We study noiseless-feedback communication over Gaussian channels with stationary autoregressive (AR) noise of arbitrary finite order. We introduce a class of Gaussian linear-feedback coding schemes based on the celebrated Schalkwijk--Kailath (SK) coding scheme, called SK(2), whose message process follows a second-order deterministic recursion, and derive a closed-form characterization of its maximal achievable rate under average power constraint $P$. As a first-order benchmark, we formulate SK(1) as a sign-consistent, branch-complete version of the real constant signed-ratio specialization of Butman's linear-feedback construction. For stationary AR($p$) Gaussian channels, optimizing over recursions with both characteristic roots outside the unit circle gives the same rate supremum as optimizing over all SK(2) recursions and yields a rate no smaller than the maximal SK(1) rate. The SK(2) scheme achieves feedback capacity for the AWGN channel, stationary AR(1) Gaussian channels, and a subclass of stationary AR(2) Gaussian channels obtained by interleaving two independent AR(1) noise processes. For the interleaved subclass, we show that a genuine SK(2) scheme strictly outperforms SK(1). Numerical comparisons illustrate further strict rate gains over a broader range of AR(2) parameters. These results show that the SK(1) scheme is not universally optimal beyond AR(1) noise and disprove a corrected form of Butman's conjecture.

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