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The Story of The Unit Simulacrum and Whole PI 
© 20042010 The names “The Bottom Up Solution”, “The Unit Simulacrum”, “Whole PI” and all equations, figures and schemes are copyrighted in 20042010 by William Craig Byrdwell. No portion of the equations, figures or schemes may be used without acknowledging the copyright holder, William Craig Byrdwell. or Byrdwell.com 
I continued working on the equations. I continued to try to find systems where I could apply the simulacrum math. I have a note at the beginning of Pad 3 : “Golden Equation applied to Spin States”. In chemistry, spin states are often represented as +1/2 and 1/2 . I represented the +1/2 state as being Case 1, and the 1/2 spin state as being Case 2. I operated ½ in the Simplified Unit Simulacrum. Then this was compared to probability, so that the simulacrum sum of probability was 1 + ½ . If A is ½ or can vary from 0 £ A £ 1, and B is 1, the (A/B) ratio would be less than 1, and would be limited to operating from 0 ® 1. Case 2 would be when B can vary from 0 £ B £ 1, and so B is limited to operating from 0 ® 1, based on observing the ratio (A/B), and the Case 2 answer is found as (1/(A/B)). The construct for any A and B have three inherent possible limits, A=0, B=1; B=0, A=1; and A=1, B=1. The Upper Critical Limit is 2, and occurs when A and B are equal to 1. Again, this showed that the Sum of all Possibilities is twice the sum of all Observations. 

By this point I was starting to think that that the Unit Simulacrum could be used any time two variables are proportional in any way. You didn’t have to even know exactly how they’re proportional, just that they’re related in some way. And anything that is proportional can be made proportional to probability. It seemed like any ratio can be deconstructed into 1 plus another ratio. I started to explore the fact that PI is also a ratio, and that perhaps it could be deconstructed and more knowledge obtained. I started to think that perhaps the PI that we observe at our level is not the ultimate proportion of PI. I have a note that says the p is the proportionality Ratio between the Dimensions. By thinking of p/2 as the next larger ratio of p above One in the third dimension, I conjectured that maybe the first occurrence of whole PI, p, is in the fourth dimension. I conjectured that perhaps p originates in the 4^{th} dimension. I tried a lot of concepts in the Byrdwell Papers. The Unit Simulacrum 1+p/2 made me think of two nested circles, one with radius = 1 and one with radius = p/2. The Probability SimulacrumThen I got back onto probability and the Unit Simulacrum. The Unit Simulacrum seemed to naturally apply to probability; since probability is also a 1based system (100% max). Since I was coming to see that you can nest Anything into anything else with One plus a ratio of Anything over anything else. So, I wrote to myself “You don’t need to know the exact ratio, but as long as you have (A/B), you can construct a simulacrum, representing the sum of One plus the Sum of other simulacra. This allows you to construct a question Matrix.” In the Question Simulacrum, you can pose a question and rate the scale of whether it occurs or not on a scale from 0 to 1. Therefore, the simulacrum sum of One plus probability is from One to Two. If something happens with zero probability (that is the ratio), there is still the One in the Unit Simulacrum that adds to the zero ratio to give a minimum simulacrum sum of 1. when all values are limited to a maximum of 1.00 (=100%), the values can be depicted by the simulated spectra shown in Figure 33. If we correlate the logical answer NO with the zero probability occurrence to the question “did it occur?”, then when we have a zero probability occurrence, then it gives a simulacrum sum of One and we know that the answer was NO, the event based on the probability did not occur. On the other hand, we can correlate the logical answer YES with an event that absolutely occurred or is occurring with 100 % probability. The Unit Simulacrum Sum with the Ratio = 1.00 (=100%) Probability is 2. So the extremes are NO an event did (does) not occur, or YES, an event occurred or is occurring. But, if neither of these extremes is true, if there is a nonzero but not 100% probability, then at least something is happening, it is first of all, a nonzero event. An event has been defined and it has a characteristic probability ratio associated with it. Probability can be correlated with some characteristic ratio that defines a Yes/No answer for a system, and when the probability is operated in the Unit Simulacrum, we see the sum of all probabilities for a system. The construct allows the ratio to go to zero without the construct disappearing. One event can go to zero but the simulacrum sum never does. The lowest Unit Simulacrum sum is the One that Always IS. The Unit Simulacrum and the First General Form of a Simulacrum both demonstrated that there are exactly Eight Possibilities for Observation of the sum of two variables. When probability is operated in the unrestricted Unit Simulacrum, there are exactly eight possibilities for perception, or observation. However, probability is a restricted system, just like mass spectrometry is a restricted system. There is the restriction on probability that no value can be greater than 1. Probability has a maximum value of 100%. Therefore, there can be no cases where the (probability/1) ratio would ever be greater than 1. Therefore, there should never be a Case 1 observation. The Probability Simulacrum is restricted to only half of the Possible Observations. Only the Case 2 possibilities will be observed. This demonstrates in a very literal way that the Sum of all Probability is equal to one half the Sum of all Possibility. In other words, the Sum of All Possibility is equal to Two Times the Sum of All Probability. This principle is also demonstrated numerically by the fact that probability has a maximum value of 1, which gives a maximum simulacrum sum of 2. The simulacrum sum shows the sum of all possibilities for observation, whether they are actually observed, or not. Of the four choices that remain for observation of probability, two solutions can be selected as the preferred solutions. The preferred solutions are the ones with Ones that are multiplied Times (One + Ratio), that can be ignored. This allows the complete simulacrum sum to be calculated simply as One Plus a Ratio. The Probability Simulacrum in Figure 34 can be seen to be solvable using the two simplest Case 2 solutions, and the One that multiplies times the One plus a Ratio can be ignored, to give the Unit Inverses (1+1/(1/S_{1})) and (1+(S_{1}/1)). 