Sorry about no formal bibliography in this version, but see, How to Ace the Rest of Calculus, and How to Ace Statistics, and their references for such information.
Cut the pack any penetration level can become any other penetration level. The mean of your basic strategy results, and the distribution of such results, does not change with penetration.
Your True Count does change with penetration as you divide by the number of decks left getting smaller and there is cumulative drives away from a zero running count. Its distribution does change with penetration. It is less precise than an exact calculation of your edge based on the current card value densities of the remaining pack at all times.
The card value densities involve a closed space and predictions of edge are manifold terms in that your measures are not giving you any information about the order of the cards (shuffle tracking can provide part of that while banger side spotting can provide absolute information on that). Your diagnosis of edge is a manifold measure where the mean edge predicted by your true count is the same mean as the overall basic strategy edge which absolutely (within the limits of statistical prediction of the binomial theorem) does not change. Just keep in mind that the cards don't know how much you bet---unfortunately the pit does however. Your true count predicts the changes in edge with penetration imperfectly compared to exact density analysis once again.
Your true count and an exact prediction of edge are topomorphic to each other. Graham-Stokes theory states that in this case (same means and boundaries are involved) that their differences must result in the integral of their differences summing to zero over the possible space they measure. The curve of the true count prediction of versus the precise actual prediction of edge must be one where equal areas of predicting too much, and too little, edge are bounded.
The curve that meets these requirements is a bow that bends more with increased penetration, for the true count versus actual edge. The basic strategy mean still does not change with penetration. All claims that this mean floats are based on an appeal to an absolute reversion to the mean density as penetration results in smaller subpacks. The defect in this claim is that the means involved are far from equal.
The Proof of Strong and Weak Means.
The complete pack has a mean that matches the mean of the mean of every subpack that can drawn from that initial pack. This is an absolute law of distributions where Abdul Jalib's True Count Theorem is a more limited version of this. The means involved for perfectly neutral subpacks, and perfectly mean subpacks for a given true count, have massively weaker probabilities for subpacks that are smaller than the largest subpack possible for neutral subpacks, or the largest mean subpack where a given true count is possible. Even when such perfectly mean subpacks occur, they are massively more probable with subpacks that form closest to the maximum size for such subpacks, where the difference between them and the largest possible mean subpacks is massively minimal.
That is absolutely the only overall floating advantage involved. This objection is further diminished by demonstrating a mechanism by which the bow-effect does occur with smaller subpacks which also shows less mean expectation than perfect mean subpacks.
By simple combinatorial analysis, using the hi-lo count--the best combination of simplicity and basic strategy edge prediction--you can show this by comparing how trading aces for tens, but still coming to the same true count, changes edges.
For every original pack of cards, a given percentage exhange of aces for tens, and vice-a-versa, is less probable, for larger subpacks than smaller subpacks. Such is more probable with more penetration. As in the proof of strong and weak means given above, the means are demonstrated to be less certain with increased penetration.
For the demonstration of bow effects with such trades, every such trade lowers the overall edge for all such similarly skewed true counts by reducing the chance of blackjacks. Every such trade also either raises ace/ace splitting probabilites, or raises the probability of holding 10/10. All such hands show reduced edges in more extreme true counts, and increased edges in more middle true counts. Use the combinatorial analyser in Proffesional Blackjack Analyser, or T-hopper's posted source code on bjmath.com's computer sims board.
The objections to the basic strategy not floating with penetration are reduced to extremely rare exceptions that have only limited changes in edge with penetration that are below statistical significance, that are instead examples of macro-dechoherence. These perfect mean effects are so improbable and otherwise limited, that they are below basic noise levels in statistics, yet their exclusion still has demonstrable effects of their removal.
So all is not lost in the false appeal of assuming a perfect reversion to the mean. In fact this does demonstrate the solution of the classic P/not P problem in that even when so improbable as to demonstrate not being able to observe such rounds at limit, as initial pack size approaches infinity, the probability of such a mean is shown to vanish, by the central limit theorem, even faster. A seperation manifold forms which exists without needing to actually observe that seperation manifold. That serperation manifold forms at that limit without ever needing to show it is dividing any set element,or subpack, from any other. The P/not P problem only requires showing set P is different from set not P, without any element of P being shown to be ever a part of set not P.
So the assumption of a reversion to absolute means has proven to be both a clear, but a suprisingly useful error!
ETF's assertion is absolutely correct that an absolute evaluation of the edge for a given subpack will not result in any bow-effect in that one and only one perfect evaluation curve can be plotted on any one manifold. Any such perfect evaluation is singular. Every additional set of gains added to basic strategy must reduce the bow-effect as well, as every such gain is intermediate between the limits of an initial linear prediction of basic strategy edge, and such an absolute evaluation.
T-Hopper is absolutely correct as well that every addtion in playing gains is more when added to also varrying you bets with the true count, than the flat bet measure of playing gains. Playing gains add a dimension to evaluating the manifold edge from the subspace set of every subpack that is more than the bivariate addition of such gains. You are always expanding the volume of that space, and not just adding gains linearly.
But neither T-Hopper's, or ETF's, assertions can be proven without the assertions I have made above.
But the recomended tactic given in opposition to attempting to exploit the bow-effect by adding more playing gains, which is to modify your betting ramps, can meet the test of Graham-Stokes theory. The betting ramp advice is based on assuming basic strategy is used. Playing gains reduce the bow-effect always--see above--and reduce the predictability of such ramp changes in that the curve of playing gains with penetration varries from the curve of betting gains possible with the true count with penetration. Graham-Stokes theory is that errors in predicted accuracy balance out across the range of such predictions. It is slippery when new interactions are added or a prediction model cannot be fully tested. That is why acceptance or rejection of any prediction model in economics is so often rationalized, using the G-S combined theory, in that seldom is an economics model every used against a total, or even near total, range of economic conditions. Not attempting to exploit the bow-effects by moving your ramp, but by attempting to maximize PEs instead, appears, at least IMHO, to involve far more predictable ingreadients, that meets the G-S theory tests far better.
To move from this to P/not P proof is thus a simple exercize in applying the Reinzi result mentioned in Peter Griffin's Theory of Blackjack, to the formation of that manifold mentioned above, showing division without movement of any P/not P element between such classes.
Of give me my damn Nobel :-)!
