Semidirect Fourier Delta Attention: Phase-Controlled Delta Memory with Constructive Chunk-WY Kernels

Linear attention replaces softmax attention's growing KV cache with a fixed recurrent state, but this compression limits exact state tracking and long-context memory. We introduce \emph{Semidirect Fourier Delta Attention} (SFDA), a phase-controlled generalization of Kimi Delta Attention that replaces real diagonal decay with block-rotational Fourier control:

$$S_t=(I-β_t k_tk_t^*)Λ_tS_{t-1}+β_tk_tv_t^*, \qquad Λ_t=\diag(α_t\odot e^{iθ_t}).$$

Our main result is a constructive chunk-WY factorization for products $A_t=Λ_t-u_tr_t^*$, giving

$$A_t\cdots A_1=Γ_t-Y_tM_tW_t^*$$

with rank growth bounded inside fixed chunks. This yields an exact affine chunk transfer, formal stability and complexity bounds, and a compact characterization of phase-plus-low-rank memory. We verify the algebra numerically and show in toy state-tracking experiments that SFDA learns cyclic memory where the phase-disabled KDA baseline remains near chance. Fused kernels and large-scale language-model comparisons are left to future work.