TEBAC Hodge Program V: Primitive Detector Singularities, Boundary Chow Supply, and Terminal Packet Rank Closure

This master preprint consolidates Module V of the TEBAC Hodge program. It unifies the terminal proof-front submodules V-A through V-W into a single synthesis devoted to primitive detector singularities, boundary Chow supply, Chow--Hilbert source realization, detector-faithfulness, moving-face escape, and finite boundary packet rank closure. The manuscript works with the terminal primitive quotient\[V_X^p\]and studies nonzero terminal detectors\[0\neq \lambda\in (V_X^p)^\vee .\] The central proof-front is the construction of boundary Chow--Hilbert packets whose specialization is visible to \(\lambda\). The resulting target chain is\[\begin{aligned}0\neq\lambda&\Longrightarrow\text{detector-normalized boundary singularity} \\&\Longrightarrow\text{Chow--Hilbert boundary source} \\&\Longrightarrow\text{finite boundary packet} \\&\Longrightarrow\lambda(\operatorname{Sp}_\tau[B])\neq0 .\end{aligned}\] The preprint develops the formal reduction from admissible normal-function singularities to boundary Chow packet visibility, including kernel descent, detector-compatible boundary evaluation, scalar boundary residues, primitive source noncancellation, moving-face detector escape, and rank-one packet generation. The terminal finite-dimensional condition is expressed as\[T_\tau a=\delta_\tau(\nu_\lambda),\qquadL_{\lambda,\tau}a\neq0.\] The manuscript is status-explicit. It does not claim that the Hodge Conjecture is already unconditionally proved. Instead, it identifies the remaining non-formal theorem target as the explicit detector-selected boundary packet construction:\[\delta_\tau(\nu_\lambda)\in\operatorname{im}T_\tau(\mathcal P_\tau)\setminusT_\tau(\ker L_{\lambda,\tau}).\] Conditional on this terminal packet construction, the established rank-closure mechanism yields the expected TEBAC terminal implication\[\operatorname{span}_{\mathbb Q}\mathcal G_{X,p}=V_X^p\LongrightarrowV_X^p=0\LongrightarrowK_X^p=A^p(X).\]