Page 27: Vince(vi) and independently Thorpv(ii) provide a solution that satisfies the Kelly Criterion for the continuous finance case, often quoted in the financial community to the effect that “f should equal the expected excess return of the strategy divided by the expected variance of the excess return:”

f = (m-r) / s^2

so it's Sharpe with variance instead of standard deviation in the denominator, correct?

An intuitive way to think about this is that Sharpe depends on the specific horizon you're using. E.g. annualized Sharpe will be sqrt(252) larger than daily Sharpe. It would not make sense to change the Kelly criterion based on a substitution of variables. In contrast variance, like returns, scales linearly with time horizon. Therefore the variance ratio is invariant to the time horizon.

Edward Thorpe and Ralph Vince both conclude that the Kelly Criterion in the continuous case is excess returns divided by variance, which is pretty close to the Sharpe Ratio, correct?

Asking to understand better, not to be combative. Your comment made it seem like that formula is way off.