Integral vanishing cycles from periods on conifold smoothings

We give a terminating construction of the marked integral vanishing-cycle homomorphism for a six-parameter family of Calabi–Yau conifold smoothings in ℙ³ × ℙ³. Its eight local spheres span a rank-six lattice in free middle homology of rank 134. Starting from algebraically selected equations, we construct a known neighbourhood in vanishing-period coordinates, a geometric symplectic marking, and absolute error bounds for transport and integration of represented cycles. Picard–Lefschetz period jumps recover the integral columns, their saturation and monodromy without a height prior or a restriction on nonzero period ratios. We verify the hypotheses of established analytic results on a possibly ineffective smaller neighbourhood. There the capacity-weighted squared norm of restriction of divisor classes from the small resolution to its exceptional curves governs the two small positive Ricci-flat two-form eigenvalues; the image of this restriction is the kernel of the recovered vanishing-cycle homomorphism. For nonflat bundles on the same fibres, a finite-dimensional invariant manifold of Hermitian–Yang–Mills metric flow is tangent at each constructed Hermitian–Einstein metric to the full low eigenspace on trace-free Hermitian endomorphisms.