In order to fairly distribute the earth’s natural resources we give every man, woman, and child a token representing one share in the earth. All resources can then be auctioned off to the highest bidder (using shares as money). Proceeds of the sales are paid to each proportional to their shares. Through this process all the world's resources can be fairly distributed.
What's the point of an auction if everybody only has 1 share to bid? The quoted text doesn't mention the time component you later expand the example to. The only way an auction makes any sense is people must have different amounts of resources to bid. The quoted text doesn't provide for that option.
OK, so move forward in time several days. Presumably people would begin trading among themselves for the things they want using their shares. In fact it will begin on the first day, leaving some people with shares and others with none before the second day arrives.
Perhaps it's due to your writing style, but I had difficulty following along at times as I read. You seem to make assumptions about this "genesis" stage, and based on several comments I see here others have noticed this also.
I find this topic interesting and I'm grateful to you for tackling it. I may have difficulty fully understanding the nuances of the problem but that is feedback to you to find a better way to explain it.
Shares are infinitely divisible. So you can bid .0001 share or .000000001 shares.
Ah yes, good point. However, such a concept is a detail of implementation that seems to go beyond the simple example stated. Funny tho that divisibility of shares didn't even occur to me.