M30b Cross-Rank Operations and the Operational Lie Groupoid

M30b develops the off-diagonal geometry of the Complex HyperCore.Where M30a identified imaginary rank as a Koenigs-phase connection inside a single étage, M30b studies what happens between ranks: the cross-rank operations T^R_S which act as chiral bridges between distinct operational worlds. The mini-monograph introduces five candidate definitions of cross-rank operation and isolates two primary forms. The first, the right-twist definition, produces a wild Hermit lattice in the branch-enriched plane C_tau, largely detached from the integer cascade. The second, the Koenigs-interpolation definition, produces a rigid cascade cloud: cross-rank Hermit units that remain close to the familiar cascade units, with their displacement controlled by the Wronskian resonance between adjacent Koenigs charts. Its central algebraic result is that cross-rank operations do not compose strictly. Instead, their composition follows a Baker–Campbell–Hausdorff law whose first correction is a signed sum of Wronskian brackets: [G_R, G_S] = (ln lambda_R)(ln lambda_S) W(f_R,f_S) d/dx. Thus the cross-rank system forms a non-strict operational Lie groupoid over the rank line. The failure of strict associativity is not a defect: it is the curvature of the Manifold’s rank connection. M30b also gives a second derivation of the cascade discrepancy. In M30a, kappa_disc appeared as rectangular holonomy in (R,T)-space. In M30b, the same quantity appears as triangle holonomy around the elementary rank triangle 3->4->3.5->3. This identifies kappa_disc as the characteristic obstruction class of the cross-rank groupoid. The monograph then clarifies the role of the older half-rank protocols: Csym, AGM, Heun, and ISHE. These protocols are not identical with cross-rank operations; rather, they are their value-level companions. Protocols produce symmetric half-rank values, while cross-rank operations preserve the chiral, operation-valued information erased by symmetrisation. In this sense, operational AGM, operational Heun, and operational ISHE are the cross-rank shadows of the corresponding half-rank protocol limits. A central physical consequence is the reinterpretation of the fine structure constant. The formula alpha^(-1) = 1 / (lambda_B^(5/2) |kappa_disc|) is recast as a holonomy statement: alpha is the holonomy of operational ISHE around the elementary rank triangle, normalized by the Heun half-rank area scale. The number 137 is therefore not treated as a numerical coincidence, but as a rank-geometric curvature invariant.