# Blog:

This blog discusses ideas, drafts and papers about the whole spectrum of systems theory and anything else that concerns the WissTec R&D Services UG.

#### 19.2.2021 - Social Systems Security 6:

###### Nazi "Logic":

(By Eike Scholz)

This is a common type of fallacy, that really should be avoided. It was prominently used in Nazi ideology, but it is much more common. Despite of Godwin's law the name is chosen to make it easy for ideologues of all sorts to drop the bullshit this kind of "logic" is.

Nazi "logic" is a deontic logic fallacy, that consists of three steps:
1.) Reasonable Judgment:
All $$x \in X$$ should $$Y$$.
2.) Ideological Narrowing:
All $$x' \in X'$$ under consideration are in $$X$$.
All $$x' \in X'$$ should $$Y$$ .
The formal problem is, that usually $\{\ x' \in X'\ |\ x' \text{ is not under consideration }\ \} \setminus X \ne \emptyset \text{ .}$ In words: Usually there are $$x'$$ in $$X'$$, that are not under consideration, not in $$X$$ and for which it is not true that they should $$Y$$. (Update: Fixed text in the formula! 😅 )

Observable Examples Cases:
• Nazis and the Jews.
• Racists and other races.
• Feminists and the "masculine".
• Communists and the bourgeoisie.
• This post. Making it a sound argument is left to the reader as an exercise.

• Why is this post in this list? Well, consider the entailed argument:
1. All ideologues committing this fallacy should stop this kind of bullshit.
2. Of all ideologues, the feminists under consideration condemn the masculine and judge, that masculine humans should atone for their wrongdoing and/or wrongbeing.
3. All feminists should stop this bullshit.
The fallacy of attribution in this case is unsound only for the reason, that in order to stop something, you need to have to be doing it. It is a kind of "When will you stop beating your wife?" situation, when the questioned actually may not beat his wife ever.

This particular example shows, that the line between Nazi "logic" and sound argumentation is sometimes just the line between careful and not careful wording and/or listening. That, given the way the Nazis rose to power, should not surprise anyone. The German voters did not listen carefully. They apparently thought, that the condemnation of the Jews by the Nazis is just lazy wording - lazy wording similar to the way feminists are put into the list above.

Thus be careful with ideological narrowing. Due to natural variation, there are too many fellow human beings in any large population, who actually like domination and causing related harm and pain. For them any kind of ideology is just a welcome excuse to indulge in their dark desires - without fear of losing their face or severer consequences. That is what one primarily observes, if looking at historical atrocities justified by ideologies. The ideologies served their purpose well in that case, just not for those who created or followed them to crate a, in their view, better world.

A certain type of follower can be worse for an ideology than any external adversary could be. Every ideology based group, that was unable to control this kind of follower, failed in the medium to long run. Often after inflicting immeasurable suffering and pain for the pleasure of those who like to indulge in that.

In an age of technological proficiency, that is easily sufficient to produce weapons of mass destruction, it will be a question of global survival to learn to handle potent ideologies with adequate respect and maturity, since those who suffer for the pleasure of others may have nothing left to lose.

The road to hell is paved with good intentions, as the saying goes.

#### 25.1.2021 - Important Announcement:

Please do not use email to contact us. Use Threema, WhatsApp or WeChat. We have severe problems with spam causing email system failures and a lot of valid communication gets suppressed as spam.

Threema: AWEAT6EM (Fingerprint: ccccc337525ee09ed331415cb9bb0ea1)
WhatsApp: +49 173 3678985
WeChat: wxid_yputii8qij1g12 (qr-code)

#### 18.1.2021 - Economic Systems Theory 1:

###### Accounting Microfoundations for Economics:

$$\def\N{{\mathbb N}} \def\Z{{\mathbb Z}} \def\T{{\mathbb T}} \def\R{{\mathbb R}} \def\W{{\mathbb W}} \def\E{{\mathbb E}} \def\X{{\mathbb X}} \def\A{{\mathbb A}} \def\ec{{\mathfrak{d}}} \def\norm#1{{\left\lVert#1\right\rVert}}$$ (By Eike Scholz)

In this post we will provide a preview for a formalism, that is currently in development. It allows to derive economic macro models from data, that is generated by actual real world accounting systems. It aims to be simplest to use with the systems buildable with our recently developed accounting backbone IT-infrastructure designs. With respect to practical problems, the formalism is required to solve stability and control problems for accounting subsystems. For example, solving inflation control problems of digitally managed currencies. Further such a formalism is needed to implement automated stabilizers effectively. With respect to zeitgeist keywords its an implementable formalism required to design stable, big-data, real-time market applications.

Update: Ongoing research made it advisable to introduce a relevant distinction between bookkeeping and accounting. When writing this article it had been used interchangeably. In the upcoming work everything that is called accounting in this post is called bookkeeping. In it accounting implies value calculations that where not sufficiently formalized when writing this blog post. The "fundamental law of accounting" in this post is actually the fundamental law of bookkeeping. The upcoming fundamental law of accounting is its equivalent regarding values.
Update 2: The law of equivalent exchange in this paper is also not the same as in the upcoming work. There the name will also refer to an equivalent with respect the value of an exchange.

Due to the focus on simplicity of use with our system designs, the formalism does not try to be similar to current ones, which whom the author is not very familiar, due to his prior work focus on modeling and simulation of systems in physics and engineering. However, the proposed formalism is generic and can be applied to any kind of problem, that needs to use mirofoundations for accounting subproblems. It has the following basic outline:
1. The economy is given as an enumeration of inventories, where each inventory is given by the amounts of goods in it.
2. Economic activity is given by exchanges of goods between inventories.
3. The evolution of the economy in time is described by a system equation, that uses the economic activity at a given time point as input. The system equation is a difference equation instead of the differential equations used in physics.
4. The accounting equations are generalized into a universal law, called the fundamental law of accounting, from which all accounting equations can be derived.
Update/Errata: That should have said accounting identity and the law is not a formal generalization. It is the "first" accounting identity which must hold for the most general accounting situations, where only few things are assumed or defined. The full work will make this clearer. The mathematics below is not affected by this.
This formalism has some similarity to physics, where the accounting equations would be conservation laws and the systems equation would correspond to the definition of force creating the system equation $$m \ddot x = f(x,\dot x)$$ where f is the sum of the effective forces and the equivalent of the economic activity in this formalism. For those not interested in the mathematical details an outlook is provided before the post that goes into the mathematical details.

Outlook:

At the moment WissTec is working to apply the formalism to the following problems:
1. Commodity currency systems and proper derivations of aggregate prices, baskets and inflation.
2. Representative currency systems, as special form of commodity currency systems trading an intangible abstract representative good as currency.
3. Compound standard currency systems, a representative currency system, that is a generalization of the historical gold standard systems. This concept is introduced by WissTec and currently being worked on. Pure fiat currency systems are a variant of compound standard currency system, that, instead of some tangible good like gold, use tokens, that guarantee to be left alone by the state as long as you pay your taxes in the given currency.
4. A subsumption formalism to rigorously derive simpler coarser economic systems from complexer finer ones. This is useful for modeling and big-data analysis.
5. Inflation control in compound standard currency systems.
6. Implementation of an internet protocol more or less directly implementing concepts of the following formalism.
Sadly, WissTec had not jet been able to secure funding to complete research on all of these topics. If you are interested in a cooperation in that regard, please contact us. Aside from doing actual research we can assist to write research grant applications or prepare material to get investor funding for commercial applications.

Mathematical Notations:

The blog post requires some mathematical maturity, that any (software) engineer and economist with sufficiently advanced training, is estimated to have. However notations differ between different fields, and the following listing should give clarity. If anything is missing please contact the author.
Functions yielding Functions:
The type $$Y\rightarrow Z$$ of a function $$g : Y\rightarrow Z$$ is identified as the set of mappings from $$Y$$ to $$Z$$ and $$X \rightarrow Y \rightarrow Z := X \rightarrow ( Y \rightarrow Z )$$ is the set of mappings taking arguments in X and yield functions mapping from $$Y$$ to $$Z$$. Thus for $$f : X \rightarrow Y \rightarrow Z$$ (or equivalently $$f \in X \rightarrow Y \rightarrow Z$$), $$x\in X$$ and $$y \in Y$$ the expression $$f(x)(y) \in Z$$ is the result of $$y$$ applied to the function resulting from $$x$$ applied to $$f$$.
Sequence Notation:
The set of all families with indices in the countable set $$I$$ and values in the set $$X$$ is denoted as $$X^{I} := I \rightarrow X \text{ .}$$ Sequences $$x \in X^{I}$$ use the usual syntax $$x_i := x(i) \text{ .}$$
Sequence Space with Counting Norm:
To avoid formal problems, for example with handling infinite sums, the following sequence space is used: \begin{align} l^1(\mathcal G) := \left\{ x\in \mathcal \R^\mathcal G \left| \norm{x} := \sum_{i \in \mathcal G} |x_i| <\ \infty \right.\right\} \end{align} It is a common type of Banach space, whose norm counts, i.e. sums the absolute values of the sequence elements.
Power Set:
The not so common notation $$2^X := \{ A | A\subseteq X \}$$ for the power set is used.
Number of Elements in a Set:
The number of elements in a set $$A$$, that is its cardinality, is denoted as $$\#A \text{ .}$$

Accounting Formalism Fundamentals:

The first thing required for this accounting formalism, is to introduce an classification of the types of goods accounted for, and define inventories and accounts with it:
Goods Classification:
A goods classification is a triple $$(\mathcal G,G,\frak G)$$, where $$\frak G$$ is the set of all goods classified, $$\mathcal G \subseteq \N$$ is a set of type ids and $$G : \N \rightarrow 2^\frak G$$ is a mapping, so that $$G(i)$$ is the set of all goods that are classifed as being of the type with index i. To be a proper classification, the mapping must satisfy the following properties: $$\forall i,j\in \N : i\ne j \Rightarrow G(i) \cap G(j) = \emptyset$$ $$\bigcup_{i\in \mathcal G} G(i) = \frak G$$ $$\forall i\in \mathcal G : G(i) \ne \emptyset \text{ .}$$ If $$i$$ is a type index, and $$g\in G(i)$$, then the short formulation "$$g$$ is of type $$i$$" is used. The set $$\mathcal G$$ is intended to be used to express how to combine different goods classifications, which is not of further interest in this introduction.
Content:
Let $$C=(\mathcal G,G,\frak G)$$ be a goods classification, then a content (of an inventory) is a vector $$a \in l_1(\mathcal G)$$ so that $$a_i\in \R$$ is the amount of the good of type $$i$$ in content $$a$$. The set of contents can shortly be written $$l_1(C_0)$$ using that the goods classification $$C$$ is a triple.
Distribution of Wealth:
Let $$C$$ be a goods classification. Then a family of contents $$w\in \W_C := \left\{\,\,w' \in l_1( C_0 )^\N \,\,\left|\,\, \norm{w'} := \sum_{i\in \N} \norm{w'_i} < \infty\,\, \right.\right\}$$ is a distribution of wealth. Distributions of wealth use the matrix syntax $$w_{gi} := (w_i)_g \text{ .}$$ in the infinite matrix space $$w\in l_1( C_0 )^\N$$.
Inventory:
Let $$C$$ be a goods classification and $$w\in \W_C$$ be a distribution of wealth, then $$w_i$$ is the $$i$$-th inventory of the distribution of wealth and $$i$$ is its inventory id.
Account:
Let $$C$$ be a goods classification and $$w\in \W_C$$ be a distribution of wealth, then $$w_{gi}$$ is the account of good $$g$$ of inventory $$i$$ from the distribution of wealth $$w$$.
In any traditional accounting process nothing must be crated or vanish. This property is integrated into the formalism using the following definition:
Fundamental Law of Accounting:
Let $$C=(\mathcal G,G,\frak G)$$ be a goods classification and $$w$$ be a distribution of wealth, then the requirement \begin{align} \forall g\in \mathcal G : \sum_{i\in\N} w_{gi} = 0 \end{align} is called the fundamental law of accounting, that is that, what is added to an inventory had to be removed from others. The law can be compactly written as \begin{align} \sum_{i\in\N} w_{i} = 0 \text{ .} \end{align}
Redistribution of Wealth:
The set of distributions of wealth adhering to the fundamental law of accounting is called the set of redistributions of wealth. Formally, let $$C=(\mathcal G,G,\mathfrak G)$$ be a goods classification, then the set of redistributions of wealth of $$C$$ is defined by $$\W_{C}^0 := \left\{\,\, w \in \W_C \,\,\left|\,\, \forall g\in \mathcal G : \sum_{i\in \N} w_{gi} = 0 \,\,\right.\right\} \text{ .}$$
Depending on how goods and inventories are grouped together, the fundamental law of accounting allows to derive any kind of accounting equation.

The above definition further allows to define the economy as a random time development of a redistribution of wealth:
Time:
The formalism uses the natural numbers as discrete time points denoted with the notation $$\T := \Z \text{ .}$$ to clearly distinguish time points from other natural numbers.
Economy:
Let $$C$$ be a goods classification and $$(\Omega,\mathcal F,\mathcal P)$$ be a probability space, then an economy of $$C$$ is a mapping $$d \in \E_{C} := \T \rightarrow \Omega \rightarrow \W_{C}^0 \text{ .}$$ In words: An economy is the time development of a redistribution of wealth as random variable. The following common notation is used: \begin{align} d_{i}(t)(\omega) &:= d(t)(\omega)_{i} \\ d_{gi}(t)(\omega) &:= d(t)(\omega)_{gi} \end{align}
The above definition of an economy uses the accountants view of an economy as a result of a random redistributive system. The woven in fundamental law of accounting states, that the total amount of content in an economy is always zero - if what is "taken" from nature is "given" back, nothing remains. This, for example, implies, that goods extracted from nature, like from mines, from farming land or from persons work hours, will be accounted for in negative amounts in corresponding production inventories. This is practical since the exact total amount of goods extractable from a resource is usually unknown and often hard to estimate. That might seem unintuitive, but has its inner logic. It is the purpose of an accounting system to track the flow of goods, without any loss or creation of goods in the accounting procedure. Its not the job of the accounting system to do capacity estimations of any kind. In the perspective of an accountant created goods come from some inventory and destroyed goods go to some inventory and remain there forever.

For the next part of the formalism economic activity is defined using a special type of redistribution of wealth:
Exchange:
Let $$C$$ be a goods classification, then $$\X_{C} := \left\{\,\,w \in \W_{C}^0 \,\,|\,\,\# \{\, i \in \N \,\vert\, w_i \ne 0 \,\} \le 2 \,\,\right\}$$ is the set of exchanges on $$C$$. The set $$\X_{C}$$ is precisely the set of redistributions of wealth affecting two or zero inventories, as the next lemma, called law of equivalent exchange, will show.
Law of Equivalent Exchange:
For all $$w\in\X_{C}$$ there exist a content $$c\in l_1({C_0})$$ and inventory ids $$k,l\in \N$$ so that $$w_i= \begin{cases} c &\text{ if } i = k \\ -c &\text{ if } i = l \\ 0 &\text{ otherwise } \end{cases}$$ where $$c$$ is called the exchanged content. In words: In an exchange one inventory loses exactly what the other gains.

Proof: Due to the definition of exchanges there exist contents $$c,d\in l_1(C_0)$$ and $$k,l\in\N$$, so that $w_i= \begin{cases} c &\text{ if } i = k \\ d &\text{ if } i = l \\ 0 &\text{ otherwise } \end{cases}$ and the fundamental law of accounting yields $0 = \sum_{i\in \N}w_i = c + d$ and thus $$d=-c$$.
Economic Activity:
Let $$C$$ be a goods classification and $$(\Omega,\mathcal F,\mathcal P)$$ be a probability space, then the set of economic activities is defined as: $${\small \A_{C} := \left\{\,\, A : \T \rightarrow \Omega \rightarrow \X_{C}^{\N} \,\,\left|\,\, \forall \omega\in\Omega : \forall t\in\T : \norm{A(t)(\omega)}:=\sum_{k\in \N} \norm{A_k(t)(\omega)} <\infty \,\,\right.\right\} }$$ An economic activity is therefore a mapping from a time point to a family of random variables yielding exchanges happening at a given time. The probability space is an implicit argument and used as source for randomness. For empirical time series analysis it usually does not need to be completely constructed, since constructing the required marginal distributions will suffice.
Economic activity, at any given time point, is usually a series with a finite amount of non-zero exchanges, despite the fact, that the above definition would allow infinite series, as long as the sum converges in the given norm. The formalism can be generalized to infinite economies with infinite economic activity, where subsets of inventories yield economies of the kind defined above. However, that is not of practical interest, since the visible universe is of finite volume. Further, the required math would need more complicated norms and definitions. But if you are, for fun, interested in asymptotic estimations of economies of hypothetically intergalactic species, you can ask the author for some recreational math.

With the economic activity formalized, it is possible to formalize the notion of an economic system and define the systems equation governing the time dependent behavior of an economy.
Economic System:
An economic system is a tuple $$(C,\mathcal A)$$ where $$C$$ is a goods classification and $$\mathcal A \subseteq \A_{C}$$ is the systems set of admissible economic activity.
Economic System Equation:
The equation $$d(t+1)(\omega) - d(t)(\omega) = \sum_{k\in \N} A_k(t)(\omega)$$ for an economic activity $$A$$ and an economy $$d$$ is called the economic system equation. It formally states, that the change in the (re)distribution of wealth at a given time point, is given by the accumulated exchanges of the economic activity at that time point.
The above formulation allows to give a closed form initial value problem solution for the economy of an economic system.
Analytic Economic System Solution:
Let $$(C,\mathcal A)$$ be an economic system, then, given an economic activity $$A\in\mathcal A$$ and the initial value $$d_0=d(t_0)(\omega)$$ for $$\omega\in\Omega$$, the economy is given by: $$d(t)(\omega) = \begin{cases} d_0 + \sum_{t'=t_0}^{t-1} \sum_{k\in\N} A_k(t')(\omega) \text{ if } t\ge t_0 \\ d_0 - \sum_{t'=t}^{t_0-1} \sum_{k\in\N } A_k(t')(\omega) \text{ if } t < t_0 \end{cases}$$
Proof: Let $$\omega\in\Omega$$, then for $$t \ge t_0$$ it holds
{\small \begin{aligned} &d(t)(\omega) - d(t_0)(\omega) \\ =& d(t_0+(t-t_0))(\omega) - d(t_0)(\omega) \\ =& \sum_{j=0}^{t-t_0-1} (d(t_0+j+1)(\omega) - d(t_0+j)(\omega)) \phantom{xxx} \text{ (as telescoping sum) }\\ =& \sum_{j=0}^{t-t_0-1} \sum_{k\in\N} A_k(t_0+j)(\omega) \phantom{xxx} \text{ (inserting economic systems equation) }\\ =& \sum_{t'=t_0}^{t-1} \sum_{k\in\N} A_k(t')(\omega) \end{aligned}} and for $$t < t_0$$ equivalently {\small \begin{aligned} & d(t_0)(\omega) - d(t)(\omega) \\ =& d(t+(t_0-t))(\omega) - d(t)(\omega) \\ =& \sum_{j=0}^{t_0-t-1} (d(t+j+1)(\omega) - d(t+j)(\omega)) \phantom{xxx} \text{ (as telescoping sum) }\\ =& \sum_{j=0}^{t_0-t-1} \sum_{k\in\N} A_k(t+j)(\omega) \phantom{xxx} \text{ (inserting economic systems equation) }\\ =& \sum_{t'=t}^{t_0-1} \sum_{k\in\N} A_k(t')(\omega) \text{ .} \end{aligned}} Rearranged and combined these equations yield the analytic solution.
Depending on the exact form of admissible economic actions, different economic systems can be investigated. To continue, it is useful to introduce some abbreviating notations. The complete set of data required to determine the time development of a system is called a setting. Data in a setting can, for example, include input data and boundary constraints. This leads to the following definitions:
Economic System Setting:
For an economic system $$(C,\mathcal A)$$ a system setting is given by $$\sigma=(A,t_0,d_0) \in \mathcal A\times \T \times \W_C^0$$ where $$t_0$$ is the initial time and $$d_0$$ is the initial state.
Economic System Solution Notation:
Let $$(C,\mathcal A)$$ be an economic system and $$\sigma=(A,t_0,d_0)$$ a system setting. The the corresponding system solver is denoted as $$\ec : \A_{C} \times \T\times \W_{C}^0 \rightarrow \E_{C}$$ and the solution $$\ec(A,t_0,d_0) := \ec(\sigma) := t \mapsto \omega \mapsto \begin{cases} d_0 + \sum_{t'=t_0}^{t-1} \sum_{k\in\N} A_k(t')(\omega) \text{ if } t\ge t_0 \\ d_0 - \sum_{t'=t}^{t_0-1} \sum_{k\in\N } A_k(t')(\omega) \text{ if } t < t_0 \end{cases}$$ is the solution of the initial value problem for economic activity $$A$$ with intitial value $$d(t_0)(\omega) = d_0$$.
We conclude the basics of the discussion with an application example called production circulation system. For brevity the random event argument $$\omega\in\Omega$$ will be omitted in the following derivations.
Production Circulation System:
A production circulation system is a tuple $$\mathcal P = (\mathcal S, \mathcal I_\circ)$$ of an economic system $$\mathcal S=(C,\mathcal A)$$ and a set of circulation inventory ids $$\mathcal I_\circ\subset\N$$, that defines a projection $$\pi_\circ^\mathcal P(d)_i := \begin{cases} d_i &\text{ if } i \in \mathcal I_\circ\\ 0 & \text{ otherwise } \end{cases}$$ that is used to define the admissible set of economic activity by $$\mathcal A = \left\{\,\,A\in \A_{C} \,\,\left|\,\, \exists (t_0,d_0) \in \T \times \W_{C}^0 : \pi_\circ^\mathcal P\left( \ec(A,t_0,d_0)\right) \ge 0 \,\,\right.\right\}$$ where the inequality in the set comprehension is to be understood point wise in time, inventory, good and random event. That is, all circulation inventories have at all times only accounts with a non-negative balance.
A production circulation system splits the inventories in two sets, so that it is possible to derive a clear conservation law for goods in the circulation inventories. However, to actually derive the conservation law the following definitions are required:
Projection Extracting Production Inventories:
Let $$\mathcal P$$ be a production circulation system, then the projection onto the pruduction inventories is defined as $$\pi_\pm^\mathcal P(d) := d - \pi_\circ^\mathcal P(d)$$ which is equivalent to the definition: $$\pi_\pm^\mathcal P(d)_i := \begin{cases} d_i &\text{ if } i \notin \mathcal I_\circ\\ 0 & \text{ otherwise } \end{cases}$$
Projection Extracting the Distribution of one Good:
The following projection extracts the distribution of one good \begin{align} \pi_g(d)_{hi} := \begin{cases} d_{gi} &\text{ if } h = g \\ 0 &\text{ otherwise } \end{cases} \end{align} and as corollary result it follows that $$\pi_g \circ \pi_\circ^\mathcal P = \pi_\circ^\mathcal P \circ \pi_g$$ as well as that it holds point wise that: \begin{align} d \ge 0 &\Rightarrow \pi_g(d) \ge 0\\ d \ge 0 &\Rightarrow \pi_\circ^\mathcal P(d) \ge 0 \end{align}
Stationary Production of a Good:
An economic activity $$A$$ has stationary production of good $$g$$ at time $$t$$ if $$\pi_\pm^\mathcal P\left(\pi_g\left(\sum_{k\in\N}A_k(t)\right)\right) = 0$$
Goods Conservation Law:
Let $$\mathcal P = (\mathcal S, \mathcal I_\circ)$$ be a production circulation system, $$t\in \T$$, $$A\in \mathcal A$$ and $$g$$ a good id, then \begin{aligned} &\pi_\pm^\mathcal P \left( \pi_g\left( \sum_{k\in\N}A_k(t)\right)\right) = 0 \\ &\phantom{xxxx}\Longrightarrow \\ &\norm{\pi_\circ^\mathcal P (\pi_g( \ec(A,t_0,d_0)(t+1)))} = \norm{\pi_\circ^\mathcal P(\pi_g(\ec(A,t_0,d_0)(t)))} \text{ .} \end{aligned} In words: If the production of $$g$$ is stationary, then its total amount in the circular inventories does not change.

Proof: Let $$a := \sum_{k\in \N} A_k(t)$$ then using the definition of $$\pi_\pm^\mathcal P$$ it yields $$\pi_\pm^\mathcal P(\pi_g(a)) = 0 \Rightarrow \pi_\circ^\mathcal P(\pi_g(a)) = \pi_g(a) \label{GoodsConservationLawProofFixPointEq}$$ and the economic systems equation further yields $$\ec(A,t_0,d_0)(t+1) = \ec(A,t_0,d_0)(t) + a \text{ .}$$ This allows to conclude the proof with {\small \begin{aligned} &\norm{\pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t+1))\right)} \\ &= \sum_{i\in\mathcal I} \norm{\pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t+1))\right)_i} \phantom{xxx} \text{ definition of the norm } \\ &= \sum_{i\in\mathcal I} \sum_{g\in \mathcal G} |\pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t+1))\right)_{gi}| \phantom{xxx} \text{ definition of the norm } \\ &= \sum_{i\in\mathcal I} \sum_{g\in \mathcal G} \pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t+1))\right)_{gi} \phantom{xxx} \text{ since  \ec_i(A,t_0,d_0)(t+1)\ge 0 for i\in \mathcal I_\circ } \\ &= \sum_{i\in\mathcal I} \sum_{g\in \mathcal G} \pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t) + a)\right)_{gi} \phantom{xxx} \text{ due to the system equation} \\ &= \sum_{i\in\mathcal I} \sum_{g\in \mathcal G} \left(\pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t))\right)_{gi} + \pi_\circ^\mathcal P\left(\pi_g(a)\right)_{gi}\right) \\ &= \sum_{i\in\mathcal I} \sum_{g\in \mathcal G} \left(\pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t))\right)_{gi} + \pi_g(a)_{gi}\right) \phantom{xxx} \text{ using the first step of the proof} \\ &= \left(\sum_{i\in\mathcal I} \sum_{g\in \mathcal G} \pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t))\right)_{gi}\right) + \left(\sum_{i\in\mathcal I} \sum_{g\in \mathcal G} \pi_g(a)_{gi}\right) \\ &= \left(\sum_{i\in \mathcal I}\sum_{g\in \mathcal G} \pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t))\right)_{gi}\right) + \left(\sum_{i\in \mathcal I} a_{gi}\right) \\ &= \left(\sum_{i\in \mathcal I}\sum_{j\in \mathcal G} \pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t))\right)_{gi}\right) \phantom{xxx} \text{ due to the fundamental law of accouting} \\ &= \left(\sum_{i\in \mathcal I} \sum_{j\in \mathcal G} | \pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t))\right)_{gi}| \right) \phantom{xxx} \text{ since  \ec_i(A,t_0,d_0)(t)\ge 0 for i\in \mathcal I_\circ } \\ &= \left(\sum_{i\in \mathcal I} \norm{\pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t))\right)_{i}}\right) \\ &=\norm{\pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t))\right)} \text{ .} \end{aligned}}

Conclusion:

The above formalism allows to express the accounting side of macro economic problems using micro foundations expressed as economic action, where the economic action is the sum of individually performed exchanges.

Further, the above goods conservation law is the first example, that simple macroeconomic properties can be derived from this formalism. The goods conservation law also states, that accounting will neither create nor destroy goods hold in circulation inventories, which are thus ideal to track intangible goods like currencies.

The goods conservation law provides a direct practical insight:

Depending on the laws and system implementation, private banks, that allow giro accounts with negative balance, may be creating money. That is, the banks internal accounting may or may not (implicitly) use negative balanced accounts in the production circulation system sense, to represent giro accounts with negative balance.
In the complete upcoming work we will show, that, even if private entities can produce money in a regulated way, the central bank interest rate can be used to destroy this kind of money, to achieve a relatively stable monetary system in some situations.
This should also make intuitively clear why the Chinese, de facto completely state owned, banking system has better control capabilities with respect to the stability of the currency, then most western systems - be it for better or for worse. The western systems are set up to not be destabilized by an ill-incentiviced elected government, trying to reward their voters with "printed" money. On the other hand: Less control capabilities for the government also implies less power to counter certain types of crisis.