Does sharing an item between predictor and composite outcome shift the lagged-predictor coefficient by $\tfrac34c-\tfrac14b$?

In a two-direction lagged mixed model fitted by ML in lme4, I found that an outcome composite overlapping with the predictor produced a spurious lagged-predictor effect. The overlapping composite was the mean of four items, $Y_\text{old} = (S + X)/4,$ where $X$ is the predictor item and $S$ is the sum of the other three. The clean composite was the mean of those three items, $Y_\text{new} = S/3,$ so $Y_\text{old} = \tfrac34Y_\text{new} + \tfrac14X.$ In the lagged model with the lagged outcome included, the coefficient on the lagged predictor was about $-0.105$ with the overlap and $-0.008$ without it. The lagged-outcome coefficient was $b = 0.394.$ The approximation $\tfrac34c - \tfrac14b,$ with $c = -0.008,$ gives ≈ $-0.105,$ which matches. These numbers come from a 0006 analysis ( $125$ participants, $40$ days of six prompts), with single-item predictor and outcome. Is this derivation correct in general? Is there a standard reference for how a shared item between predictor and outcome biases lagged and autoregressive coefficients?