I was not aware that this is what defines gambling. And "no service produced" is certainly wrong by accepted economic theory - arbitrageurs provide a price discovery service for everyone; they get rewarded for exposing the inefficient prices, even though it is done through market mechanics rather than a specific customer.
OP appears to be a statistical arbitrageur - which is the same concept, except that it includes a shift in time or space (and incurs risk). You might not be interested in this price discovery service, but other people are paying for it with their wallet. (And it's mostly the market makers who pay for this with reduced profits)
> one trader's gain being another trader's loss (relative to market returns).
That's not true in investing in general - when shares have time to appreciate or depreciate, it is definitely not a zero sum game. Everyone can win, or everyone can lose, or anything in between (it all depends on your time range, and your measure of loss or profit. The "non-zero-sum" element arrives partly from companies using operating profit to buy back their own shares).
> Because there's a commission on trades, and because you pay taxes on net gains but your minimum tax is zero, high frequency trading by its very nature must a loss for most players.
That's only true if all players are hf players. If there is sufficient non-HF activity, then the zero-sum argument does not hold.
(I'm not saying that it's not a good approximation - in most time scales, in most scenarios, it is - but it is not the mathematical truth you imply it is)
Again, it's a great approximation most of time and over most time periods and asset classes, but it is NOT axiomatic in the way most people believe it is.
Remember: as long as there is a way to inject or withdraw more capital into the system (through whatever asset class, as they are all interconnected), the sum is not identically zero.
Just assume one of the stocks is a gold mining company that works efficiently. The share value rises, and the shares are redeemable for the gold, without anyone having to lose anything (except mother earth)