This preprint develops an axiomatic framework for continuous-time linear causal measurement and analyzes optimal memory selection in the narrowband Gaussian–Markovian regime. The first part shows that any linear, bounded, time-translation-invariant, causal observation operator must be a retarded convolution with an L¹ kernel, making a standard linear-systems result explicit in a measurement-oriented setting. The second part studies continuous-time Kalman–Bucy filtering of a narrowband signal with Ornstein–Uhlenbeck phase noise and proves that the optimal memory depth τ* obeys an inverse coherence scaling τ* ∝ 1/Δω, where the dimensionless product τ*Δω lies within an order-unity interval across broad parameter ranges. The work formulates clear falsification criteria (scaling exponent, interior optimum, adaptive vs. fixed kernels) and provides both synthetic and circuit-QED experimental designs suitable for testing memory–bandwidth coordination. Claims are restricted to the narrowband Gaussian–Markovian setting, and no universality beyond this regime is asserted.