56 days is not much time to get beans... but most of the bush beans should make it. All the beans listed are bush beans, except "Broad Windsor" (fava), "Scarlet Runner" (pole runner bean) and "Blue Lake" (pole snap bean).

It will be difficult to meet all the criteria listed with a single variety. "Scarlet Runner" is probably the best candidate for tallest stalk & largest leaves, but it will form beans later than the bush varieties... and will not form beans at all without bees. "Blue Lake" would be my next best choice, and it will form pods indoors.

Conditions are a major factor; when & where will the beans be grown? Provided that you must have a "bean stalk", you are restricted to the pole varieties, and to the fava, which will form a tall bush. The fava is unlikely to form pods in the limited time allowed.

I would grow a Sword Bean. But it takes about four months to form pods. The pods are a foot long, 3 inches wide, and 5/8 inch thick and the plants can reach 30 feet tall.

You have listed several varieties that have limited potential to reach your objective. Blue Lake is the only pole bean in the list. Here are some varieties that are more likely to please.

Emerite - pole bean with exceptional sweet flavor. It is a reasonably rampant grower easily reaching 12 feet high in 80 to 90 days.

Musica - a very rampant pole bean that will reach 16 to 20 feet tall under optimum conditions. It makes beans 11 inches long by 1 inch wide by 3/8 inch thick.

You ask about growing them the fastest and use of chemicals. Beans are rampant growers in high quality organic soil. The best I've seen is composted rabbit manure. If you want to grow really tall bean plants, then you should get a pot that will hold a minimum of 15 gallons of soil and fill it as much as possible with composted manure (2 parts) and peat moss (1 part). This mix will be low on nitrogen so you will have to add a nitrate source. I use chicken manure for the extra nitrogen because I have it available. Given circumstances, you might be better off getting one of the commercial fertilizers like Miracle Grow.

Your major limiting factor will be light exposure. Beans grow in direct proportion to the amount of light they receive. If you want to skunk the competition, get a good grow light and put it on the plant 24 hours a day.

Last suggestion is to plant at least 3 beans. At least one of them will grow a bit faster than the others. You can then pull out the slower growing plants and keep only the largest one.

DarJones

I should also mention that you could grow:
Blue Lake
Scarlet Runner
Broad Windsor

as your three varieties. This is three different species and will give a very different perspective on plant phenotype and production potential.

The Blue Lake and Scarlet Runner are rampant climbers so you will have to give them something to grow on like a bamboo pole.

DarJones