This link was originally provided by T-Hopper to a discussion by Don Schlesinger and Brett Harris about the floating advantage, that was first discussed here and then became part of a thread on bjmath.com: the link from T-Hopper. The followup sections on bjmath.com have been retitled, but involve a very similar post to ML's.
Both the Harris/Schlesinger and ML posts made the very same mistake in reasoning, but let's dispell one point raised by ML first: there is no one way or convention on styles for proofs. ML's demand I follow his style is a demand that I follow his error in fact.
Both posts took the true count prediction of edge as fully representative of the edge in the subsets of an initial pack. This is simply not the case as differing populations are involved.
The true count predicts the edge for all possible packs that can have a given true count. For every true count, at any given specific number of cards left in a pack, there is a most probable pack composition. The mean edge of that composition is the mean of the edges of every possible such pack composition.
This relates to Theorem II as given by ML, which can be better stated as: the mean of a given pack is also the mean of the means of every subset of that pack.
But the compositions of a pack have a different distribution and composition, where Theorem II applies not just for the true count, and all possible packs with that TC, but to the subsets that each subsequent subset is drawn from. You are dealing with, instead of the set of all packs that can have a given true count, with a given number of cards left, the set of all subsets of prior subsets that can have a given true count with a given number of cards left.
That is an entirely different population with an entirely different set of applicable means. The true count is a less accurate measure of the edge than the actual composition of such subsets. Thus these are two different measures for predicting basic strategy edge that are topomorphic to each other as per standard Graham-Stokes equation tools for evaluating these different measures.
The bow effect results from the binomial theorem predictable decrease in the probability of subsets having means relating to the entire spectrum of possible true counts sets, relative to the set involving the available spectrum from the actual card subsets.
The best resolution of this is to look again at Theorems I and II, from ML. Theorem I is that the mean basic strategy edge is invariant with penetration in that any penetration, for beginning a round, can be transformed into any other penetration by simply cutting the starting pack to bring that point into play.
That is in conflict with Theorem II, as presented by ML with the assumption that the population of all possible true count subsets has the same means as the population of true count subsets that is available from the subsets of the original pack (from now on subsets_(TCG), and subsets_(TCS) respectfully).
The mean edges of subsets_(TCG) increase in that the mean of each TC, at each penetration point, for subsets_(TCG), does show an increase in edge, in that the mean composition is smaller in each case. But the composition of subsets_(TCS) shows a change from compositions from that mean composition of subsets_(TCG). And their edges thus differ as well.
Were ML's viewpoint to be true, Theorem I could not be true. Theorem I is true however if the mean edge of subsets_(TCS) is equal to the mean edge of the the largest subset_(TCG) of the original pack, where the original pack is the largest such subset for neutral pack subsets (ie for basic strategy the largest neutral pack subset is the starting pack). That requires some mechanism that is within the observed characteristics of the bow effect and results in the mean of subsets_TCS being less than the mean of subsets_TCG).
The alternative is the contradiction that rounds begun at various penetration points have equal means, if you examine the cards used to complete each round, and unequal means if you consider that each round is drawn from smaller and smaller subsets of the pack overall.
That mechanism is found in the general demonstration that the mean of subsets_(TCS) is less than the mean of subsets_(TCG) in ML's "proof" post. It is also found by simulation of the effects of skewing away from mean compositions using hi-lo as a simple example count. Trading aces for tens, and tens for aces, results in lower overall mean edges subsets_(TCS) for various true counts, but higher edges in true count ranges near zero, for packs composed of such skews.
The resolution is to be found in my postulate of a Theorem III, which I gave below in another thread, which is:
within an original pack of cards the probability of the appearence of subsets with mean compositons, and thus mean expectations, is reduced, and the probability of non-mean compostions enhanced, which have generally reduced expecations, such to reduce the expectations of such subsets to the same values as the similarly measured largest possible subsets of the original pack of cards.
The exception is the rare and statistically insignificant random occurences of subsets with mean compositions. This is of the same significance as the specific results of the binomial distribution vanishing at the limit which forms the normal distribution.
Since from ML it has become a matter of easy disposal of alternative ideas that don't match certain demands of certain posters for certain style formats I will leave a formal proof of this to others.
In summary:every form of claiming that the edge in blackjack, with a given true count, is the same as if the number of cards left were the number of cards started with, involves a conflict between obviously true Theorems I and II, except to postulate The Theorem III that I have presented
A further summary is my postulated Theorem III.
Still another point for summary is that with increased penetration, there is always a skewing away from the means of the original pack or larger pack subsets. This is true for both observed true counts and actual compositions. This is an obvious corollary of the binomial theorem.