Is there a proof that tax lot optimizer is really optimal?

Their engine is the same steps that you would use if you had all your trades in a spreadsheet and knew that you needed to sell 172 shares today.

Remember that is only one of the methods they provide/allow:

• FIFO (first-in, first-out)
• LIFO (last-in, first-out)
• Highest cost
• Lowest cost
• Specific lot
• Tax efficient loss harvester
• Average cost

The section you quoted has to do with long positions, they have a slightly different approach for short positions.

They also have an important note:

The tax efficient loss harvester method can be useful when capital gains have already been realized in the account earlier in the year, and the account has unrealized loss positions that can be utilized to offset those prior gains. Please note: This method does not factor in the possibility that a lot sold via this method will cause a wash sale, and therefore disallow the loss on the trade.

It doesn't know the future. It doesn't know what you have done or will do the rest of the year. They just have some software that will do the math for you.

There is no proof that it's optimal, because it's not optimal.

Suppose that the current date is December 16. There's a stock whose last trade price is $100, and you have the following tax lots:

• 100 shares purchased at $50.01 on January 1
• 100 shares purchased at $50 on December 1

Then suppose you want to sell 100 shares. TD Ameritrade's algorithm will see that these are all short-term gains, and so it will sell the lot with the highest purchase price: the shares purchased at $50.01 on January 1.

However, given that you want as much as possible of your realized gains to be long-term gains, it's probably better to sell the shares that you bought on December 1. That way, your remaining short-term unrealized gains will turn into long-term unrealized gains much sooner.