Geometric strain and two-cycle closure in a surface-tilt model

A signed surface deformation and a strain-sensitive carrier can have different temporal periods.
Motivated by a proposed surface interpretation of spin, this paper constructs a classical mechanism
for that distinction. Projection of the von Kármán strain energy onto one in-plane and one transverse
mode gives a cubic coupling and a quartic transverse term, with a coefficient constraint inherited
from positive elasticity. Under an assumed near 2:1 resonance, averaging yields a known conversion
Hamiltonian. The in-phase and antiphase locked branches have local stability in the symmetryreduced
averaged dynamics. Reflection and time reversal provide sufficient shooting conditions for a
family of two-cycle orbits in the unaveraged two-mode equations. Numerical shooting resolves such
orbits for selected parameters and measures the difference between the averaged and unaveraged
branches. A transverse bias preserves the continued periodic orbit while breaking its half-period
reflection relation. The construction supplies a geometrically derived modal mechanism and tests for
an analogue realization. It does not determine the near-resonant frequency selection or identify the
internal motion with physical spatial rotations. Electron spin and spatial dimensionality therefore
remain outside the demonstrated results.