What Floats---and why ML does this

P1 The bow-effect exists without any significant change in basic strategy expectation with penetration.

P2 While highly correlated to pack composition, the true count is seperate parameter from pack composition.

P3 As a seperate parameter the true count has bow effects with different ones for pack composition.

Theorem I: the basic strategy edge of a shuffled pack does not change with penetration. This is demonstrated by being able to to cut the pack and tranform any penetration (by cutting the pack) into any other.

Theorem II: Theorem I applies to the total remainder subsets of any pack in that the penetration of a remainder can be transformed as well.

The bow-effect is recognized to be the rise in mean edge versus true count in middle count ranges and the drop in more extreme true count ranges. This bending is also recognized to increase with penetration; as the amplitude of the TC grows with penetration, more bending is required.

Theorem III is that the mean composition of a pack has the same expected value as the mean of the expected values of all the variations of that pack with the same parameters. It is by this Theorem that true counts are evaluated by examining the the mean expected value of mean compositions for a true count.

Theorem IV is a restatement of the distribution about prior outcomes theorem of general statistics for blackjack: distribution about any result is conserved from the original distribution about prior sample points. For blackjack this requires that the distribution from a prior point, or mean is **not replaced by a distribution about a new, most probable, mean for the same parameters**. The mean pack composition of an initial pack of 8 decks is not replaced by the meann pack composition of 3 decks at the 3 decks left penetration point. Otherwise Theorem I would not be true as the mean expecation of shuffling 3 decks together is higher than that of 8 decks shuffled together. The subsets of all the cards used per round total the original pack---not some smaller pack; 8 decks has subsets totalling 8 decks, not subsets used through 3 decks totaling 3 decks within an 8 deck original pack!

Theorem IV also applies to Theorem II as well; at each penetration level a true count has a most probable configuration, but the distribution conserved from that of the original pack and all prior composition subsets. So the most probable composition for each TC, at each penetration, **differs within a pack**, from what is most probable from the specified TC and penetration point **alone**.

The bow-effects arrise from the mean composition predicted by the true count at each penetration level differing from the edge prediction from the actual pack composition, from Theorem IV. One restriction imposed by the Graham-Stokes equation is that the best adjustment for a parameter, like the true count, used to measure a population as it is sampled, must result in errors that cancel out over the measurement range.

Since the true count overestimates edge at more extreme TCs, it must underestimate edge at more central TCs, near zero as well.

The problem is that some take the intersection of the most probable zero count pack with the most overall probable pack, at each penetration point, to imply that the basic strategy edge floats along with the true count. Theorem III is used without regard to Theorem IV or the fact that two population subsets are involved. A pack with 4/13th tens, and 1/13th all other cards, is the mean composition for **both** the most probable TC=Zero compositions and **all** compositions from a complete pack.

**The variations conserved by Theorem IV, for all true counts, keeps the basic strategy edge invariant with penetration, while the variations that are excluded by having an actual TC=0 leave the variations that involve and cause the bow-effect.**

This post had to be made in this informal fashion in that the conventional notation, demanded by ML, does not not allow expression of the exclusion effects of stating the TC is actually zero--with similar for other near zero TCs--in plain text.

Cleaverly or accidently (both are somewhat embarising) ML was demanding that a formal style be used that either would involve an extended set of special symbols this board does not allow in posts, or a set of alternative abreaviations that would have made this post look like a badly written classified ad with equations glopped onto...say for example a personals ad with bizzare abreviations.

This demand of his is straight out of Orwell's newspeak in that he either knew that this would prevent opposing his errors, or that he himself was trapped by his imposed limits.

Such is all too common with all of the clones, and needs to be opposed!