From reduced phonon observables to an invariant local saddle: canonical saddle-center transport and an intrinsic threshold law

Reciprocal-space phonon spectra are commonly used as material descriptors rather than as coordinates of a dynamical space. We show that a finite retained set of normalized phonon and neutron-weighted spectra defines an observable quotient of microscopic descriptions and, for three selected endpoint materials, an explicit two-dimensional simplex of spectral mixtures in retained-observable space. The simplex is a reduced model: its vertices are realized material observables, whereas no microscopic source description is assumed to realize every interior mixture. For the B2 TlBr–CsBr–RbI trio, the square-root immersion of this simplex has a positive-definite pullback metric. Comparison with an affine density-[Formula: see text] reference yields a coordinate-invariant logarithmic contrast field with a boundary-controlled saddle at barycentric weights [Formula: see text]. Using this contrast as the reduced potential and the pullback metric as the kinetic metric defines a natural saddle–center Hamiltonian on the cotangent bundle of the observable simplex. Its quartic canonical normal form has hyperbolic rate [Formula: see text], elliptic angular frequency [Formula: see text], and quartic action coefficients [Formula: see text]. Restriction of the quartic normal form to the transition-state normally hyperbolic invariant manifold (NHIM) fixes the fourth-order excess reduced-energy law [Formula: see text], where [Formula: see text] is the elliptic action. Trajectories of the complete fourth-order local Hamiltonian validate nonrecrossing transport, the signed threshold, and the predicted logarithmic delay. The saddle-weighted full-Brillouin-zone phonon density-of-states (DOS) centroid, [Formula: see text] meV, fixes the Hamiltonian time unit to [Formula: see text] ps. An exhaustive census finds saddles in 28 of 56 eligible B2 triples, while matched B1/B3 and elemental face-centered-cubic (FCC) controls show that the construction can be applied across distinct symmetries without implying universal saddle occurrence. The resulting dynamics belongs to the constructed observable-mixture space and is not an atomistic anharmonic lattice Hamiltonian.