Colour and Generations as Geometry: The CP¹ × CP² Architecture and the Two-Threes Conjecture

The first two papers of this series established a conditional chain from Riemannian geometry to Born weights and instantiated it, in closed form, on the three-sphere - reading a U(1) gauge structure off the Hopf fibration in the process. This paper confronts the criterion those papers set for any geometric account of the strong sector: Kill Criterion KC-1, the two-threes test, which demands that colour-3 and generation-3 arise from two independent topological invariants. We exhibit an architecture that passes the test kinematically, with no discretionary integer. On the internal geometry CP¹ × CP², colour is carried by the CP² factor - the triplet is the index of the Picard generator through the canonical spin^c structure that topology forces twice over - and the family count is carried by the CP¹ factor as the index of its own tangent bundle through its own Dolbeault operator: χ(CP¹, T) = 3, the infinitesimal automorphisms of the sphere. Both threes are forced; neither is chosen. The obligatory back-reaction of the forced connections yields a coupled Einstein-Maxwell system whose solutions exist on a band of a single dimensionless dial, with the tangent twist landing on the widest band in the landscape; a shape-stability reconnaissance finds the band everywhere unstable in the classical product manner, with the least-unstable locus at the band edge. The architecture's charge lattice fails the Standard Model hypercharge table in its two-charge form - a no-go proved here - and passes exactly in three charges once the spatial Hopf charge of Paper 2 participates: Y = k/3 + q reproduces the table, and a chirality audit finds the spectrum exactly chiral, with the Standard Model's forty-five Weyl states and no mirrors. What remains open is stated with equal precision: the uniformity meta-rule behind the canonical assignments, the full stability analysis, and everything the non-claims box excludes. The route by which the endpoint was reached - including the selection routes that died and the audit that corrected the paper twice - is preserved in full.