Atlanta Braves First-Place Eliminiation Scenario
Atlanta must win 14 more games (66 total) to avoid elimination from first place.
Here is a scenario in which Atlanta can finish in first place with 66 wins.

To construct the scenario, we start with the current standings in the division:
Team             W	L	GL	PCT	GB
New York         55	49	58	0.529	0.0
Philadelphia     52	53	57	0.495	3.5
Atlanta          52	54	56	0.491	4.0
Washington       49	56	57	0.467	6.5
Miami            44	61	57	0.419	11.5

Next, we show how the remaining series of games plays out in the scenario:

Atlanta wins 1 games against Miami and loses 5.
Atlanta wins 0 games against New York and loses 3.
Atlanta wins 3 games against Philadelphia and loses 0.
Atlanta wins 9 games against Washington and loses 0.
Atlanta wins 0 games agains teams outside the division and loses 0.

Miami wins 5 games against Atlanta and loses 1.
Miami wins 7 games against New York and loses 7.
Miami wins 6 games against Philadelphia and loses 0.
Miami wins 4 games against Washington and loses 5.
Miami wins 0 games agains teams outside the division and loses 0.

New York wins 3 games against Atlanta and loses 0.
New York wins 7 games against Miami and loses 7.
New York wins 0 games against Philadelphia and loses 6.
New York wins 0 games against Washington and loses 11.
New York wins 0 games agains teams outside the division and loses 0.

Philadelphia wins 0 games against Atlanta and loses 3.
Philadelphia wins 0 games against Miami and loses 6.
Philadelphia wins 6 games against New York and loses 0.
Philadelphia wins 7 games against Washington and loses 0.
Philadelphia wins 0 games agains teams outside the division and loses 0.

Washington wins 0 games against Atlanta and loses 9.
Washington wins 5 games against Miami and loses 4.
Washington wins 11 games against New York and loses 0.
Washington wins 0 games against Philadelphia and loses 7.
Washington wins 0 games agains teams outside the division and loses 0.

Combining the current standings with the wins and losses from the scenario results in the following final standings:
Team             W	L	PCT	GB
Atlanta          66	96	0.407	0.0
Miami            66	96	0.407	0.0
New York         65	97	0.401	1.0
Philadelphia     65	97	0.401	1.0
Washington       65	97	0.401	1.0


Next, we explain why Atlanta cannot win the division with fewer than 66 wins.

There are a total of 0 games left in the following series of games between teams in the National League East:

For each team listed above, the table below shows the current number of wins in the (W) column.
The next column shows the maximum number of its remaining games that each team can win assuming that
Atlanta finishes in first place with a total of 65 wins.

Team             W	Max Additional Wins
Total            0	0

Since the total of the Max Additional Wins column is less than 0, it is not possible for Atlanta to win the division with 65 (or fewer) total wins.