RANDOM VARIATIONS ABOUT THE MAXIMAL PATH
Although chaotic, trendless swings of karma – as illustrated in the diagrams above – make little sense from an evolutionary standpoint there is some merit in the argument that karma might move randomly about the maximal path. In other words, although the long term trend in karma will be positive in order to reflect the upward spiritual ascent of the personality, changes in karma from one incarnation to the next could be negative. Thus, an evil personality could succeed a good one! Given the propensity of human beings to behave in erratic and unpredictable ways, such a scenario is not merely possible but highly likely. Under this assumption, the karmic path would look as follows:

The actual karmic path Kr(x) - drawn in black - moves randomly about an upward path of spiritual improvement. This upward path is the maximal path K(x) and is drawn in red.  It will be seen that, whereas the maximal path “hits” the incarnation axis only once, the actual path “hits” it fifteen times. Hence, it might be thought that the chances of eliminating karma are increased and that the Impossibility Theorem is invalidated. Actually, this is not the case. In fact the chances of eliminating karma are decreased. The rest of this section will be devoted to proving this assertion. To fully understand the proof the reader will need to have studied Chapter 4 of Refuted which develops the mathematics of karma and the notion of a karmic probability density function. For those who wish to familiarize themselves with the elements of probability theory and mathematical statistics, I would recommend John Freund’sMathematical Statistics (Prentice-Hall) by Irwin Miller and Marylees Miller; and Paul G. Hoel’s  Introduction to Mathematical Statistics (John Wiley & Sons).  For more advanced reading I would suggest Introduction to Mathematical Statistics (Collier-Macmillan) by Robert V. Hogg and Allen T. Craig.

In Refuted it is shown that the maximal path is described by the following equation:

where K(x) is the value of karma in incarnation x, and A
1 and A are the magnitudes of positive and negative actions respectively in the first incarnation. These are measured as a proportion of their upper bounds and will be numbers between 0 and 1. The other functions are as follows:

The parameters λ and µ are the coefficients of adjustment to positive and negative karma respectively. Typically, λ is a small positive number such as 0.05, and µ is a small negative number such as -0.06.  The magnitudes A
1 and A2  are determined randomly and the parameters λ and µ are characteristics of the series of personalities and will, in general, vary from one series of personalities to another. Since A1 and A2 are random variables, the maximal path – which  depends on A1 and A2 - is randomly determined. It follows that for any given x - the number of the incarnation - the value of karma in that incarnation is also a random variable. In Refuted it is shown that the probability density function (pdf) of karma has the following graph:

The area under the graph between any two values of karma, measures the probability that karma will fall between those two values. Thus, the area under the graph between any two small values of karma about zero i.e. -δ and δ (e.g.  δ = 0.01) measures the probability that karma will be so small it can be regarded as eliminated.

As the number of the incarnation increases the magnitude of p(x) increases and the magnitude of  q(x)  decreases. Hence, the graph becomes “flatter” and moves to the right as shown below:

Consequently, the area under the graph between any two small values of K will decrease. The following diagram makes this clear:

In other words, as the number of the incarnation increases, and  the personality becomes more spiritually enlightened, the probability of eliminating karma decreases. This proves that the doctrine of karma as a system of ethics is fundamentally incoherent  because no one can escape the wheel of rebirth by being a good and compassionate human being. Escape is only possible if bad karma happens to be balanced by good karma.

The mathematics is as follows. When q(x) ≥ δ (the case in the diagram above) the probability of eliminating karma for any given x (calculated either by geometric methods or by integration of the relevant parts of the pdf ) is:

When q(x) ≤ δ the probability is:

By inspection of the function P2(x) it can be seen that as x → infinity, P2(x) → 0. These two results can be combined to form a probability incarnation graph illustrated in the diagram below. Here,  λ = 0.05, μ = -0.06, δ = 0.01.

Caution: P(x) is not a probability density function since it does not describe a random experiment in x.

RANDOM VARIATIONS

When karma varies randomly about the maximal path – the principal topic of this section –  the equation of karma becomes:

where u is a random disturbance. We shall make the following general assumptions:

The first assumption means that the values of the disturbance are equally likely, and the second that the expected ( i.e. mean)  value of the disturbance is zero. The last assumption ensures that the disturbance is proportionate to the level of karma i.e. as the level of karma increases, the range of the disturbance also increases. Under these assumptions, it can be shown, by standard methods of mathematical statistics, that the pdf of karma changes its shape with the number of the incarnation approaching a stable trapezoidal form as the number of the incarnation  increases. The sequences of graphs below makes this clear. Karma first appears in the second incarnation (x =2) because there are no actions before the first incarnation. The initial pdf of karma has a smooth bell-shaped distribution centred on zero.

As the number of the incarnation increases, the pdf becomes trapezoidal in shape and more spread out.

As before, the probability that karma is eliminated (i.e. falls close to zero) decreases with x. The corresponding mathematical equations are as follows. When q(x) ≥ δ the probability of eliminating karma for any given x is:

and when q(x) ≤ δ  the probability is:

In the denominator of the R.H.S. of the above equation, the term 'p(x) squared'  dominates so that as x → infinity, Q2 (x) → 0. Combining these results we obtain the probability incarnation graph for λ = 0.05, μ = -0.06, δ = 0.01

Superimposing the two graphs shows their relationship:

It can be seen that the probability incarnation graph for karma with random variations about the maximal path is lower for all values of the incarnation x i.e.

for all x ≥ 2.

This  result was obtained for λ = 0.05, μ = -0.06, δ = 0.01  but it is true  in general - as we shall now show.

For q(x) ≥ δ we  have, after some algebraic manipulation, the following ratio:

Since q(x) ≥ δ,  p(x) > 0,  q(x) > 0, it follows that:

Hence, in the above fraction, the numerator is less than the denominator i.e.

For q(x) ≤ δ,  we have, after some algebraic manipulation, the following ratio:

Since q(x) ≤ δ, p(x) > 0,  q(x) > 0, it follows that:

Hence,  in the above fraction, the numerator is also less than the denominator i.e.

Combining both results, we have:

for all x ≥ 2. [Q.E.D.]

Conclusion: When karma moves randomly about the maximal path, the probability that karma is eliminated is decreased. Hence, the Impossibility Theorem applies a fortiori. More precisely, if
pr is the probability that a series of personalities eliminates karma when karma moves randomly about the maximal path, then:

where
ԑ  is the error in the operation of karma (see above). This means that the Impossibility Theorem  sets an upper limit to the probability that reincarnation is a fact i.e.

for any  N series of personalities moving randomly about the maximal path. If m series of personalities are following the maximal path and series of personalities are moving randomly about the maximal path, the inequality  would be:

where m + n = N. This proves that reincarnation cannot happen.

FURTHER DISCUSSION
The careful reader will note that there are two methods of calculating the probability of liberation when personalities follow the maximal path. The first method – the x-method – calculates the probability of the maximal path “hitting” the incarnation axis at a point close to an integral value of x i.e. it calculates the probability that:
n – ε < x < n + ε

where ε – the error in the operation of karma - is a small number and n is a whole number greater than 1. In this case, the probability of liberation is equal to ε and is the same for every series of maximal personalities provided the adjustments to positive and negative karma are small. The second method – the K-method – “averages” out the probabilities of eliminating karma in each incarnation x for a fixed value of δ i.e. it calculates the average of the probabilities that:
-δ < K < δ

where δ is a small number.
Whereas the x-method uses the probability density of x (see Refuted), the K-method uses the probability incarnation curve (discussed above). Both methods ought  to produce the same value for the probability of liberation. However, since ε and δ are measured in different units - ε is measured in terms of the ‘standard unit’ (see above) and δ in terms of karma - each method will produce a slightly different result if ε and δ are assigned the same value e.g. 0.01. Hence, a value of δ must be chosen to ensure that the two methods do give the same result; this value can be found by trial and error. The following table shows the “correct” value of δ for different values of ε:

TABLE 1

Thus, whichever method is used, the probability of liberation will equal the error in the operation of karma, ε.

The utility of the K-method is that it provides an easy way of calculating the probability of liberation when karma moves randomly about the maximal path. (The x-method does not give an analytical solution).  Using the K-method, the following table shows the probability of liberation for randomly moving karma for various values of ε and the corresponding values of δc :

TABLE 2

It can be seen from Table 2 that when karma moves randomly about the maximal path, the probability of liberation is slightly less than half the error in the operation of karma.

* * * * *

The system of karma here presented incorporates two basic features:

(a) the upward ascent of the personality in the long run expressed as the maximal path.
(b) the erratic and unpredictable behaviour of the personality in the short run expressed as random fluctuations about the maximal path.

A karmic path can, therefore, be envisaged as the sum of two components: an evolutionary, maximal component associated with the personality’s behaviour in the long run, and a random component associated with the personality’s behaviour in the short run. (For a maximal series of personalities the random component is zero). Although specific values of λ and µ have been used for illustrative purposes, it is not necessary to know their values in order to derive the central proposition of the analysis. All that is required is that λ and µ be small and unchanged  to reflect the slow, positive ascent of the personality in the long run. The vagaries of personal behaviour in the short run are summarized by the random component. Crucially, whatever the values of λ and µ - and they will vary from one series of personalities to another -  the probability of liberation will not exceed
ε. This is an intrinsic and invariant property of karma.

The error in the operation of karma does not, of course, refer to errors in the workings of nature.  We are here concerned only with the analysis of a doctrine and its mathematical representation. The concept arises because it’s unreasonable to insist that karma must be exactly zero for liberation to be possible. Suppose, for example, the magnitude of bad karma is:

5.102738478378273834561897673563728

If this had to be counter-balanced by an equal quantity of good karma, then liberation would not be possible if it was

5.102738478378273834561897673563729.

The error in the operation of karma permits small differences in good and bad karma to be disregarded.

It is important to note that, since the karmic  graph “hits” the incarnation axis at a point between two whole numbers, the error in the operation of karma cannot exceed 0.5. Hence, unless it “hits” the axis at the midpoint, it will always be nearer one number than the other. The nearest whole number defines the incarnation of liberation (because karma will be smaller). If the error relative to one incarnation number is increased to a figure above 0.5, it will ipso facto be decreased to a figure below 0.5 relative to the other incarnation number. This latter incarnation will then define the incarnation of liberation and the error will accordingly be less than 0.5. The probability of  the graph “hitting” the incarnation axis at the midpoint, or an integral value of x, is precisely zero. Although 0.5 is the upper bound for
ε, the error will, of course, need to be much smaller than 0.5 to ensure that residual karma is negligible.

On the question of the so-called "Noble Eightfold Path," it is a mistake to assert that the personality can control and eliminate karma by “right” living, meditation and general renunciation of the world. The reasons are twofold:

(1) the character traits needed to engage in such activities are themselves determined by karma and can be traced back to the actions of the first

incarnation. Since the actions of the first incarnation are the product of chance, the character traits needed to eliminate karma are also the product

of chance.
(2) the actions of the personality can have unintended consequences and, therefore, unintended karma. If karma is eliminated, it will be due to a

chance combination of factors and not to the efforts of the devotee. As we have seen, the probability of this happening can be calculated.

In the case of point (2) unintended consequences of actions will add to the fluctuations of karma about the maximal path and, accordingly, reduce the probability of liberation. In many ways point (2) is typical of the undeveloped personality. If A shoots and kills B but didn’t mean to kill him, A’s actions will have karmic consequences over and above the intended action.

* * * * *

Random motion about a maximal path – which includes the special case of motion with a random disturbance of zero - describes karma in its most general setting. As we have seen, other descriptions are possible but are grossly inconsistent with karma’s function as a facilitator of human spiritual evolution.  The inability of karma to determine the personality’s actions in the first incarnation coupled with the erratic and unpredictable behaviour  of the personality in the short run means that liberation from the cycle of rebirths is a chance event – an event that is beyond the control of the personality. It is quite remarkable that, whatever the response to karma, the probability of liberation can never exceed ε – the error in the operation of karma. An immediate consequence of this result is that the smaller the error, the smaller the probability of liberation - approaching zero in the limit. It is the universalist nature of karma that is its undoing because it permits of no exceptions: every series of personalities without exception must eliminate karma in order to escape the cycle of rebirths. As we have seen, the probability of this happening is so small it can be equated to zero. Consequently, some series of personalities will be infinite in extent; and a series of personalities that is infinite in extent has no final personality.  In short, karma cannot describe anything that happens in nature. Without karma, the  notion of a “stream of consciousness” linking a group of non-contemporaneous personalities is entirely gratuitous. One might just as well speak of a stream of consciousness between a person and his ancestors; or a stream of consciousness between a group of trees; or a stream of consciousness between a group of any living things – non-contemporaneous or otherwise. If a group of non-contemporaneous personalities have nothing in common other than the fact that they are personalities dwelling on the same planet, then it is meaningless to apply the term ‘rebirth.’ Nothing is reborn because every personality is unique unto itself.  # Reincarnation Refuted ONLINE Logic And Mathematics            (a) the moral and spiritual qualities of the present personality's behaviour determine the quality of life of the new personality in the next incarnation. Expressed simply: actions determine karma.

(b) The quality of life of the present personality determines the moral and spiritual qualities of the personality's behaviour in the next incarnation. Expressed simply: karma determines actions.

Conflicting ideas about the self and the nature of the reincarnating entity are the source of much confusion in the reincarnationist world. In Hinduism, for example, the reincarnating entity is the eternal Self. In Buddhism, no such entity exists; instead, a chain of personalities are linked together by a "stream of consciousness." Between these two extremes are many variations reflecting different historical and cultural traditions. This disparate state of affairs has led some people to argue that refutations of reincarnation are essentially meaningless because it's unclear what is being refuted. In reply to this - using the Four Postulates - it can be proven that the self does not incarnate more than once which is to say it does not reincarnate. Discussion of reincarnationist belief is, therefore, unnecessary. For those who don't accept Postulate III, such as Buddhist monks, the analysis of karma, alone, is sufficient to show that single incarnation is the only  viable alternative. Once again, discussion of reincarnationist belief is unnecessary.

In the analysis of karma, the concept of a reincarnating entity is dispensed with entirely. Instead, one considers a series on non-contemporaneous personalities whose purpose is the elimination of karma. The underlying principle linking these personalities may vary from one system of thought to another but the concept of a temporal series of connected personalities does not. Crucially, there must be a reason for these personalities to exist - they do not appear by chance - and this reason is the motive force for reincarnation. This reason may be grounded in choice or compulsion but, ultimately, the distinction is irrelevant because all reasons are functionally equivalent to karma - only the language varies. Thus, a series of non-contemporaneous personalities exist to eliminate karma or the need to learn new lessons.

One of the most surprising features of karma is its ability to be quantified and analyzed with the tools of mathematics. What is hidden beneath a jungle of metaphysics can be dug up, dissected and demolished by mathematical methods well within the grasp of any first year university student in the physical or social sciences. (Mathematically inclined readers may consult Refuted for details.) In the first instance it can be shown that liberation from the cycle of rebirths is a chance event, and this leads to a  remarkable theorem - the Impossibility Theorem - which states that the probability of every series of personalities eliminating karma and achieving liberation from the cycle of rebirths is zero. Consequently, at least one series of personalities will be infinite. Since this is impossible reincarnation cannot be a fact of nature. This demonstrates that the doctrine of karma is fundamentally incoherent i.e. internally self-contradictory. The contradiction arises because karma is supposed, ultimately, to liberate everyone from the cycle of rebirths through a process of learning and spiritual evolution but liberation for everyone is impossible. This incoherence shows up in another way: the more evolved a personality becomes, the less likely the personality will achieve liberation - the reverse of what is supposed to happen. This latter proposition is proven in Refuted.

The formula in the panel above is based on the assumption that every series of personalities follows a slow, consistent path of moral and spiritual improvement. Given any linked series of non-contemporaneous personalities, adjacent personalities will not be very different from one another in terms of their moral and spiritual qualities. The particular roles in life may vary considerably e.g. one personality may be a king and the next a janitor, but their moral and spiritual qualities will not be very different. However - and this is very important - the later personality will be slightly better (morally and spiritually speaking) than the earlier one. Of course, this scenario does not cover all possibilities. It is conceivable for an evil personality to succeed a good one, or even an animal incarnation to succeed a human one. As we shall see, however, large, irregular and possibly inconsistent changes in the moral and spiritual qualities of personalities between adjacent incarnations, reflected as large, irregular and inconsistent swings of karma, will decrease the chances of attaining liberation. To illustrate, a man carrying a rifle moving around quickly and haphazardly on rough, uneven terrain is less likely to hit a given target than a man walking slowly and progressively uphill. Any departure from this slow, upward path will decrease his chances of hitting the target. We shall refer to a series of personalities that is following a slow, consistent path of moral and spiritual improvement as a maximal series of personalities and the path followed a maximal path.  This terminology is used because a path of slow, consistent improvement maximizes the chances of eliminating karma - an assertion we will justify below. Suppose the probability of a maximal series of personalities attaining liberation is 1 chance in 100, then the probability of a non-maximal series attaining liberation would be smaller e.g. 1 chance in 150, or 1 chance in 200 etc. depending on the degree of variation from one incarnation to the next. The difference between a maximal and non-maximal path can be illustrated graphically:

The question is: what determines the karmic path? Before we consider this question let's clarify the meaning of the term 'actions'. In the doctrine of karma actions relate not so much to the outward behaviour of the personality but to the moral and spiritual qualities of the personality's  behaviour. For example, firing a gun at a wall is very different from firing a gun at a person. The outward behaviour may be identical but the motive is vastly different. Actions can, therefore, be good or bad. Karma, on the other hand, refers to the quality of the personality's life and can also be good or bad. If, for example, a child is murdered or born handicapped, or someone's house is repeatedly burgled or burns down - that's bad  karma. If someone wins the national lottery, or lives a life of ease and luxury, or is highly intelligent - that's good karma. Actions and karma interact with one another from one incarnation to the next in the following way:

There is no circularity here because present actions determine future karma, and present karma determines future actions. For example, someone who murders (bad actions) in the present incarnation will experience a poor quality of life (bad karma) in the next incarnation. Actions in the following incarnation will then change for the better. This means that future personalities will learn from the mistakes of previous personalities. If this were not the case then reincarnation would be pointless because no lessons would ever be learned. Different modes of interaction are possible but anything more complicated is likely to produce a non-maximal path. (The reader may refer to Refuted for further discussion of this point.) Now, if the actions of one incarnation determine the karma of the next incarnation, and the karma of the next incarnation determines the actions of the incarnation after that, what determines the actions of the first incarnation? They cannot be determined by the karma of the previous incarnation because there is no incarnation previous to the first one. This means that initial actions cannot be determined within the system of karma, and if this is the case then they must be the product of chance. The subsequent path of actions and the corresponding path of karma must, then, also be the product of chance. What is supposed to be a system of justice turns out to be a rather arbitrary system of punishments and rewards.

Those familiar with engineering systems will recognize the above scheme as an input-output system with a feed-back loop:

To transform the input-output scheme into a mathematical model, it is necessary to quantify actions and karma, and in Refuted  I spend some time showing how this can be done. Since the present discussion is expository we shall take quantification as given. Now, a mathematical model in which the quantity of something in the present time period depends on the quantity of something else in a previous time period is known as a difference equation, and anyone familiar with difference equations will recognize that the scheme outlined above can be so represented. Now, the solution of a difference equation can, in many cases, be represented graphically, and this turns out to be possible in the case of a karmic path. For illustrative purposes, a karmic path can be shown as a smooth or jagged line depending on whether it is maximal or non-maximal but the true graph will be a series of points - each point corresponding to a particular incarnation. (The gaps between the points correspond to the gaps between incarnations). Karmic paths will, therefore, look like the ones below. The points have been joined to show the general trend.

A maximal karmic path can be thought of as a single entity that is randomly determined by some hypothetical experiment like the throw of dice. The dice are thrown in the first incarnation, the actions are set and the entire karmic path is determined. (Precisely how the path is determined depends on the response to karma). If the actions had been different, the path would have been different. The following diagrams show maximal paths corresponding to different initial actions (diagram on the left) and maximal paths for the same initial actions but different responses to karma (diagram on the right). For ease of viewing, points on the graphs are not shown. Note than karma begins in the second incarnation - hence the gap between the graphs and the vertical axis. No significance should be attached to the numbers; the scale is shown to indicate that the graphs are mathematically correct representations of karma (drawn in MathCad).

In mathematics, smooth, continuous curves are easier to handle than discontinuous points, and in Refuted  I introduce a "smooth" approximation to the series of points that define the karmic path. This replaces the path of points with a smooth, continuous curve like the ones shown in the diagrams above. Given that the maximal path can be represented by such a curve, we need to find the probability that the curve will "hit" a point on the incarnation axis close to a number that defines an incarnation. When this happens, the karma in that incarnation will be negligible and liberation from the cycle of births will be possible. A direct "hit" occurs if the curve crosses the incarnation axis at integral value (i.e. whole number); karma will then be zero. Note that the graph of a maximal path never "hits" the incarnation axis more than once. This means that if liberation is achieved, the incarnation of liberation is uniquely determined.

In the following discussion, we shall refer to the distance between two incarnation numbers as the standard unit. The Greek symbol, ε (epsilon) shown in the diagrams above is defined as the error in the operation of karma and sets the limit to the permissible deviation of the curve from the number of the incarnation. The error is indeterminate but needs to be fairly small to ensure that residual karma is negligible e.g. one hundredth of the standard unit. In the diagrams above, the error is one-fifth. If the error wasn't small, liberation would be possible with non-negligible karma. In the following diagram the error is two-fifths of the standard unit and residual karma is much larger.

Let n be the number of the incarnation, then liberation is possible if the curve "hits" the incarnation axis anywhere from n - ε  to n + ε where n can be any integer greater than 1. (n must be greater than one for reincarnation to happen.) Using standard techniques in mathematical statistics it's possible to calculate the probability that this will happen and it turns out to be equal to ε - the error in the operation of karma. This probability is the same for every series of maximal personalities.

Let's suppose that the error is 0.1. Then, statistically speaking 1 series of personalities in 10 will eliminate karma - where eliminating karma means reducing it to zero or some negligible quantity - and achieve liberation. But, the doctrine of reincarnation requires that every series of personalities eliminate karma otherwise reincarnation for some would be infinite. Since all karmic paths are determined independently - they are the product of a hypothetical random experiment in the first incarnation  - to find the probability of, say, two series of personalities eliminating karma one simply multiplies the individual probabilities together i.e. (0.1) x (0.1) = 0.01 - one chance in a hundred. The probability of three series of personalities would be (0.1) x (0.1) x (0.1) = 0.001 - one chance in a thousand. It is evident that the probability of eliminating karma diminishes rapidly with the number of series of personalities. In mathematics multiplying a number by itself is called "raising" it to the power of 2; multiplying it by itself three times is raising it to the power of three. Multiplying it by itself any number of times is raising it to the power of that number. Thus,

0.1 raised to the power of 2 =  (0.1) x (0.1) =  0.01

0.1 raised to the power of 3 = (0.1) x (0.1) x (0.1)  = 0.001

0.1 raised to the power of 4 = (0.1) x (0.1) x (0.1) x (0.1)  = 0.0001

etc.

The probability of a million series of personalities eliminating karma would be:

0.1 raised to the power of one million = (0.1) x . . . . . . . . . . . . . . . . . . . . (0.1)  ≈ 0
one million times

The wavy equal sign means almost equal to. But, of course the number of series of personalities must be counted in billions, say 10 billion. We should, therefore, write:

0.1 raised to the power of 10 billion =  (0.1) x . . . . . . . . . . . . . . . . . . . . (0.1) = 0
10 billion times

which can be equated to zero because anything with that probability will definitely not happen! Generalizing, if ε is the error in the operation of karma and N the number of series of personalities, the probability of every series eliminating karma is ε raised to the power of N i.e.

Since ε is small in relation to the standard unit and N is a very large number, this quantity is, for all practical purposes, equal to zero. This means that it's virtually certain that at least one series of personalities - in practice it could be billions - will not eliminate karma and, consequently, reincarnate an infinite number of times. Since this is impossible the doctrine of reincarnation cannot describe anything that happens in nature. This is the essence of the Impossibility Theorem.

On the question of non-maximal paths - though conceivable - they make little sense from an evolutionary point of view. In biological evolution organisms progress from simple to more complex forms producing an orderly, progressive series -  not a random, haphazard one. The same must apply to any theory of spiritual evolution - including the doctrine of reincarnation - because a series of personalities that swings endlessly from one extreme to another in a random, purposeless way is anything but evolutionary. Indeed, any departures from the maximal path may be considered counter-evolutionary. Therefore, it is  highly reasonable to assume that personalities follow a path of slow, consistent and progressive improvement, each personality learning from the mistakes of  previous ones. (However, random variations about the maximal path itself, in which one personality may or may not progress from the previous one, is probably a more reasonable assumption. Though counter-evolutionary in the short-term, such variations are consistent with evolutionary change in the long-term. See the discussion below for the analysis of karma in this scenario.)

It is easy to show how large departures from the maximal path decrease the chances of eliminating karma. In the diagram below the non-maximal path "hits" the incarnation axis more than once and, in one case, comes within a distance ε of an incarnation number. (The karma for this "hit" is the vertical distance from the incarnation number to the point b.) But, because the variation is relatively large, the corresponding karma is relatively large and is, therefore, inadmissible. For the maximal path, the karma is the distance from the incarnation number to the point a, which is relatively small. It will be noticed that at high levels of positive karma - where the graph becomes steep - the maximal path will also produce "hits" that are inadmissible, and explains why liberation becomes more difficult as the personality develops.

Of course, one may design  a non-maximal path that scores direct "hits" or close "hits" such as the one below. But because this path is only one of an infinite number of non-maximal paths, the probability of this happening by chance is precisely zero.                   Let R be the probability that reincarnation is a fact, ε be a small error in the operation of karma, and N be the total number of series of personalities, then        The Impossibility Theorem