At least some work must have started and project must be funded and on track
Currently the bridges we have are
San Mateo - Hayward bridge, route 92, connecting Foster City/San Mateo to Hayward in the central bay
Dumbarton Bridge - Connecting East Palo Alto to the south east bay
San Francisco - Oakland Bay Bridge
Golden Gate bridge - Connecting SF to Marin County in the north
Richmond Bridge connecting the Marin town of San Quentin to Point Richmond in west oakland
This claim is that there will be a new one.
EDIT: 9/4: The inoperative and only partially complete Dunbarton Rail Bridge, which hasn't been in service since 1982, is not counted as a current bridge. So if it or some variant of it begins to be constructed according to the criteria above, then this would qualify as a new bridge. --END EDIT
The limit of the bay for the purpose of the question will be everything below Richmond bridge. So it counts if any point in Marin is connected to east bay or SF Peninsula land, or any point in the peninsula is connected to any part of the east bay, below the current Richmond bridge.
Stepping across islands is fine and doesn't change the situation. For example, the SF-Oakland Bay bridge steps over Yerba Buena island but it's still considered a bridge. For our purposes, as long as a span of at least 1000 feet is new and also the connection satisfies the above, it can count, or if it's entirely new between Marin and SF Peninsula, it counts even if shorter.
Tunnels don't count, nor does traveling through the air with mechanisms such as a controlled catapult. Train only, pedestrian only, car only, hyperloop, biking only, etc can all count as long as they travel above the water and it's conceivable that someone climbing/walking could progress across them (even slowly or requiring great skill)
If a current bridge is destroyed & rebuilt, that would count. If it's improved/expanded/retrofitted, it doesn't count.
There’s some discussion of rehabilitating the rail bridge south of San Mateo bridge - it wouldn’t be new construction but I think it would need work to get trains to run on it again. Facebook was involved in this effort and then withdrew. Assuming that would not count.
@KevinBurke Based on satellite maps, it looks pretty incomplete now. See image:
From this viewpoint, it seems to me that completing this and making it work again would count as a new bridge. From Wikipedia it looks like it hasn't been operational in 41 years. I will edit description to include this information.