Proposition: Finite resolution and a noisy blend reference

For $0<\alpha <1,\alpha \ne 1/2$, the minimum separation between the four scalar mixture values is

Nearest-value decoding recovers both signs if scalar observation error is strictly less than ${\Delta }_{\alpha }/2$.

If independent Gaussian noise of standard deviation $v>0$ is added to each blend coordinate, define its one-coordinate class densities

where ${\varphi }_v$ is the centered Gaussian density. Then

For $v>0$, this accuracy is strictly less than one.

Proof. Sort the four noiseless scalar values and inspect adjacent gaps. Nearest-value decoding is correct below half the minimum gap. For Gaussian noise, condition on the visual sign and average the two distractor signs. Coordinate likelihood decisions are independent. Both densities are positive everywhere, so their overlap has positive integral and coordinate success is strictly below one.