I think it's a mistake to apply ideas from the mathematics of functions (other than statistical functions) to "trendlines" observable in financial market data. The idea of local maxima, for instance, suggests to most people who've heard the term that you could find a maximum by finding roots of a derivative function, the proof of which is based on the dynamics of the underlying processes being discernable or model-able as a function.

To convey the intuition I'm trying to describe as quickly as possible, I think you're thinking of market data being useful to help you find the Old Faithful geyser, and once you find it you just need to hang around and scoop up the reliable expulsions. Coupled with the even worse idea that you can see it where nobody else can :) To continue the analogy, I think it's not Old Faithful you found, but a solar flare on the surface of the sun, and while waiting for the next one you are more likely to get scorched.

We aren't going to debate technical analysis here (or resolve it), the point I'm making overall is that people persistently believe that financial markets are leaving money lying around, that these people can see where nobody else can see it, and that is easy to bend over and pick up. I'm trying to use some fairly simple logic and explanation to say, "it's not that easy; if it was, how can you think an entire financial industry somehow can't see what you can see? Wall street hires people just like you."

I'm not sure where you're going with this, but it's digressed far from what I was talking about. To paraphrase your argument, you're talking about assuming convexity, which is a big deal in financial modeling.

If convexity isn't assumed, you're right back to my original point.

the observation that "the markets are too short term focused" is not a useful starting point for a moneymaking strategy because it can be shown with a simple explanation to a layperson that it isn't even an observation, it's a myth... is where I started and where I'm going.

I don't think using terms such as convexity explains anything to anybody, but if it indicates to you that the market is too short term focused, you should just print money with a derivative security that prefers companies that lose money in the short term because that will with no effort uncover all the long term strategies...