A Compact Gauss Field with a Native Clock: Ground Energy and Planar Strain

A compact square couples the complete eight-level clocks to unrestricted integer electric flux. At specified couplings, positivity, a complete finite eigenvalue count and an infinite-tail estimate enclose the normal ground energy near 1.6373452821 and give an excitation gap above 0.0046547168. Differentiating the same action gives controlled nonzero planar-strain forces. The calculation specifies the square geometry, electric law and field-only strain coupling.