D12RG Paper 5.2: Connected Verlinde Completion, Baryonic Dual-Return, and the Visible Ten-Sector Readout

We identify the connected ten-label source datum used in the D12RG CFT framework with the standard rank-one lattice fixed-point VOA \(V_{\sqrt6\mathbb Z}^{+}\) / zorb3, compute the corresponding Verlinde fusion tensor in the Paper 5.2 source order, and transport the resulting fusion ring into the visible/readout basis used in Paper 5. The category itself is therefore known; the new contribution is the D12RG derivational and readout route into it. The source datum satisfies \( (S^{\rm full}_{10})^{\dagger}S^{\rm full}_{10}=I_{10}, \) \( (S^{\rm full}_{10})^2=C_{10}, \) \( (S^{\rm full}_{10}T^{\rm src}_{10})^3=C_{10}. \) The associated Verlinde tensor is nonnegative integral and gives a connected ten-label source fusion ring. In this datum the fusion matrices are multiplicity-free: their entries are all \(0\) or \(1\). This is a standard special case for fusion rings, not a general consequence of the Verlinde formula, whose coefficients are nonnegative integers in the usual semisimple setting \cite{Verlinde1988,EtingofNikshychOstrik2005,VOADatabaseZorb3,Abe2001,AbeDongLi2005,CuntzGoff2008}. This source ring is then related to the Paper 5 visible sector by a typed transport map, rather than by identifying source labels or source phases directly with physical readout data. The modular identities also give a compact return structure. Since \(C_{10}^2=I_{10}\), one obtains \( (S^{\rm full}_{10}T^{\rm src}_{10})^6=I_{10}, \) while the source phase matrix satisfies \( (T^{\rm src}_{10})^{24}=\operatorname{diag}(I_6,-I_4), \) \( (T^{\rm src}_{10})^{48}=I_{10}. \) Thus the connected source datum displays three return ladders: \(S\) at powers \(2/4\), \(ST\) at powers \(3/6\), and \(T\) at powers \(24/48\). Under the shell correspondence \((ST)^n\leftrightarrow\Phi^{2n}\), the midpoint identity \((ST)^3=C_{10}\) is the \(\Phi^6\) dual-return layer, while \((ST)^6=I_{10}\) is the first full source return at \(\Phi^{12}\). The paper therefore separates three layers that were previously easy to conflate: the hidden/readout construction of Paper 5, the connected source modular datum, and the transported visible fusion structure. Its role is to give the connected modular and Verlinde bridge between Paper 5 and the later shell, baryonic, and lifted-closure analyses.