Three Filters, Four Emails

A spam filter is a single neuron with a sigmoid on its output:

$z = w_1x_1 + w_2x_2 + b, \qquad p = \sigma(z) = \frac{1}{1+e^{-z}}$

Here $x_1$ is a suspicious-words score and $x_2$ is a known-sender score, each between $0$ and $1$ . An email is flagged as spam when $p \ge 0.5$ and let through otherwise.

The filter in production is P. A retraining run has produced two candidates to replace it, Q and R:

filter	$w_1$	$w_2$	$b$
P (production)	$2$	$-1$	$-1$
Q	$6$	$-3$	$-3$
R	$6$	$-3$	$-1$

On today's four emails, the three filters output these values of $p$ :

email	$x_1$	$x_2$	P	Q	R
A	$1$	$0$	$0.731$	$0.953$	$0.993$
B	$0.2$	$0.6$	$0.231$	$0.027$	$0.168$
C	$0.9$	$0.2$	$0.646$	$0.858$	$0.978$
D	$0.1$	$1$	$0.142$	$0.004$	$0.032$

All three flag A and C as spam and let B and D through.

The team cares about one thing only. The replacement must make the same spam / not-spam call as P on every email it will ever see, whatever that email's two scores are.

Which of the candidates is guaranteed to do that?

Three Filters, Four Emails

Hints