Essay. The “state space” of The Process is the σ-algebra, (N), of payables and receivables that are generated by the companies in some “market” such as all the companies in the Dow Jones Industrials, or all the companies in the S&P 500, and although we can start with just one company, typically, other companies will be drawn-into (N) through the trading connections which will include its employees, customers and suppliers, and sources of financing.

In (N), we can expect to find sets of payables (a) and receivables (b) that share a common modality, 0<α<+∞, and which we say are “in-process” with that modality and, in particular, any company in (N) will demonstrate the properties of its calculated modality, α=R/P, where R is what is owed to the company and P is what the company owes, both of which can be calculated from any balance sheet, and although the company will demonstrate the properties of its calculated modality, it will not actually “have” that modality because the tabulated sets of its payables and receivables are countable, whereas the modality for (a) and (b) in-process requires fractal sets (a) and (b) that exist only in (N) and for which the elements are necessarily uncountable.

For example, if (a) and (b) are in-process with modality α, and measures a=p((a)) and b=p((b)), respectively, they satisfy the 1st- and 2nd E-conditions with a×log(a)=α×log(b) and b×log(b)=α×log(a), respectively, and, therefore, all such (a) “cover” all such (b) with a “fractal dimension” that is proportional to the measures a and b; that is, if d=log(b)/log(a), then a=α×d and b=α×(1/d), and, similarly, all such (b) “cover” all such (a) with a fractal dimension, 1/d, that is proportional to b; please see our Post “(P&I) The Process – Commensurability” for more information.

Whereas the “state space” can have an arbitrary extent, the “configuration space” is limited to the unit square, 0≤a,b≤1, and it is 2-dimensional, consisting of pairs (a,b) that are the measures of payables sets (a) and receivables sets (b) that are in-process with the same modality.

The geometry in Figure 1 above, suggests “continuity” and “differentiability” of a=p((a)) with respect to b=p((b)), and vice versa, but that is not the case; for example, if we apply the chain rule of differentiation to the 1st and 2nd E-conditions, then log(a) + 1 = α×(1/b)×db/da, and log(b) + 1 = α×(1/a)×da/db and, therefore, α²= a×b×(1+log(a))×(1+log(b)) if neither a nor b=1/e; but for every α there is a set of payables (a) and receivables (b) for which the equation fails, and for α=1/e (Company D) it fails for α=a=b=1/e, and it fails for all a,b<1/e and, therefore, for all companies that demonstrate the Company A modality; please see Figure 2 on the right (and click on it, to make it larger if required); for example, the implementing set for a=a(left)<1/e is at p((b))=b>1/e, so that 1+log(a)<0 but 1+log(b)>0.

Moreover, there is no useful topology on (N) that would allow us to talk about “continuity”, and if (a’) and (a”) are two sets of payables in-process with (b’) and (b”) with different, or even the same, modalities, there is no assurance that the sets of payables (a’)∪(a”) or (a’)∩(a”) are in-process with anything; if the modalities are the “same”, they could belong to different “companies” that are represented in (N) at different stages of their “in-process”.

However, the diagram, Figure 2, also shows that for the same implementing set (b) for a(left), there is another implementing set with the same measure as (b) at a”>1/e and we might conclude that the effect, or “resultant”, of (b) on (a), is to move a(left)→a”, and then b to b(up) and a” to a'(left), for which both 1+log(a'(left))<0 and 1+log(b(up))<0, but a'(left)>a(left), signifying an “advance” in the progression of a=a(left) towards a=1 and the delivery of product; figuratively, the collection, or existence, of receivables at (b) has allowed the “company” to increase its “payables” and, therefore, advance the production of its “products” which are nothing but “payables” and “receivables” in this context, and in the “real world”, too, despite tangible evidence of a “product” that emerges from payables and receivables which are exchanged, pro rata, at a=b=1, with “profit”; please see our Post “(P&I) The Process End-Of-Process” for more details on this.

That also provides us with an opportunity to define db = (1/α)×b×(1+log(a))da as a derivative of Radon-Nikodym-type with respect to the two measures, b((a)) and a((b)), defined on payables sets (a) that are in-process with receivables sets (b), and which requires a piecewise continuous step-function, χ(α;a)≥1, in the integrand in order to select the right (a) and, for that (a), the right (b) to move the process forward; and, similarly, for da = (1/α)×a×(1+log(b))db.

We also note that if (a) is in-process with (b) with modality α<1 (Company A and Company B modalities, and possibly Company D with α=1/e) and a=a(left)<1/e, then there is always an a=a(right) for which the “implementing set”, b(right), has the same measure b=p((b))=p((b(left)))=p((b(right))), but there is no reason that b(left) and b(right) should be the same sets, even though they have the same measure, because the payables set a(right) is much larger than the payables set a(left); however, for such a(right), there are also always two implementing sets b(right)(up) and b(right)(down) that are different from b(right) if α<1; please see Figure 2 above, but that is not the case for a(left) if a(left)<exp(-1/(α×e)) which is the maximum extension of the 2nd E-condition, b×log(b)=α×log(a), into the (a)-space and which occurs at b=1/e.

In the geometry of the “configuration space”, which we call the “modal geometry of the firm“, the points a=1/e and b=1/e, and a=b=1, are “critical points” for all 0<α<+∞, and the geometry is different for each of the “companies”, Company A (0<α<1/e), Company B (1/e<α<1), Company C (1<α) and Company D (α=1/e), to which we also add Company E (α=1) because it demonstrates neither of the properties of Company B or Company C in this context; and the geometry also changes for Company A in the case of “extreme alpha” for α < γ/2e ≈ 0.1047236… where γ is the Euler–Mascheroni Constant γ=0.57721…; please see The Process for more details.

The system is also “autonomous”, that is, it is “self-controlling” or “self-describing”, and there is no sense of “time” as in the ticking of a clock; however, every set of receivables, (b), that is in-process with a set of payables, (a), has an “entropy” that is defined by the “functional”, Ω(b) = -∫ b(a)×log(b(a))da = -∫α×log(a)da = -α×[a×log(a) – a] = α, where the integration is over the interval, [0≤a≤1], and b(a) is the measure of p((b)(a)) for the implementing sets for (a) with a=p((a)) from the “beginning of process” at a=0, to the “end of process” at a=1, and it depends only on the modality, α, which then functions as an “arrow of time” in a self-describing system; we note, however, that the measure, da, needs to be replaced by, χ(α;a)da, in order to account for what actually happens, and since χ(α;a)≥1, invariably, the realized entropy in-process is always greater than the modality, α.

However, in the sense of Nernst (please see Figure 3 above and click on it to make it larger if required), the only “pure states” that occur, occur at the end of process at a=b=1, and the delivery and receipt of product, both of which engage the entire σ-algebra, (N), with Ω(N)=α.

For every modality, 0<α<+∞, there is only one process as (a)→(b)→(a), but the process (a)→(b) which converts payables to receivables, is not the same as the process (b)→(a) which converts receivables to payables because the former occurs in the “process space” whereas the latter occurs in the space of the “trading connections”; if there were an inspired “management” in the latter, with respect to all of them, it would be some “principle” that is common to all of them, and it might not be the same as the former, but they both end-up with the same modality if they are trading partners; for more details, please see our Posts “(P&I) The Food Chain” and “(P&I) The Process Discordant“.

However, the “one” process, (a)→(b)→(a), and (a)→(b), and (b)→(a), may be implemented in different ways that depend on the σ-algebra, (N), and, therefore, its generators which could be one or more companies, or a market, or an entire economy.

We also know that what we have called the “end of process” (please see our Post “(P&I) The Process End-Of-Process“) occurs at a=b=1 in two steps: the “delivery of product” which we can describe as an “operator”, or transformation, S:(N)→(N), and the “receipt of product” which is also an operator, T:(N)→(N), and if N’=S‾¹(N) and N”=T‾¹(N) are the inverse images of (N) by these operators, and p((N’))=p((N”))=p((N))=1, even though the sets are quite different, and it’s noteworthy that the “delivery of product” and the “receipt of product” engages the entire σ-algebra, (N), if the modality is to be maintained; nor is “time” a factor because the transformations are “state transformations” and effected by the “implementing sets” according to the modality; moreover, the entropy that is realized at a=b=1 will be decreased because the “delivery of product” and the “receipt of product” are done in-process and the resulting a=p((a)) and b=p((b)) will both be less than one.

The M-condition for the “delivery of product” is log(a) = lim (1/(b^α))×log(1+b^α) as b→0; there are no such sets (a), but we know that for each b, as b→0, there is a set (a’) which satisfies the 2nd E-condition, b×log(b)=α×log(a’) and a’=p((a’))→1 and 1+log(a’)= lim (1/(b^α))×log(1+b^α) as b→0 where the logarithm is required because we know that log(1+x)/x→1 as x→0 (L’Hôpital’s Rule) and if the latter exists, then so must the former and they are same.

Similarly, the N-condition for the “receipt of product” is 1+log(a”) = -lim (1/(b^α))×log(1-b^α) as b→0; in both cases, a’=p((a’))=a”=p((a”))=1, but a’ and a” are not necessarily the same sets because for each b>0, log(a’)-log(a”)=(1/(b^α))×log(1+b^α) + (1/(b^α))×log(1-b^α) =(1/(b^α))×log[(1+b^α)×(1-b^α)]=(1/(b^α))×log[1-(b^α)^2]<0 so that the (a’)-sets are always “smaller” than the closing (a”)-sets, which change is due to the “receipt of product”.

Both of these limits need to be derived, or “enabled”, in the pre-image, N’ and N”, respectively, and the limiting sets, (a’) and (a”), are never actually obtained, but we can discover the values of a’ and a” from the implementation sets that are used to effect the limit.

What’s missing is a map of the “configuration space”, but it depends critically on the “selection functions”, χ(α;a)da and χ(α;b)db, and it can be likened to a “pinball”-game with three “throbbing” orbs that are changing in size depending on a, b and the modality for Company A through E. It’s “hard, man, really hard” and there’s no room for “subjectivity” or “randomness”, but we know that it’s computable, and doable in a qualitative way, at least, and what else should we work on while “The Market” is still sleeping on its “risk-adjusted” returns?

For more applications of these concepts please see our Posts which rely on the 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 demonstrated societal norms of risk aversion and bargaining practice. And for more on The Process, please see our Posts The Food Chain and The Process End-Of-Process.

And for more information on real “risk management” and additional references to the theory and how to read the charts and tables, please see our Post, The RiskWerk Company Glossary; we’ve also profiled hundreds of companies in these Posts and the Search Box (upper right) might help you to find what you’re looking for.

And for more on what risk averse investing has done for us this year, please see our recent Posts on The S&P TSX “Hangdog” Market or The Wall Street Put or specialty markets such as The Dow Transports & Utilities or (B)(N) The Woods Are Burning, or for the real class actionLa Dolce Vita – Let’s Do Prada! and It’s For You, Dear on the smartphone business.

And for more stocks at high prices, The World’s Most Talked About Stocks or Earnings Don’t Matter – NASDAQ 100. And for more on what’s Working in AmericaBig OilShopping in America or Banking in America, to name just a few.

Postscript

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