Observable Kinetic Quotients and Sixth-Order Constrained Bifurcation

Finite-time kinetic records usually determine a data fibre rather than a unique microscopic state. We construct one finite-dimensional observable quotient and use it, without changing the retained variables, for stable coordinate recovery, Q-exact Hamiltonian reduction, relative-entropy contraction, and local constrained bifurcation. Q-exact means constant on every fibre of the selected quotient map. For symmetric Hilbert-Schmidt tensors (square-integrable kernel tensors) through degree six, orthogonal projection gives the unique best scalar Hamiltonian Taylor jet that factors through the quotient. On the same coordinates, a bounded minimal exponential family defines an analytic relative-entropy section, while an application-supplied regular constraint map reduces the microscopic variational problem to a finite-dimensional scalar. We derive recursions for the curvature of the constraint manifold and for variational Lyapunov-Schmidt elimination, an explicit sixth-order reduced jet, and an inertia-splitting formula that distinguishes a reduced minimum from a minimum of the pre-reduced constrained problem. In a three-dimensional critical space with signed-permutation symmetry, a nonzero quartic anisotropy excludes a primary reduced two-axis minimum. Such a branch is first admitted by a sixth-order balance, either on an exact quartic-degeneracy locus or inside an explicit parameter-dependent wedge. Quantitative estimates transfer tensor residuals through the entropy section, constraint chart, and range elimination, but do not automatically transfer a critical parameter or reoptimize invisible variables. The final section specifies the response operator, constraints, critical representation, direct vertices, complement inertia, remainder, and residual margins required from a concrete Vlasov-Volterra model.