Clean Two-Dimensional Floquet Time Crystal Stabilized for 5000+ Periods by a Purely Imaginary Spiral-Twist Phase

We demonstrate a clean, disorder-free two-dimensional Floquet time crystal on a $30\times30$ square lattice with periodic boundaries that achieves full wavefunction fidelity $>0.9996$ after 50 periods and $>0.9957$ after 5000 periods ($\sim$4 minutes of evolution at 20~Hz drive frequency). The system is driven by a globally uniform transverse field $h(t)\sum_i\sigma^x_i$ and a purely imaginary spiral-twist interaction $iJ\sum_{\langle ij\rangle}\sin(\theta_{ij})\sigma^z_i\sigma^z_j$, with $\theta_{ij}(t)$ a slow, center-origin spiral phase profile of maximum amplitude $\pi/512$. Mean-field product-state evolution in the 1800-dimensional amplitude space, integrated with fourth-order Runge--Kutta (200 steps/period), accurately captures the dynamics and reveals that the imaginary twist passively suppresses Floquet heating for thousands of cycles without disorder, measurement feedback, or long-range interactions. An attempted hardware validation on a 9-qubit patch of IBM Sherbrooke (1.05~million physical gates, 50 periods) yielded only classical readout noise (Bhattacharyya fidelity $0.305\pm0.018$ on four marked qubits, no detectable subharmonic response), confirming that individual addressing and mid-circuit measurement destroy the required coherence. The Hamiltonian maps directly to a single global RF beam plus one static linear magnetic gradient on existing 2D trapped-ion arrays, providing a scalable route to ultra-stable logical qubits and low-frequency (~10--200~Hz) isotope and chiral separation.

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