Rather than either shares based on Bitcoin snapshot or shares to miners, I suggest a rewards of Truthcoin in proportion to total coin (Bitcoin + Truthcoin) time participation (which is proportional to Bitcoins bet on the wager). Having consistent rules independent of time or block number would avoid accusations of preming or unfair early adopter rules. Opportunities for participation will increase as the gambling increases, so although early adoptiong will have advantages, early adoption before gamblers become aware of the network will have only marginal advantage. I imagine initial Truthcoin shares would be sent to the Truthcoin address corresponding to the participating Bitcoin address. Voting/participating both your Bitcoin and Truthcoin shares would be easy because they are the same address on different networks, thus have the same keys and the same signature, making Truthcoin share payouts easy to compute. Bitcoin payouts of commission earning to Truthcoin shareholders would be accomplished simply by sending Bitcoin to the Bitcoin address
identical to the Truthcoin address. Mining rewards likewise would be sent to the Bitcoin address that the Truthcoin miner indicates - making Truthcoin among the first mergemining operation to pay miners in Bitcoin.
So we get the recursive formulas. If
coin@n = (Bitcoin@n +Truthcoin stake @n), and
participation@n = (+/-)wagers@n, and
Truthcoin@(n+1) = Truthcoin@n + (coin@n)*(participation@n)
Truthcoin@(n+1) = Truthcoin@n + (B@n +Truthcoin stake @n)*((+/-)wagers@n)
Thus for the special case of constant wagers (wagers@n = w for all n = 0,N, otherwise wagers@n=0 for n<0) for each block, with no share transfers at a particular address so Bitcoin held @ n = constant B:
Truthcoin @ 1 = B*w
Truthcoin @ 2 = B*w + (B + Truthcoin@1)*w
= B*w + (Bitcoin + Bitcoin*w)*w = 2*B*w + B*w^2
Truthcoin @ N = N*B*w + (N-1)Bitcoin*w^2 + ... + 2*B*w^(N-1) + B*w^N
Since I can't do infinite series as well as I could in college, I will now resort to proof by Wolfram:http://www.wolframalpha.com/input/?i=B*sum+(N*w%5E(N%2B1-n)),+n%3D1+to+N
Truthcoin @ N = B*N*w*(w^(N-1))/(w-1)
(lim n->infinity)(w/(w-1)) = 1
Thus for large w, Truthcoin @ N ~ B*N*(w-1)^N, so "exponential" growth
For small wagers, w = 0.5, Truthcoin @ N ~ N. (Exercise for the reader - I just verified this with a calculator at a few values of N.)
So basically participation has diminishing returns of shares with time unless wagers become at least 0.5 Bitcoin per block, which is where share growth linearly increases with time. The exact threshold can be changed by throwing in some adjustments, but accusations of premining shares become provably false because participation becomes increasingly valuable not only with persistence (large N) but even more so with large wagers. You can't really premine or even get an unwarranted early adopter advantage as long as wagers are still increasing and your Bitcoin value is constant.
So what is the incentive for early adoption then? Because the assumption of constant Bitcoin is false. Early adopters will get dividends in Bitcoin, which increases their Bitcoin share, giving them a further advantage over late adopters. This only works though if they have significant savings, that is they only spend a fraction of their income. If they plan to spend all their earnings, they will asymptotically approach the interest share of late adopters with the same Bitcoin holding. Also, Bitcoin is supposedly deflationary against fiat currency so people who are late adopters to both Bitcoin and Truthcoin are at a double disadvantage.
I believe this type of allocation will easily make the Truthcoin and Bitcoin networks self-reinforcing and make betrayal dis-incentivized. The time-invariance of rules is very important because it makes altcoins or alt-side-chains uncompetitive against an established network unless they have real added economic value by specializing. If rules have some explicit time dependency or "premining"/"preallocation" then a community of sufficiently wealthy investors might be motivate to make an independent network rather than sink capital into acquiring interest in an existing network. Early announcement doesn't eliminate that possibility, it only reduces it in the short term.
Additionally, if there are transaction fees for converting Bitcoin*participation->Truthcoin, it will encourage people to hold their Bitcoins in a single output. This will unburden the Bitcoin blockchain of unnessary transactions and untrimmable junk from multiple address/multiple output wallets. Outputs that are economically combinable into one output will get combined by Truthcoin shareholders seeking more Truthcoin, rather than waiting indefinitely long for "free" transaction eligibility. And this doesn't just happen once, but every block that pay Truthcoin dividends will incentives a transaction to combine multiple Bitcoin outputs into single Bitcoin outputs. Truthcoin will add a time value and transaction fees to Bitcoin beyond whatever time value and transaction fees it already has. This will benefit miners greatly - even if they irrationally don't participate in merge mining.