No, by digging the mine, you alter the land. You thereby gain ownership of the land. The ore, both in the vein and dug up, is part of that land
In other words, your argument that homesteading is just a way to let you keep the product of your labor is false.
Care to explain how the first person gets most of the reward? Even if he rents it, He won't be able to charge more than the renter is willing to pay, and if the area is rich in ore, prospective renters may value that already dug hole less than the money he's charging for it, and go stake their own claim.
Once all land is owned, the only limit to what renters are "willing" to pay is what they
can pay.
Land doesn't work like the products of human labor. If there was a sudden demand for, say, ice cream, the price increase would be temporary. Eventually, the increased demand would be balanced out by increased production, or if it wasn't people would look for alternatives. There is no way to increase production of land, and there are no alternatives. Land value tends upwards, absorbing all the new wealth gained through increased productivity.
In a truly free market, unskilled labor can only receive a mere subsistence after rent. We don't need to speak in hypotheticals, this exact situation happens all the time throughout the developing world and has happened through history. I hate to speak in favor of regulation, but that's the only reason why we see it less in the west.
So drawing a line on a map is less arbitrary? hmm, no. And your system has the same problem. How much land do you charge him for, if he breaks a trail trough the woods? All the woods he's thus altered? (ask FirstAscent exactly how) Just the trail? some area (line of sight?) around the trail?
I didn't say it was less arbitrary, I said it was equally arbitrary.
You don't generally charge him after the fact, he would reserve a section to work on ahead of time. If he blazes the trail without renting it, then it would be a special situation to be handled on a case-by-case basis. You'd try to charge him by the amount he reduced the land's value. If he increased it, then nothing.
You're aware that most land in the US was homesteaded? And I've explained a transition like this before. Gov't land is up for grabs, with the current users of it getting first "dibs" (the President would get some sweet digs), Private land, unless clearly stolen is assumed to be legitimately gained. Do some people get screwed? Maybe. But a lot fewer than the current system, and fewer than yours, too.
The White House is used by many more people than the President, how do you decide who gets priority.
Who exactly gets screwed with geoism?
Well, I never had my neighbor's yard, but I did legitimately gain my house. I do have it. Their claim conflicts with mine. That's why I care.
It doesn't though. You bought it with the full understanding that it would fall under the jurisdiction of the government. If the seller led you to believe it wouldn't, then your beef is with them. You have the right to occupy, modify, and resell that land, but the government's legal sovereignty over it was recognized since long before you were even born. You never had sovereignty, so by your logic you are wrong to demand it.