benchmark

This preprint is Volume IV of the Asymptotic Sufficiency Depth series and develops a fully marked geometric–analytic benchmark beyond the finite-state and path-space settings of Volumes I–III. The fixed polynomial is f(X,Y,Z) = X^6 + Y^6 + Z^6 - X^4 Y^4 Z^4. In the determinant-character sector of its finite phase symmetry group, the study connects a primitive rank-six integral vanishing-cycle lattice to absolutely normalized rapid-decay periods and to a directionally summed ordinary-Borel Stokes system. The declared integer marking yields explicit intersection and Seifert forms. A common rapid-decay sector gives six scalar periods forming a fundamental system of a sixth-order differential equation. With the branch, path, basis, and lateral conventions fixed, the twelve candidate Stokes directions consist of six nontrivial rank-one factors and six identity factors on the declared direct Borel branches. Their ordered clockwise product agrees, in an explicitly identified basis, with normalized boundary monodromy. The reverse jump vector (-i,-i,8i), the forward normalization, and the direct zero factors are also recovered by an independent local analytic route based on Mellin transforms, a Meijer G kernel, and hypergeometric connection formulas. The analysis further distinguishes direct-branch disappearance from global analytic absence: specified continued branches can acquire nonzero minors where the corresponding direct branches are regular. The result is benchmark-specific. It does not classify a general (p,q) family, all Borel sheets, or a universal correspondence between Borel minors and hidden Kullback–Leibler loss. All matrix and Stokes statements refer to the explicit markings, paths, branches, and normalization conventions defined in the paper. The accompanying reproducibility archive contains exact-arithmetic and symbolic checks for selected finite identities. These checks complement the continuous geometric and analytic arguments given in the article; they are not substitutes for those proofs. Text, figures, documentation, metadata, and non-code data are released under CC BY 4.0. Original project code is released under the MIT License.