I got into an argument with somebody about the following hypothetical situation:

You have $500,000 to invest as a lump sum in a low cost S&P 500 index fund (not dollar cost averaged, but invested entirely in 1 day at closing price). This could be a fund like FXAIX or VFIAX.

For long term investing (let's pretend 25 years), is it statistically more likely to have a higher ending portfolio balance if I invest it all today regardless of market performance, or wait until the next down day?

I argue that today/now is better because it's impossible to time the market and a waiting until a down day could lead to further lost potential gains. My friend argues that waiting until a down day increases the likelihood of getting a better value/price on your investment.

Has anybody run a simulation like this using past market performance or have mathematical evidence that one strategy is better than the other?

Edit: Can anybody prove with math/statistics/data that one method has higher probability of being better than the other?