To take the mean value m of a special or universal collection of catenals as the neutral value is tantamount to assuming that k(m) = 0, that

 ______               ___
  k(v) = 0  and that   v^ = 0

[v with a caret over it is the symbol for the catena value].
It cannot be proved that the mean value is always the neutral value, and vice versa. And to be able to disprove it, one would first have to make clear whether such a claim is a linguistic, psychological, physical, normative or other kind of claim (or all at once). Moreover, one would have to agree on the type of catenization function and the exact form of catenality to be chosen in each case. Here we shall treat the equality of neutral and average or mean value merely as a postulate, albeit a challenging one indeed. The simplest mathematical form of this hypothesis of mean-neutrality is:

  v^ = 0

(the mean catena value is 0).
It applies to every independent form of catenality, and both to universal and to certain special catenizations. For special catenizations the distinction between catenals taken into consideration (and included in the mean) and all other catenals (not included in the mean) must be relevant. As already suggested, one may suppose that such a distinction is relevant if the catenals belong to a closed system.

The hypothesis of mean-neutrality can be read in two directions: (1) given the neutral value (that is, its empirical equivalent), the mean is equal to it in a closed system or in the world at large; and (2), given the mean, the neutral value is equal to it. We also assume that the derivations are nonfactitious, if possible; this in accordance with the rule of nonfactitious priority. It should be interesting to first have a look at one or two cases in which the neutral value is given, before turning our attention to cases in which the mean is given or calculable.

With regard to the energy increase catena increase of energy (or gaining energy) is positive, decrease of energy (or losing energy) negative, and neither gaining nor losing energy, but having a certain amount of energy nevertheless, neutral. According to the hypothesis of mean-neutrality no energy will 'melt' into nothingness in a closed system and no energy will 'spring' from nothingness, since the mean energy increase in such a system has to be 0. The independent form of catenality concerned is, then, not energy as distinct from mass but energy inclusive of mass. If there is energy (including mass) which does seem to disappear into or to arise from nothingness, the physical system in question is simply not closed. This impossibility of a closed system's increase-catenary mean not being 0, and thus of energy melting into or springing from nothingness, is precisely what the physical definition of closed system rests upon. The hypothesis of mean-neutrality (or a more specific derivative postulate) thus underlies the very principle of the conservation of mass and energy, and similar, physical principles of conservation.

It might be objected that altho the hypothesis of mean-neutrality may hold in certain cases it does not hold in other cases. Yet, if someone believes to know a counterexample, it is imperative that all the basic assumptions are checked again. Take, for example, the suggestion that it would be false or absurd to assume that in a closed system all the objects are on the average (if the catenization is linear) at rest. Why could there not be more movement in one coordinate direction than in the opposite coordinate direction (granted that a closed system need not be spatially closed in all directions)? The flaw in this reasoning is that the movement of objects in the closed system is related to an external system, that is, a system in which we live ourselves, or a system of which the first, closed system would form part. But motion catenization with respect to that different or larger system (even if 'universal') is not relevant if the first system is a closed system at all. If it is, then rest in this system is the mean displacement per second; and the mean displacement per second in the two directions of a coordinate is then 0 in this particular system with respect to the coordinates of this system. That the average change of place may not be 0 with respect to a system in which we ourselves call things "at rest" or "in motion" is just not relevant if we are really talking about a closed system.

©MVVM, 41-67 ASWW

Model of Neutral-Inclusivity
Book of Instruments
Catenas of Attributes and Relations
The Scope of Catenization