Given the count and number of cards left there is a pack composition that most perfectly matches that information, and has a mean that is the mean of all the possible packs with that information applying, but---
---it is also definite that that composition is not representative for the compositions of those packs that actually have such numbers.
Contradiction?
Actually not. In all such cases the compositions are known to be different from the perfectly mean composition, and each departure from that composition is the starting point of all the distributions of compositions that can result. The mean composition covers all, but is known to be off for any given actual pack.
That is why I based my proof on the True Count Theorem, where when a pack is not defined by the most probable composition given only your main count(s) and the cards left, you have to add various side counts, where the true count of each added side count is also subject to the TCT, and thus limits further distribution. You are increasingly pulled away from the mean most perfect compositions, making them: representative for all possible packs given the count and cards left, but not representative for any given initial pack dealt to that number of cards left and having that count.
This can alternatively be illustrated by using Griffin's Unit Normal Loss Integral tables from The Theory of Blackjack. Every starting pack has an expected average departure from any prior mean that is basically a function of the number of cards from the start of the pack to the current penetration in that pack. This gives another model for what I described more simply in my proof Blame Woggy etc. Every such interval of pack penetration gives you a prediction on deviation from such perfect compositions, where the apparent contradiction above, that a composition can be simulaniously representative of all packs given the count and cards left, but not representative of all probable packs given the number of cards initially, and the count, and the cards left.
What ML discusses is a false definition that is so false his errors in this must be deliberate. He omits how, see the chart in TOB on the spectrum of opportunity given different sized initial packs, having X cards left out of 8 decks initially, is different from having X cards left out of 2 decks initially. If you apply the formula from Thorp and Walden's paper on the spectrum of opportunity from counting, (see bjmath.com) the difference in the opportunity is closely related to the difference in basic strategy numbers, where the effects in being pulled from the mean composition(s) exactly cancel any gains from the mean composition(s) resembling initially smaller packs. It is a bit advanced to post on th free pages and the usual suspects would lock out any posting on this on the closed boards.
But the best retort to this is still my proof with perhaps this addition for ML:
As you require new counts to define the actual composition the actual compositions more closely resemble packs relating to the extreme subsets for single denomination counts.
This would be added to the proof section for Postulate IV for my proof. It would show how bow effects are amplified not just by having more extreme true counts overall, but also are amplified for the side counts (as per my proof) being driven into more extremes as well.
But ML's view is that the definition of mean composition, derived from representing all possible packs given count and cards left, is absolute, and leads of course to the absolute absurdity that he keeps posting.
At this point I have to leave the absurd to the absurd. There is no convincing the clones. Alas this is not the first time, or the first area of expertise, where I have been ahead of the curve and smitten by the blows of those whose mental abilities required the curve be instituted in the first place. I simply must refuse to play or post at their level of understanding, or refusal to understand.
