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 mechanismfor that distinction. Projection of the von Kármán strain energy onto one in-plane and one transversemode gives a cubic coupling and a quartic transverse term, with a coefficient constraint inheritedfrom positive elasticity. Under an assumed near 2:1 resonance, averaging yields a known conversionHamiltonian. The in-phase and antiphase locked branches have local stability in the symmetryreducedaveraged dynamics. Reflection and time reversal provide sufficient shooting conditions for afamily of two-cycle orbits in the unaveraged two-mode equations. Numerical shooting resolves suchorbits for selected parameters and measures the difference between the averaged and unaveragedbranches. A transverse bias preserves the continued periodic orbit while breaking its half-periodreflection relation. The construction supplies a geometrically derived modal mechanism and tests foran analogue realization. It does not determine the near-resonant frequency selection or identify theinternal motion with physical spatial rotations. Electron spin and spatial dimensionality thereforeremain outside the demonstrated results.