This is a reason I haven't made active use of DCA. But DCA doesn't give up much, either, so I'm fine with "passively" engaging in DCA-like behavior by haphazardly dumping spare money into index funds after min("significant" drop, time limit) instead of purely maximizing time in market.

The time-weighted average price was $37/share.

Yes, that exactly. It's a psychological trick (if you have all the money upfront) and not a financially sound one.

Math citation needed - the noise function you're hypothesizing could just as easily drive the average up as down, unless you have a reason to think it should go down, in which case you're timing the market and there's a better future time to go 100%.

If you commit to the strategy (buy monthly for 10 months) then you're just choosing an unknown price over a known price.

And if you don't commit to the strategy, then you're actively trading and not DCA'ing.

Another thought: if the thesis is that the price will go up, then whatever sum you have and want to invest, should you invest it all now - given that the price is supposed to go up?

A financial hedge is, generally, something which reduces risk or improves on the risk:reward ratio in a mathematically demonstrable way. Generally this means bundling uncorrelated (or anticorrelated) assets. This concept won a Nobel for Econ in 1990(?), and is the foundation of modern portfolio theory.

Using "hedge" in a more colloquial way, it can mean "a way to protect yourself from an unexpected poor outcome". But by that definition this also fails, since you're just as vulnerable to outlier events, but the nature of those outlier events is different (and perhaps less intuitive).