The Hyperfine Structure (21-cm Line) as Discrete Phase-Locking: Replacing Magnetic Dipoles with Hardware Clock Incompatibility

The astrophysical 21-cm hyperfine line of hydrogen is traditionally described as the energy splitting arising from the continuous magnetic dipole-dipole interaction between the electron and proton spins. In this paper, we expose a fundamental methodological trap in digital physics: attempting to simulate macroscopic continuous fields within a pixelated substrate is an ontological contradiction. We demonstrate that within Computational Finitism, spacetime is a discrete causal graph governed strictly by Modulo-9 (ℤ 9) constraints, and macroscopic "continuous fields" are representational illusions. By re-engineering our simulation to utilize Additive Modular Convolution rather than rigid boundary overwrites, we formalize spin not as a vector field, but as the timing of Topological Redundancy Buffer (TRB) bit circulation. We prove that the 21-cm energy splitting is a discrete Topological Phase-Locking phenomenon: the Parallel (in-phase) configuration forces local hardware saturation and zero-divisor accumulation, while the Anti-Parallel (out-of-phase) configuration achieves stable vacuum closure. The resulting energy difference is a bounded, rational invariant, completely eliminating the need for continuous magnetic dipoles.