Is your question why do prices exist? Or what is it precisely?
Do you not understand the LTV? Do you think it is wrong?
I've already explained how the "determination" of prices worked in my long post here >>656650
>But that was only possible because labor time value theory was violated in the first place.
No. You can't "violate" the LTV.
Reminder that Adam Smith and David Ricardo, among other important economists believed in a LTV, although in much cruder forms. Marx's LTV is logical conclusions of the previous flawed forms. If you think they dedicated their lives to the research of economics, working on the shoulders of giants, but some rando on the internet will BTFO them with REASON and LOGIC, you are probably too arrogant for your own good.
Following the temperature analogy, which is a physics model and not a social mechanics model, but whatever...
I am trying to tell you why and how the individual velocities oscillate around a predictable velocity. You are saying "but what if there were one particle who's velocity is 900 km/hr". To which we're responding "yes, it's possible, but it's highly fantastical/improbable. And even then, it doesn't trump the theory that temperature is a good indicator of expected particle velocity". To which you respond, "if your theory can't explain why this particle is going at 900 km/hr, then it is useless and must be discarded". The analogy being that price is velocity, temperature is value, and the particle in motion is the commodity.
Then you give the example of the single particle in a vacuum. You ask, what is the temperature of a single particle in a vacuum?. This doesn't make sense, nor is it useful, since temperature is defined as part of a system, of particles in relation to each other (without getting into quantum shit). The Boltzmann distribution is no longer a useful model. It doesn't mean that the Boltzmann distribution is now cancelled and should be ostracized. You simply need to know for what your model is useful and use it appropriately. In our topic at hand, I believe we have explained why the model doesn't apply to individual prices. It would help a lot if you read the first chapter of Capital, it explains it much better than I can, and in much more detail.
I'll try to explain anyways... This doesn't work because exchange value is measured in relation to other commodities, just as temperature is defined in the relation of particles as a whole. So, my 8 hours worth of corn are equivalent to your 8 hours worth of fish. This "8 hours worth of fish" presupposes that fish is something that is frequently sold and it's worth is known. This doesn't mean it took me 8 hours to harvest corn, or it took you 8 hours to fish. For all it matters, I could've found 8 hours worth of corn on the ground, and exchanged it for 8 hours worth of fish. Going back to the analogy, the commodities cannot have this "hourly value" if they are not already in a relation with all commodities in the market, just as particles in a system cannot have temperature, unless they are in a relation with all particles in it. The analogy is not perfect, of course. If I grab a liter of gas, and ask "how much energy is here" I could use the temperature to give a rough estimate of how much energy is in that enclosure, I do not need to know all the individual particle speeds. Similarly, 8 hours worth of corn, when taking into consideration the whole market, will have a rough standard exchange rate. Some might exchange corn very expensively, some very cheaply, but when taken as a whole and abstracted, a "commodity value" is established, socially.
If there's a surge in demand for whatever reason, perhaps corn becomes more valuable. In that case, the price could be above the "man hour value" it had originally. Here, the corn producer would be getting a lot of profit. That is fine, that does not break the LTV. This is simple bartering. When you get to capitalist commodity production, things start to get more stable. For example, the price of cereal at the supermarket is generally stable all year long. Or for example, when products ran out at the supermarket, most products retained their prices, even when they could have gouged them. Price gouging is harder in the capitalist generalized commodity production. Due to competition, you'd probably need capitalist collusion, which is real and does happen, despite being illegal. Or you'd need monopoly protection, such as patents, land, property rights, industry secrets, etc. Otherwise, it can be very hard to keep prices gouged.
Do note that this is a simplification. Not all man-hours are measured equally by society. You have to understand the basic model first before you start tacking on complexity.
It's nice that people are explaining shit with machines and shit, but I personally think it's counter productive, because you haven't understood the basic model yet and more complexity will not help your understanding. Read my post that tries to explain it with prices instead of labor hours. Hopefully it will give the idea of where this is all coming from. As long as you're willing to learn/teach in good faith, I'm willing to discuss. So feel free to ask any question of something you didn't understand, or any concept that is still unclear.