The Geometry and Dynamics of Spiral Minimal Products

We study spiral products $G_\gamma(t,x,y)=(z_1(t)f_1(x),z_2(t)f_2(y))$, formed from spherical $\mathscr{C}$-totally real immersions $f_i:M_i^{k_i}\to S^{2n_i+1}$, with $k_1+k_2>0$, and a profile $\gamma=(z_1,z_2)$ in $S^3$. The product is minimal precisely when the factors are minimal and $\gamma$, after reparametrization, is a geodesic of $\bar g=|z_1|^{2k_1}|z_2|^{2k_2}g_{S^3}$ wherever $\bar g$ is positive definite. The profile flow is Liouville integrable. On each regular two-turning component of the doubly spiral parameter domain, the complete-cell phase map is real analytic with open dense full-rank locus; hence ordinary-closing profiles of arbitrarily large primitive order are dense. At fixed regular nonzero momentum, an exact Routh completion reduces the profile Hessian to a scalar Sturm form plus two nonnegative squares. For compact minimal inputs and an ordinarily closed profile of primitive order $m_\gamma$, $G_\gamma$ satisfies $\operatorname{Ind}(G_\gamma)\geq\operatorname{Ind}(f_1)+\operatorname{Ind}(f_2)+2m_\gamma-3$. At the contact momentum level, factor-adapted selection produces, from any prescribed pair of compact connected embedded special Legendrians, compact embedded special Legendrian products of every sufficiently large prime closing order. Their second fundamental forms are uniformly bounded, while volume and normal Morse index grow at least linearly with the order. Real spherical minimal embeddings yield analogous families. Applied to finite-holonomy horizontal lifts, it yields Delaunay-type minimal Lagrangian immersions in complex projective spaces. For canonical lifts of compact embedded inputs and a regular ordinarily closed contact profile, the primitive spherical quotient is embedded; its Hopf quotient is embedded exactly when the reduced relative winding number is $1$ and satisfies an additive Hamiltonian-index bound.