(P&I) The Process End-Of-Process

The End-Of-Process
The 1957 Cadillac & An Ice Cream Cone

Essay. We know that The Process never ends but we have no idea of how it begins. On the other hand, we also know that at the end of a process involving one of the companies A, B or C and their trading connections, there is an exchange event which has extraordinary consequences.

For example, we would not think that buying an ice cream cone for our daughter from a street vendor would “change the world” but it does; for one thing, the candy-maker is out of pocket and so too their suppliers, and so forth. It is also true that The Process is exacting within its modality; for example, throwing another slab of metal on a finished car is unlikely to increase its value (although in some economies, it does) whereas a finished car without wheels is unlikely to find a buyer in-process but may do so in some other modality.

The implementation of The Process at the “end of process” has an orientation or “distinction” between the seller who “produces” payables (the product) and who we think of as being “proactive” in developing the trading connections and therefore the sets of the σ-algebra and embedding space (N), and the buyer who “produces” receivables and receives “product”.

That distinction is not necessary and difficult to “see” if the producer is a government, for example, and we have to wonder what the product is but it helps us to think about it that way even if the development of the modality that is required between the “producer” and the “producee” (so to speak) in the E-conditions makes no such distinction.

It is also a helpful point-of-view in understanding modalities that are far from α=1 which is essentially an “exchange economy” and becomes indispensable if the modality of the “producer” is not aligned with or does not cause an alignment with the modality of the “producee” such as in a×log(a)=α×log(b) and b×log(b)=β×log(a) with α≠β which we might expect in economies in which self-interest and the societal norms of risk aversion and bargaining practice with which we are familiar are trumped by some other factors such as armed force; please see our Post “The Food Chain” for more details on this.

In this Post, we will show that the M-condition, log(a) = lim (1/(b^α))×log(1+b^α) as b→0, is both necessary and sufficient to explain the development of the implementation sets that are required at the end-of-process if the modality is maintained but it is not sufficient to explain which of those sets are required in order to complete the exchange of payables and receivables or equivalently, the payment of product by the producer and the receipt of product by the trading connections.

In order to complete the exchange and explain the “state” of the payables and receivables sets after the exchange, we need a new but similar condition to the M-condition, the N-Condition (Goetze 2006) log(a) = -1 – lim (1/(b^α))×log(1-b^α) as b→0 at the “end of process” which effects the exchange of payables and receivables for payments and receipts at a=b=1 even as b→0 as noted. It is also noteworthy that it is the “implementation sets” that give meaning to simple algebraic statements such as “subtract one from one and get one” but not zero which we usually associate with the concept of infinity such as the even numbers are neither more nor less numerous than all the numbers and subtracting one from them just gives us the odd numbers and the disjoint union of both the odd and even numbers is no more numerous than either.

The M- and N-Conditions At End-Of-Process

The In-Process Game

We know that the In-Process Game never ends except in the one case in which n=p+r exactly where n is the number of elements in the embedding space and σ-algebra (N) and p=p((p)) and r=p((r)) are the measure or probability of the payables set (p) and receivables set (r) that we find there in whatever “unit” we’re using (and the measure of the sets in (N) depends on the units); if n>p+r even by one, the game never ends unless there is a 4th jar in which we can put “beans” that are neither (p) nor (r).

However, we note that if n=p+r, then 1=(p/n)+(r/n)=a+b where a=p((a))=p/n is the measure or probability of the payables set and b=p((b))=r/n is the measure of the receivables set and that is always true except that we don’t know how that is maintained if (a) and (b) change or how that change is effected if n>p+r and we are dealing with a specific modality and subsets of (N) which will affect all of (N) even if the σ-algebra is not minimal and has sets that are not connected to the process and its trading connections.

But we must also have that 1 = (p/n)+(r/n) = (-1+p/n) – (-1 -r/n) and 0 = (-1+p/n) + r/n and both expressions are meaningful and are the same but have different meanings at the end-of-process in which we need to subtract one from 1=p((N)) twice; once for the delivery of product by (p) and again for the receipt of product by the trading connections in (r) because the “product” is implicit to (r) if we assume the orientation implied by a “producer” and a “producee”; and at the end of all that we still have 1=p((N)) to which there is a path that we need to negotiate from there (1) to here (0) without losing our modality or “falling-off”, so to speak.

The Principia Mathematica 1910-1913

Nor is this the first time that the world has encountered the problem of 0≠1; in fact, it can be said that all of mathematics depends on the distinction of two ciphers 0 and 1 that are not the same because one is deemed to be a successor to the other and mathematics itself provides a solution to the problem of explaining succession in the form 0=1+exp(πi) and i^2=-1 (which is to say that (i^2)+1=0 too) which describes a reflection in the origin or a rotation in a higher dimension where there is more room or space to subtract or add one.

Are You The One?

And, of course, we spend most of our lives as zeros looking for the one, don’t we? And then as nominal zeros, we need to wonder how we can afford them.

We noted in The Process that if there is a modality established by the E-conditions then it cannot be the case that r→0 as p→n because the expansion of the set (p) in (N) is governed by the Stirling Approximation (17th century) as p×log(p) = p + Σlog(k) + O(log(p)) and therefore log(p) = 1 + (1/p)Σlog(k) + O(log(p)/p) which looks familiar from the game as an average of the outcomes for large n and p; and the same can be said for p as r→n.

Moreover, we know that 0=log(1)=lim log(1+x)/x as x→0 (L’Hôpital 1796) and therefore that we should expect as a necessity that log(a) = lim log(1+b^α)/(b^α) as b→0 because that relationship respects the two E-conditions at the boundary a=1 and b=0; however, such limit for a = exp(log(a)) is always greater than 2 but cannot exceed the exponential e for 0<a<1 and is reduced to a result of 0<a<1 by subtracting 2 which we can understand as the exchange at a=b=1 and the delivery of product by (a) (nominally, subtracting 1 from a) and receipt of product by (b) (subtracting 1 from b) causing both of the sets (a) and (b) to be “harder to find” but “needing to be found” and the limit still respects the relationship between log(p) and the average search times for the set (b) with measure b=p((b)) that is required by the Stirling Approximation and any such solution that is based on counting must do that.

Figure 1: M-Condition For (a) And α = 0.2 For the “Delivery of Product”

In other words, the M-condition for (a) is necessary and sufficient to establish the end-of-process for (a) with respect to (b) subject to the modality of the E-conditions; please see Figure 1 on the right which is reproduced from The Process.

The N-Condition For 0=1

However, the M-condition is not sufficient to explain the end-of-process for (b) nor is the obverse condition log(b) = lim log (1+a^α)/(a^α) as a→0 because (b) is in the set of the trading connections and the trading connections don’t “deliver product”, they receive it, and the delivery of product is only implicit in (b) and not explicit as in the case of (a).

The implementing sets for the trading connections on the receipt of product (rather than the payment of product) have a different behaviour that is well-described by the N-condition (Goetze 2006) log(a) = -1 – lim log(1-b^α)/(b^α) as b→0 where we note that formally -1 = lim log (1-x)/x as x→0 and it is also noteworthy that -1=log(1/e) because b=1/e defines the maximum extension or push-back of the trading connections into the (a)-space and hence the N-condition allows that log(a)=0 and a=1 as b→0.

We must also deal with the fact that we are in-process and respect the E-conditions so that it is not possible that both a→0 and b→0 because if a→0 then a^a=b^α from the 1st E-condition implies that b→1; and if b→0 then b^b=a^α from the 2nd E-condition implies that a→1; moreover, the notion that “a should be 1 to deliver 1” is not necessary to complete the process and cannot be implemented in any case because we cannot actually have that both a=1 to deliver the product and b=1 to receive the product at the same time.

Figure 2: End of Process α=0.2 < 1/e Company A Modality

We have then (please see Figure 2 on the right) the following diagram of the 2nd E-condition (the 1st is its reflection in the diagonal line a=b), the 1st M-condition and the 1st N-condition at the end of process effecting a=1 and b→0 but not actually getting there because the process does not end with the delivery of product.

At the end-of-process and on the delivery-of-product and the receipt-of-product, the implementing set occurs at a” which is the intersection of the 1st M-condition (and therefore in the correct modality) with the 1st N-condition and the “profit” (please see Figure 1 above) is increased by a reduction of the payables set (a”) and an increase in the excess of the receivables set at b=b(up) that is required to “receive the product” as b→0 within the implementing sets of the established modality.

We also note that the diagram (Figure 2) projects the “height” of the M-condition and the “height” of the N-condition both of which occur as measures of sets in the (b)-space, into the (a)-space by a translation of (-2) in the case of the M-condition and a reflection in the case of the N-condition because b→0 happens only as a→1 in the (a)-space.

Figure 3: Extreme Alpha (α=0.02)

As α→0, the M-condition also increases in b and becomes a near straight line at b=e-2 and the N-condition also increases in b and becomes a near straight line at b=1/e although both “lines” begin at a=b=0 and end at a=1 and b=0 so that “beginning of process” at a=0 and the “end of process” at a=1 are explosive events for extreme modalities below α=γ/2e ≈ 0.1047236… where γ is the Euler–Mascheroni Constant γ=0.57721…(please see The Process for more details) ; moreover, as α increases above α=1/e the relative position of the M-condition above the N-condition reverses and both become near flat lines with height b near zero as α increases and α→∞.

Company D & The Death Embrace

We called Company D with modality α=1/e the “death embrace” because there are no implementing sets that would advance a=1/e past a=1/e and that would also be more or less true in a neighborhood of α=a=b=1/e because everything more or less collapses into a single “point” or “event” in (N) and if we think of the sets with a=p((a)) and b=p((b)) and each with measure 1/e=0.368… the event cannot be said to be “small” basically taking with it 74% of (N).

Figure 4: The Death Embrace

We also noted in The Process that new companies are often financed at that level with α=1/e=R/P representing 40% equity and 60% debt but few companies are financed below that level such as 30% equity and 70% debt.

Although companies don’t have the same precision that we might expect in “nature’s modality”, the energy-mass-light modality at e/m=c^2=1 in the right units, we still need to respect the law of averages and demonstrated conventions.

The “jiggle” room that allows companies (and possibly nature) to escape the death embrace is provided by the implementing sets that are defined by the N-condition; please see Figure 4 above.

As the modality is quietly modified in a neighbourhood of α=1/e, the M-condition and N-condition are also modified but move at different rates and eventually in different directions as α progresses through α=1/e and takes the M-condition with it and then above it but the N-condition can never exceed b=1/e and therefore there are always implementing sets that allow the process to continue.

We’ll be back.

We also note in passing that although the process never ends, our end is already on the way as a burst of γ-radiation from some dying star that will blow-away our atmosphere with the same certainty and cause but not reason as we might blow out a candle into that goodnight. But the process never ends and we’ll be back.

For more applications of these concepts please see our Posts which rely on a Theory of the Firm developed by the author (Goetze 2006) which calibrates The Process to the units of the balance sheet and demonstrates the price of risk as the solution to a Nash Equilibrium between “risk-seeking” and “risk-averse” investors within the societal norms of risk aversion and bargaining practice.

Postscript

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