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If the 'far' switch closed then wires blue and red at the timer are connected. Far switch open then blue and red are not connected. The only way the circuit made sense to me (eventually) was that the timer was sensing whether red-blue were open or closed. It could use the battery to sense open/closed. And that is consistent with the other 2 items I mentioned, which don't make sense if the 'far' switch is being used in a power circuit.
It doesn't make sense that the 'far' switch is in a power circuit, so there must be another answer. IMHO the power circuit is the timer switches black-blue on or off. (In a single switch configuration that is what the timer does.)
Alternate ideas? There is always the gnome theory of cause and effect (everything is caused by gnomes).