*5.3. Measurement*

The described above numerical ⎧ ⎩Ξ ⎫ ⎭-version of the ⎧ ⎩Ξ ⎫ ⎭-brace " <sup>∪</sup>-phenomenology" makes it possible now to preliminarily formalize a concept, the absence of which deprives the theory of its basis. Namely, *measuring statistics by observation* A *over* S:

$$\text{QM-meassurement:} \qquad \left( [\lambda\_{1\prime}\mu\_1]\_{\prime}[\lambda\_{2\prime}\mu\_2]\_{\prime} \dots \right) \longmapsto \left( \mathfrak{f}\_1, \mathfrak{f}\_2, \dots \right) . \tag{48}$$

That is, the [*λ*, *μ*]-collection gets mathematically mapped into the f-statistics. This is a maximum of information provided by observation A . Mapping (48) annihilates the pairs [*λ*, *μ*]. Therefore, the inheritance/homomorphism of operations ∪ and 0 onto anything at all is eliminated. Upon operation (48), both the (<sup>f</sup>,<sup>κ</sup>)-sets and ∪-unions thereof, 0- operations, and the semigroup G per se disappear. As a well-known result, the distinctive feature subsequently referred to as a superposition will also disappear after measurement. The new numbers {<sup>f</sup>*s*} may be "added up" only as required by the different, i.e., the classical rule: forming the convex combinations (45). We note that the formalization of the measurement does not now depend on how the mathematical map [*λ*, *μ*] (f) would be further implemented—it is a separate job [6]—or how the *t*-dynamics would be introduced.

**Remark 9.** *The incorporation of t-dynamics into the theory is still impossible due to the absence of mathematics to be applied to instants t*<sup>1</sup>*, t*<sup>2</sup>*. Accordingly, no physical t-process, a temporal imitation of the measuring, or its dynamical description may correspond to the mathematical mapping shown in Mapping* (48)*. The known "conceptual" problems with the collapse dynamics [1,13,18,45,115] are actually non-existent [15,87,93]. More precisely, they stem from the blurring of meaning that we typically give to the words "states" (what is that?), "ensembles" (what are they comprised of?), and "dynamics/collapse" (of what?). In regard to the latter, the authors of the book [58] speak out in a most definitive manner—the "fairy tales". See also Section 10.3 further below.*

In Section 2.1, the fundamental premise of the *α*-symbol-based distinguishability ≈ was the foundation of the entire subsequent language; "two clicks are never identical" ([126], p. 761). One then observes that the measurement or its outcome will essentially remain a vacuous term "for microsystems nothing can be directly measured" ([92], p. 304) until it invokes the concept of a QM-state, i.e., the ⎧ ⎩Ξ ⎫ ⎭- and *α*-objects. In the following, we shall see that, as a rough guide, everything that is observable whatsoever is a function of the state and of the state space.

Once again, it is stressed that the concept of the state must precede the notion of measurement, rather than the reverse. "[J.] Bell fulminated against the use of the word "measurement" as a primary term when discussing quantum foundations" ([30], p. 262). See also the entire chapter 23—"Against "measurement'"—in ([28], pp. 213–231).

#### *5.4. Covariance with Respect to Observations ("the same")*

Up to this point, we had had no need for the matching of observation A with observation B, although it is clear that a description based on a certain specified A will inevitably be non-invariant with respect to the tool {A , B, ...}—"observation space"—and unacceptable (pt. R) due to the impermissible exclusivity of the set {*<sup>α</sup>*1, ...}. At the same time, we do not have anything but {A , B, ...} and micro-acts (12) (pts. T and M). In the brace, this fact has already been present; transitions A are combined into integrities (24). Logically, however, the ⎧ ⎩Ξ ⎫ ⎭ A -, ⎧ ⎩Ξ ⎫ ⎭ B-objects are incomparable and isolated from each other as carriers of statistics of different origins.

On the other hand, "*the same* is observed by instrument B, as by instrument A ". Although this context has not ye<sup>t</sup> been invoked, without it, the application of set-theoretic constructs to physics is devoid of meaning, just like the union of the speeds of an electron and of the Moon into a set {*<sup>v</sup>*e, *<sup>v</sup>*M}, with the subsequent creating a certain physical characteristic of this "two-body system"—say, the mean velocity 1 2 (*<sup>v</sup>*e + *<sup>v</sup>*M). Indeed, "The statements of quantum mechanics are meaningful and can be logically combined *only* if one can imagine a *unique experimental context*" ([40], p. 115).

Thus, the global structuredness is required in the set of various ⎧ ⎩Ξ ⎫ ⎭-data according to the context "the same, identical" or its negation. Apparently, this addition implies such entities as "the same particle", "in the same preparation/state", "under the same temperature and **M**-environment (12)", "the same closed system S", "in the same external field", "in the same interferometer" with "the same detectors/solenoids", the like [33,40,44]; the short and generalized notation S, **<sup>M</sup>**,... . All the notions here, including the state, are physical conventions, ye<sup>t</sup> their formalization and modeling are called for the creation of a theory (Section 2.4).

The notion "with the same initial data" falls under the same category, if the intention is to use the term time *t*. Again, the very creation of the ⎧ ⎩Ξ ⎫ ⎭-brace as a set "by the piece" is from the outset thought of as (Section 2.4) a creation on the assumption of common S, **M**, ... . For instance, the A -statistics ⎧ ⎩Ξ ⎫ ⎭ A are gathered within "the same" S, **<sup>M</sup>**,... as the B-statistics ⎧ ⎩Ξ ⎫ ⎭ B. On its part, any variation is sufficient to obtain "not the same", even if we "*envision it as null*" in the spirit of the widely known "without in any way disturbing a system" ([131], p. 234). To take an illustration, equipment of interferometer (Section 6.5) with additional "which-slit" detectors is already at variance with the notion of "the same

S". In similar cases, we end up in situations similar to Case (23) since the detectors cause an *α*-distinguishability.

Notice that the notions "the same" and "distinguishable" (Remark 2), while antonymous, mutually exclude each other. Semantically, one without the other makes no sense, which closely resembles Bohr's conception of complementarity [78].

It follows from the above that in order to match A and B, the *metatheoretical* [149] category S, **<sup>M</sup>**,... is required; however, we are only in possession of the ensemble brace ⎧⎩Ξ⎫⎭A and ⎧⎩Ξ⎫⎭B (pt. T). On the other hand, without joint consideration of *the two instruments*, i.e., without introducing a mechanism for the mathematical matching ⎧⎩Ξ ⎫⎭A ⎧⎩Ξ ⎫⎭<sup>B</sup>, ⎧⎩Ξ ⎫⎭A ⎧⎩Ξ ⎫⎭<sup>B</sup>, . . . , the segregation of the ⎧⎩Ξ⎫⎭-objects is absolute. (It is clear that the matching of single micro-events Ψ <sup>A</sup> *αs* and Ψ <sup>B</sup> *βj* is also futile.) It is impossible to associate physics with the abstractly segregated ⎧⎩Ξ⎫⎭A -brace. Otherwise, the solitary object ⎧⎩Ξ⎫⎭A , generating nothing more than statistics provided by the single instrument A , would yield a description of everything, which is absurd by pt. R••. The physical contents (to come) arise precisely through the above-mentioned matching (see Section 6.4 below).

As a result, we adopt a kind of the relativity-principle analogue—a tenet on the quantum observational covariance.

III Theory should introduce a means of equating the macro-observations (pts. O + M) by differing instruments {*<sup>α</sup>*1, ...}<sup>A</sup> = {*β*1, ...}<sup>B</sup> under a common (the same) experimental environment S, **<sup>M</sup>**,... .

> (*The third principium of quantum theory*)

Cf. [22] (p. 632) and mathematical analogies [85,86].

#### 5.4.1. Semantic Closedness and the Equal Sign =

We are currently returning once again to Section 2.1, falling into a situation when the case in hand does not just entail fundamental theory in the form of ,math- + ,physical "blabla-bla"-, while, continuing on an informal note, the mathematics of physics—quantum mathematics—is being created "from scratch". When building up this math, it is impossible to forego the physical conventions S, **<sup>M</sup>**,... ; meanwhile, any preliminary and the formal characterization for S, **<sup>M</sup>**,... is ruled out.

Indeed, the attempts to mathematically formalize the physical context of observation, rather than observation itself, will not logically manage without another "more fundamental" observation, in this case, of the very experimental environment. The semantic cycling is apparent here, and any of its mathematization will lead to a retrogression of definitions into infinity, which is known as the "von Neumann catastrophe" ([80], pp. 158– . . . )) or as "trying to swallow itself by the tail" ([28], p. 220). Which is why, once again, the "Box (6) method" prohibitions are required. See also a paragraph containing the capitalized emphasize "CANNOT IN PRINCIPLE" on p. 418 of the work [15]. Sooner or later, it will have to be declared that mathematics will be created for *the* convention S, **<sup>M</sup>**,... and that this mathematics will be a mathematical model for this S, **<sup>M</sup>**,... . The analogous argument—"mathematics is there to serve physics, and not the other way round" ([16], p. 242; L. Hardy)—has already long been met in the literature [23,33,40]. In connection with the "general contextual models", see the books [64,150] (the Växjö-model, "quantum contextuality") and bibliography therein.

**Remark 10** (semantic)**.** *To avoid the just mentioned linguistic closedness—a kind of mathematical "pathology" of the physical and natural languages—a description that lays claim to the role of an unambiguous/rigorous theory requires a careful separation of the object- and meta-languages. For more detail, see [105] (Sections 14–16), [106] (Section V.1), and [136] (Section 3.9). For this reason, the constructs should track the blending of the object QM-domain (syntax) and the meta-domain (semantics) and, more generally, the penetration of extra-linguistic elements of thinking [148] into QM. The notion of "the same* S*,* **<sup>M</sup>***,... ", which is intuitive in the natural language, should explicitly be indicated as an external and fundamental category (pr.* III*), and its circular re-interpretations/retranslations within the theory should be banned. That is, re-stating "the sameness* S*,* **<sup>M</sup>***,... , the*

*identical* S*,* **<sup>M</sup>***,... " by way of word or of the equality symbol* = *between some other entities is forbidden.*

• *"The same" may no have a definition in terms of anything else. It exists prior to theory and has only a meaning (*= *verbal context), though its natural-language descriptions may be of great variety and be "presented to us in wildly different ways" ([86], p. 2 and the whole of the Section "The awkwardness of equality").*

*One could, e.g., accept the typical verbal vehicle "a complex of conditions, which allows of any number of repetitions" (quotation from the literature). It is clear that the words "complex . . . allows . . . repetitions" here are just another semantic equivalent to the word "the same* S*,* **<sup>M</sup>***,... ". The physics terminology per se (Sections 6.4, 6.5, and 9.1) will become accessible when physical concepts are introduced via the originating—and obligatorily very ascetic—quantum-mechanics language. See also the selected thesis on page 70.*

It is crucial to immediately note that, in the same manner, the classical description contains the cited arguments in their entirety. It is easy to convince that such a description also implies implicitly that which is designated above as S, **<sup>M</sup>**,... ; otherwise, the physical reasoning would be entirely impossible. "[W]e often prefer to regard a number of outcomes of distinct physical operations as registering the same property, . . . representing the same measurement. . . . permitting an unrestricted identification of outcomes would lead to "grammatical chaos"" (Foulis–Randall ([111], p. 232)). More to the point, the physics and mathematics not merely have been closely interwoven with each other. Any recursive procedure of definitions will inevitable result in either a cyclic definition at some level, or a definition that refers outside not only of the physics but even of the math. Hence, the hierarchical arrangemen<sup>t</sup> of notions/. . . /definitions—a property that is frequently uncontrolled and violated in the human thinking—can only be meaningful if at least one knot in the definition network is externally defined. In this work, that basic points are, as a rough guide, the brace Ψ <sup>A</sup> *α* and the notion of "the same S, **<sup>M</sup>**,... ", motivated in Remarks 2 and 10, respectively.

**Remark 11.** *Here, the situation is similar to the role of the axiom of choice in the ZF-system [134,147]. It has been well known for a long time that the axiom is often subconsciously implied ([149], Chs. II, IV); it can also not be either circumvented or ignored. Another counterexample to "infinite retrogression and circularity" in logic comes from the very same system. This is a ban on infinite chain of set memberships* ∈ *on the left*

$$\|\cdot \cdot \cdot \in X\_n \in \cdot \cdot \in X\_2 \in X\_1 \in X\_0$$

*(the regularity axiom* [∀*x* ∈ *X*, *x* ∩ *X* = ∅] ⇒ [*X* = ∅]*) under the permissibility of the infinite* (∈)*-continuing to the right:*

$$X\_0 \in X\_1 \in X\_2 \in \cdots \in X\_n \in \cdots \in \cdots$$

*(not rigorously, the infinity axiom) [120,134].*

*The obvious parallels here are the famous Russell paradox [149] or a chaos in the computer file system when the "hard links" from a folder to the parent folder are allowed. Thus, the relations* ∈ *"downwards" to the left and necessarily terminates in something, i.e., in a set X*0 *that contains nothing:* ∅ = *X*0 ∈···∈ *Xn* ∈···*. Therefore, one needs to give "meaning" to the only set—the empty one* ∅*. Incidentally, it is these axioms that guarantee the existence of infinitely many ordinal numbers* (106) *and the uniqueness of this structure. The ordinals and numbers have yet to be dealt with further below in more detail.*

All that remains is to add that no theory in physics is feasible without re-calculations of physical units and of vectors/tensors without transformations in the fiber superstructures over manifolds, etc. Accordingly, the considerations on invariance and on transformations should be present in the quantum case as well, but it, which is its principal difference from

the classical case, still lacks the concepts of physical quantities/properties (see Section 6.4). Therefore, such argumentation may only be applied to those objects that we have at our disposal, i.e., to the ⎧ ⎩Ξ ⎫ ⎭-brace. The renunciation of pr. III would actually be tantamount to the inability to make the physics theories whatsoever.

Now, pr. III and the "quantum diversity of the reference frames" {A , B, . . . } require a kind of factorization of the entire family { ⎧ ⎩Ξ ⎫ ⎭ A , ⎧ ⎩Ξ ⎫ ⎭ B, ... , ⎧ ⎩Ξ ⎫ ⎭ A , ⎧ ⎩Ξ ⎫ ⎭ B, ...} with respect to the conception S, **<sup>M</sup>**,... , i.e., the introduction of an operation of equating the results ⎧ ⎩Ξ ⎫ ⎭ A , ⎧ ⎩Ξ ⎫ ⎭ B that came when observing S. ⎧ ⎩Ξ ⎫ ⎭ A = ? ⎧ ⎩Ξ ⎫ ⎭ B should not be immediately put since these braces are simply different sets. That is why, with isolated semigroups

$$\{\{\{\bigoplus\_{n'}^{\bullet\_{\omega}}\{\Xi\_{\iota}\}^{m\_{\bullet}}\{\Xi\_{\iota}\}^{m\_{\bullet}},\dots\}\},\quad\{\{\{\widehat{\#}\}^{m\_{\bullet}}\{\Xi\_{\iota}\}^{m\_{\bullet}}\{\Xi\_{\iota}\}^{m\_{\bullet}},\dots\}\},\quad\dots$$

at our disposal, we have to conceive of them as elements of a new set H of objects having a single nature, 1) to carry out the mapping {<sup>G</sup>A , GB , ...} %→ H, assigning new representatives |**<sup>Ξ</sup>**A ∈ H to the ⎧ ⎩Ξ ⎫ ⎭-brace, and 2) to equip H with an equivalence relation |**<sup>Ξ</sup>**A ≈| **<sup>Ξ</sup>**B (the concept "the same" above). Let us implement all of that by the scheme

$$\begin{array}{rcl} \{\Xi\}\_{\mathcal{A}} := \left[ \begin{smallmatrix} \boldsymbol{\mu}\_{1} \\ \boldsymbol{\lambda}\_{1} \end{smallmatrix} \middle| \, \underline{\mathbf{a}}\_{1} \oplus \stackrel{\scriptstyle \mathsf{f}}{\operatorname{\bf curl}} \begin{smallmatrix} \boldsymbol{\mu}\_{2} \\ \boldsymbol{\lambda}\_{2} \end{smallmatrix} \middle| \, \underline{\mathbf{a}}\_{2} \oplus \stackrel{\scriptstyle \mathsf{f}}{\operatorname{\bf curl}} \cdots \quad \longmapsto \left[ \begin{smallmatrix} \boldsymbol{\mu}\_{1} \\ \boldsymbol{\lambda}\_{1} \end{smallmatrix} \middle| \, \boldsymbol{\mu}\_{2} \right\rangle \bigoplus\_{\begin{smallmatrix} \boldsymbol{\mu}\_{2} \\ \boldsymbol{\lambda}\_{2} \end{smallmatrix}} \left| \, \underline{\mathbf{a}}\_{1} \right\rangle \bigoplus\_{\begin{smallmatrix} \boldsymbol{\mu}\_{2} \\ \boldsymbol{\lambda}\_{2} \end{smallmatrix}} \left| \, \underline{\mathbf{a}}\_{2} \right\rangle \bigoplus\_{\begin{smallmatrix} \boldsymbol{\mu}\_{2} \\ \boldsymbol{\lambda}\_{1} \end{smallmatrix}} \left| \, \underline{\mathbf{a}}\_{1} \right\rangle \bigoplus\_{\begin{smallmatrix} \boldsymbol{\mu}\_{2} \\ \boldsymbol{\lambda}\_{1} \end{smallmatrix}} \left| \, \underline{\mathbf{a}}\_{2} \right\rangle \left| \, \underline{\mathbf{a}}\_{1} \right\rangle \left| \, \underline{\mathbf{a}}\_{2} \right\rangle \left| \, \underline{\mathbf{a}}\_{1} \right\rangle \left| \, \underline{\mathbf{a}}\_{2} \right\rangle \left| \, \underline{\mathbf{a}}\_{1} \right\rangle \left| \, \underline{\mathbf{a}}\_{2} \right\rangle \left| \, \underline{\mathbf{a}}\_{1} \right\rangle \left| \, \underline{\mathbf{a}}\_{2} \right\rangle \left| \, \underline{\mathbf{a}}$$

 .........

In this, the new addition + must of course homomorphically inherit operations 0 A , 0 B , . . . , and the extension of this definition throughout H is then made with the aid of the very equivalence ≈:

 ......

$$|\Xi\_{\boldsymbol{n}}'\rangle \oplus |\Xi\_{\boldsymbol{n}}''\rangle = \left| |\Xi\_{\boldsymbol{n}}''\rangle \approx |\Xi\_{\boldsymbol{n}}''\rangle \Rightarrow \right| = |\Xi\_{\boldsymbol{n}}'\rangle \oplus |\Xi\_{\boldsymbol{n}}''\rangle = |\Xi\_{\boldsymbol{n}}'\rangle \oplus |\Xi\_{\boldsymbol{n}}''\rangle \dots$$

The negation ≈ of the relation <sup>≈</sup>, e.g., |**Ξ** A ≈ | **Ξ** A , is exactly the very same distinguishability that was discussed in Sections 2 and 3.

For the sake of convenience, we adopt the regular sign = for ≈ in order not to introduce ye<sup>t</sup> a further homomorphism, which are already numerous, with more to come. In other words, the physics S, **<sup>M</sup>**,... is "concentrated" in the sign =, turning the empirical structures (49) into the A -, B-implementations of the object |**Ξ** ≡| **Ξ**A = |**<sup>Ξ</sup>**B under construction. The adequate term for it—the Info/Data-Source or "representative of information" (C. Brukner (2014))—corresponds to the preliminary prototype of the concept ˇ of a state, but we will remain within the standard term, disregarding its variance.

#### **6. Quantum Superposition**

.........

How come the quantum? . . . No space, no time—J. Wheeler (1989)

. . . postulation of something as a Primary Observable is itself a sort of theoretical act and may turn out to be wrong—T. Maudlin ([151], p. 142)

#### *6.1. Representations of States*

Let us simplify notation according to the rule [ *μ λ*] =: a. The sought-for relationships between A , B, . . . then turn into the key point of further construct—the equalities

$$\left[ \begin{array}{c} \text{representation} \\ \text{of } |\boldsymbol{\Xi}\rangle \text{-state} \end{array} \right] : \qquad \mathsf{a}\_{1}|\mathsf{a}\_{1}\rangle \oplus \mathsf{a}\_{2}|\mathsf{a}\_{2}\rangle \oplus \cdots \xleftarrow{\text{ $\mathsf{a}\_{\mathsf{the}}$ }} \mathsf{b}\_{1}|\beta\_{1}\rangle \oplus \mathsf{b}\_{2}|\beta\_{2}\rangle \oplus \cdots = \cdots \ \cdot \ \ \text{ (50)}$$

They furnish *representations* |**<sup>Ξ</sup>**A *,* |**<sup>Ξ</sup>**B *, . . . of quantum state* |**Ξ** *of system* S. By design, this DataSource object |**Ξ** carries data ⎧ ⎩Ξ ⎫ ⎭ A , ⎧ ⎩Ξ ⎫ ⎭ B and, more generally, ⎧ ⎩Ξ ⎫ ⎭-data (47) from arrays of *any* observations, including the imaginary ones. That is what eliminates

the initial need for the ⎧⎩Ξ⎫⎭A -brace (24) to come from the observation A , which is reflected in the shortening of the term "representation of state" to simply "state" |**Ξ** . It should be added that the straightforward storing of objects {|**<sup>Ξ</sup>**A , |**<sup>Ξ</sup>**B , . . . } in a certain set H, but with the independence of operations { <sup>+</sup>(<sup>A</sup> ), <sup>+</sup>(B), . . . } preserved, would not differ from the tautological substitution of symbols. Accordingly, the semantic autonomy of ⎧⎩Ξ⎫⎭-brace would also be inherited, whereas covariance III requires an elimination of precisely this autonomy. What is more, the set-theoretic original copy for operations {0A , 0B, . . . } and + is one and the same—the union ∪.

The symbols |*αs* ⎫⎭ and |*βs* ⎫⎭ in (50) are no more than symbols. Hence, the objects' property (50) of being identical must be reflected in terms of their coordinate a, b-components (pt. R•). This means that any aggregate (<sup>a</sup>1, a2, . . . ) is unambiguously calculated by means of a certain transformation *U* into any other (<sup>b</sup>1, b2, . . . ) when the two aggregates represent a common |**Ξ** :

$$(\mathfrak{a}\_1, \mathfrak{a}\_2, \dots) = \widehat{L}(\mathfrak{b}\_1, \mathfrak{b}\_2, \dots) \dots$$

.

The *U* then becomes an isomorphism between these aggregates (a preimage of the future unitary transformation ([6], p. 14)) and, accordingly, their lengths must coincide. This length—a certain single constant—will be symbolized as D.

#### *6.2. Representations of Devicesand Spectra*

Naturally, the instrument is converted to the H-structure language along with ⎧⎩Ξ⎫⎭- objects. It is a set of symbols {|*<sup>γ</sup>*1 ⎫⎭, |*<sup>γ</sup>*2 ⎫⎭, . . . } in place of the previous {*<sup>γ</sup>*1, *γ*2, . . . }. As has just been shown, their number for any C -instrument should be equal to D. However, generally speaking, |TA | = |TB| since TA and TB are assigned in an arbitrary way (pt. O). Therefore, if we take an illustration A {*<sup>α</sup>*1, *<sup>α</sup>*2} and <sup>B</sup>{*β*1, *β*2, *β*3}, then H-representation of instrument A should appear at least as {|*<sup>α</sup>*1 ⎫⎭, |*α*2 ⎫⎭, |*α*3 ⎫⎭}. Clearly, the already present distinguishability *α*1 ≈ *α*2 (Section 2.2) is automatically converted into an abstract distinguishability of new symbols |*α*1 ⎫⎭ = |*α*2 ⎫⎭, and empirical A -distinguishability is confined exclusively by these two symbols. In that case, for the purpose of noncontradiction, the added third symbol |*α*3 ⎫⎭, as an adjunction to the abstract relations |*α*3 ⎫⎭ = |*α*1 ⎫⎭ and |*α*3 ⎫⎭ = |*α*2 ⎫⎭, should be complemented with the notion of its physical *in*discernibility from |*α*1 ⎫⎭ or |*α*2 ⎫⎭. By an extension of this argument, one obtains that every A -instrument should be endowed with the (non)equivalence relation (/) in terms of the H-structure by its formal {|*<sup>α</sup>*1 ⎫⎭, ...}- representations. How do we do this?

Let us proceed further from a self-suggested extension of pt. R. Let us declare—and it is more than natural—that the number representations *αs* are linked not only to observations but to instruments as well. Each *αs* is the new object of a numerical type: a number or a collection of numbers. Then, indiscernibility, say |*α*3 ⎫⎭ |*α*1 ⎫⎭, is recorded by coincidence of the numeral labels *α*3 = *α*1 attached to the symbols |*α*3 ⎫⎭ and |*α*1 ⎫⎭, respectively. The abstract ("old") distinguishability |*α*3 ⎫⎭ = |*α*1 ⎫⎭, meanwhile, remains as it is. From here, we have the following formalization of the relationship between and = by means of dropping/adding the brackets | ⎫⎭:

$$\begin{aligned} \left| \mathfrak{a}\_{\ast} \right\rangle \neq \left| \mathfrak{a}\_{k} \right\rangle & \iff \left| \mathfrak{a}\_{\ast} \neq \mathfrak{a}\_{k} \right\rangle \\ \left| \mathfrak{a}\_{\ast} \right\rangle \cong \left| \mathfrak{a}\_{k} \right\rangle & \iff \left| \mathfrak{a}\_{\ast} = \mathfrak{a}\_{k} \right\rangle \end{aligned} \quad \text{under} \quad \left| \mathfrak{a}\_{\ast} \right\rangle \neq \left| \mathfrak{a}\_{\ast} \right\rangle \,. \tag{51}$$

Call the quantity *αs* (numerical) the *spectral label/marker* of eigen-element |*αs* ⎫⎭. Then, by the H-*representation* **[**A **]** *of instrument* A , we will mean the set of objects |*<sup>α</sup>*1 ⎫⎭,..., |*<sup>α</sup>*D ⎫⎭ supplemented with the spectral structure (51):

$$\left[\omega'\right] := \left\{ \left|a\_1\right\rangle\_{\left|a\_1\right\rangle}, \left|a\_2\right\rangle\_{\left|a\_2\right\rangle}, \dots \right\} \,. \tag{52}$$

 It is not difficult to see that if |*α*1 ⎫⎭ |*α*2 ⎫⎭, then either |*α*3 ⎫⎭ |*α*1 ⎫⎭ or |*α*3 ⎫⎭ |*α*2 ⎫⎭. Otherwise, spectral markers 1*<sup>α</sup>*1 = 1*<sup>α</sup>*2 should coincide, and primary primitives *α*1 ≈ *α*2 lose

their empirical distinguishability in contrast to (7). The multiple coincidence of 1*<sup>α</sup>s*-markers is admissible.

In the presence of relations (51), it is natural to state that instrument A is coarser (more symmetrical) than B and, terminologically, to declare that the degeneration of the spectral-label values takes place. In cases of embeddability such as <sup>A</sup>2{*<sup>α</sup>*1, *<sup>α</sup>*2} ⊂ <sup>A</sup>3{*<sup>α</sup>*1, *α*2, *<sup>α</sup>*3}, instrument A2 can even be called the same as (coinciding with) A3, but with a more rough scale. Conversely, A3 is a more precise extension of A2. In particular, the natural notion of a device resolution fits here.

All instruments may then be mathematically imagined as having the same resolution, but, perhaps, with degeneration of spectra. The non-coinciding instruments may be interpreted as non-equivalent reference frames A = B in an observation space. According to pts. R•• and III, they are mandatorily present in the description. The spectral degenerations are also always present since element *α*1 can always be removed from TA , and there are no logical foundations to prohibit an observational instrument with family TA − {*<sup>α</sup>*1}. Hence, it follows that introducing the spectra—instrumental readings—is required even formally, without physics. It is of course implied here that spectral (in)discernibility is realized in the same manner as its statistical counterpart in Sections 2.6 and 4, i.e., by numbers. Incidentally, such a property of 1*<sup>α</sup>s*—i.e., of being a numerical object—is not at all necessary at the moment. The spectrum {1*<sup>α</sup>*1, 1*<sup>α</sup>*2, ...} may be thought of as an abstract set of labels attached to the eigen-elements. As numbers, it is introduced for the subsequent creation of models to classical/macroscopic dynamic, and they are numerical.

Returning to D, we note that, in any case, the toolkit {A , B, ...} =: O in real use has always been defined, fixed, and is finite. Consequently, the constant

$$\mathbf{D} \not\ni \mathbf{2} \tag{53}$$

has also been defined and fixed, and it becomes the globally static observable characteristic— an empirically external parameter. Meanwhile, the entire scheme internally contains the natural method of its own extension D %→ D + 1, and the potentially all-encompassing choice D = ∞ may be considered the universally preferable one in QT. By freezing the different D < <sup>∞</sup>, the theory makes it possible to create models, and they are not only admissible but also well-known. Their efficiency is examined in experiments. Once again:

• The D-constant concept of spectra and their degenerations is created by the (A , B)- covariance requirement, i.e., by principium III.

As a result, the structure of H-representations of states and of instruments are liberated from the arbitrariness in assigning the subsets TA in (8). The statistical unitary pre-images (34) and H-elements of the form <sup>c</sup>|*<sup>γ</sup>s* ⎫⎭can be associated with any "eigen symbol" |*γs* ⎫⎭. They are always available because every possible brace (32) is known to contain subfamilies when ongoing Ψ, Φ-primitives ge<sup>t</sup> to a single one, e.g., to *γ*1. Therefore, every representation a1 |*α*1 ⎫⎭ + ··· is always equivalent to a ⎧⎩Ξ⎫⎭C -brace for some observation C with a homogeneous outgoing ensemble {*<sup>γ</sup>*1 ··· *<sup>γ</sup>*1}. That is, one may always write

$$\mathfrak{a}\_1|\mathfrak{a}\_1\rangle \oplus \mathfrak{a}\_2|\mathfrak{a}\_2\rangle \oplus \cdots = \mathfrak{c}\_1|\gamma\_1\rangle \oplus 0|\gamma\_2\rangle \oplus \cdots =: \mathfrak{c}\_1|\gamma\_1\rangle \,. \tag{54}$$

while naturally referring to c1 |*<sup>γ</sup>*1 ⎫⎭ as one of the *eigen-states of instrument* C , with an appropriate adjustment of the similar definition in pt. O. The construction of the representation-state space is far from being complete since it is still a "bare" semigroup H.

#### *6.3. Superposition of States*

Since writings (50) exist for any ensemble ⎧⎩Ξ⎫⎭-brace, let us consider the following two representations:

$$\begin{aligned} \mathfrak{a}\_1|\mathfrak{a}\_1\rangle \xleftarrow{} \mathfrak{a}\_2|\mathfrak{a}\_2\rangle &= \mathfrak{b}\_1|\beta\_1\rangle \xleftarrow{} \mathfrak{b}\_2|\beta\_2\rangle \xleftarrow{} \cdots \ , \\ \mathfrak{a}\_2|\mathfrak{a}\_2\rangle &= \mathfrak{b}\_1'|\beta\_1\rangle \xleftarrow{} \mathfrak{b}\_2'|\beta\_2\rangle \xleftarrow{} \cdots \ . \end{aligned} \tag{55}$$

Comparison of these equalities tells us that the second one is a solution of the first one with respect to a2 |*α*2 ⎫⎭. Hence, the semigroup operation + admits a cancellation of element a1 |*α*1 ⎫⎭. This means that there exists an H-element a˜ 1 |*α*1 ⎫⎭ such that

a˜ 1 |*α*1 ⎫⎭ <sup>+</sup> a1 |*α*1 ⎫⎭ + a2 |*α*2 ⎫⎭ = a˜ 1 |*α*1 ⎫⎭ <sup>+</sup> <sup>b</sup>1 |*β*1 ⎫⎭ <sup>+</sup> b2 |*β*2 ⎫⎭ <sup>+</sup> ··· ⇓ <sup>0</sup>|*<sup>α</sup>*1 ⎫⎭ <sup>+</sup> a2 |*α*2 ⎫⎭ = a˜ 1 |*α*1 ⎫⎭ <sup>+</sup> b1 |*β*1 ⎫⎭ <sup>+</sup> b2 |*β*2 ⎫⎭ <sup>+</sup> ··· ⇓ (due to (54)) a2 |*α*2 ⎫⎭ = b1 |*β*1 ⎫⎭ <sup>+</sup> b2 |*β*2 ⎫⎭ <sup>+</sup> ··· ⇓ ⇓ |0⎫⎭ := <sup>0</sup>|*<sup>α</sup>*1 ⎫⎭ = a˜ 1 |*α*1 ⎫⎭ <sup>+</sup> a1 |*α*1 ⎫⎭, a˜ 1 |*α*1 ⎫⎭ <sup>+</sup> b1 |*β*1 ⎫⎭ <sup>+</sup> ··· = b1 |*β*1 ⎫⎭ <sup>+</sup> ···

where |0⎫⎭ stands for a zero in the semigroup H (image ⎧⎩0⎫⎭ of the finite-length brace ⎧⎩Ξ⎫⎭) and 0 in <sup>0</sup>|*<sup>α</sup>*1 ⎫⎭ is a symbol of its [*λ*, *μ*]-coordinates. By canceling out a*s* |*αs* ⎫⎭, one by one, if necessary, one deduces that any element of H does have an inversion. That is, H is actually a group. We re-denote inverse elements a˜*s* |*αs* ⎫⎭ by (−a*s*)|*<sup>α</sup>s* ⎫⎭ and inversions of sums are formed from ( +)-sums thereof. Moreover, all the [*λ*, *μ*]-pairs turn into a set {<sup>a</sup>, b, ...} equipped with the above-mentioned composition <sup>⊕</sup>, which follows from an obvious property of unitary brace:

$$
\mathfrak{a}|\mathfrak{a}\_1\rangle \oplus \mathfrak{b}|\mathfrak{a}\_1\rangle = (\mathfrak{a} \oplus \mathfrak{b})|\mathfrak{a}\_1\rangle \tag{56}
$$

,

(inheritance of clossedness under the <sup>∪</sup>-operation). This composition is also a ⊕-operation of a group and of a commutative one:

$$\mathfrak{a}\circ\mathfrak{b}=\mathfrak{b}\circ\mathfrak{a}, \qquad \left(\mathfrak{a}\circ\mathfrak{b}\right)\circ\mathfrak{c}=\mathfrak{a}\circ\left(\mathfrak{b}\circ\mathfrak{c}\right), \qquad \mathfrak{a}\circ 0=\mathfrak{a}, \qquad \mathfrak{a}\circ\left(-\mathfrak{a}\right)=0. \tag{57}$$

Therefore, the group nature of semigroup H and the group (57) come from the scheme

$$\begin{array}{rcl} \left\lceil \begin{array}{c} \text{single observations} \\ \omega', \beta \theta, \dots \end{array} \right\rceil & \Rightarrow & \left\lceil \begin{array}{c} \text{semigroups} \\ \mathfrak{G}\_{\omega'}, \mathfrak{G}\_{\omega}, \dots \end{array} \right\rceil & \rightarrow \\\\ \left\lceil \begin{array}{c} (\omega', \beta \theta)\text{-covariance}, \\ \{\mathcal{S}, \mathfrak{M}, \dots\} \text{ and principal III} \end{array} \right\rceil & \Rightarrow & \left\lceil \text{group } \mathcal{H} \right\rceil \end{array}$$

and, technically, from equatings/identifyings (50), i.e., from conception "the same" (Section 5.4). For its part, it is this very structure of algebraic operations—the two- and three-term (and nothing else) axioms of commutativity/associativity, i.e., the group—that comes from properties (39). All of this provides an answer to the key question: where do the (semi)group and the minus sign come from and why?

Thus, handling the |**Ξ** -objects breaks free from its ties to the notion of observation, and the objects admit the formal writings <sup>a</sup>|Ψ⎫⎭ <sup>+</sup> <sup>b</sup>|Φ⎫⎭ <sup>+</sup> ···. Call them *superpositions*. However, as soon as they or the state are associated in meaning with the word "readings" (this is discussed at greater length in Sections 6.4 and 6.5), this term should be replaced with a non-truncated one, i.e., a representation of the state with respect to a certain observation. Specifically, the statistical weights f*j* are extracted from such expressions only after their conversion into a sum over eigen-states of the form (50); a task of the subsequent mathematical tool.

No superposition <sup>a</sup>|Ψ⎫⎭ <sup>+</sup> <sup>b</sup>|Φ⎫⎭ <sup>+</sup> ···, including (54), has any physical sense in and of itself [5] (p. 137), [12] nor is it preferable to any other one. It merely mirrors the closedness of states with respect to operation + since any |**Ξ** is re-recorded as a sum of various {a|Ψ⎫⎭, <sup>b</sup>|Φ⎫⎭, . . . } in a countless number of ways and is linked to any other such sum. Without a system of |*<sup>α</sup>s*⎫⎭-symbols for instrument A , nothing observable is extractable out of the aggregate of coefficients {±<sup>a</sup>, ±b, . . . } (and, of course, of the |Ψ⎫⎭-letters themselves) in any imaginable way. Accordingly, it is incorrect to speak of—a widespread misconceptionthe destruction of the superposition or of the "relative-phase information" ([119], p. 253), associating the word destruction with the physical/observational meanings or processes.

As a result, *even without having a numerical theory yet* and without recourse to the concept of a physical quantity, superposition may not address whatever physical concepts, we arrive at the paramount property, which characterizes the most general type of microobservation's ensembles (17).

•*Superposition principle*

A ( +)-composition of quantum states a|Ψ⎫⎭ and <sup>b</sup>|Φ⎫⎭, which are admissible for system S, is an admissible state

$$
\mathfrak{a}|\Psi\rangle \oplus \mathfrak{b}|\Phi\rangle = \mathfrak{c}|\Xi\rangle\tag{58}
$$

and, with that, the set a|Ψ⎫⎭, <sup>b</sup>|Φ⎫⎭,<sup>c</sup>|Ξ⎫⎭, ... =: H forms a commutative group with respect to operation +. The family {<sup>a</sup>, b, ...} of coordinate R2-representatives of states (50) is also equipped with the same group structure under the ⊕-operation (57) and with the rule of carrying the operation + over to ⊕:

$$
\mathfrak{a}|\Psi\rangle \oplus \mathfrak{b}|\Psi\rangle = (\mathfrak{a} \oplus \mathfrak{b})|\Psi\rangle \,. \tag{59}
$$

Let us clarify the transferring of (56) to (59). The union of the state prototypes <sup>a</sup>|Ψ⎫⎭, b|Ψ⎫⎭ ∈ H is known to belong to G. Thus, the composition a|Ψ⎫⎭ + b|Ψ⎫⎭ should be identical to a certain element c|Ψ⎫⎭ ∈ H. It is clear that c depends on a, b and, hence, a|Ψ⎫⎭ + b|Ψ⎫⎭ = <sup>c</sup>(<sup>a</sup>, <sup>b</sup>)|Ψ⎫⎭. The exhaustive properties of dependence <sup>c</sup>(<sup>a</sup>, b) are given by Formulas (57) and (59) under notation <sup>c</sup>(<sup>a</sup>, b) =: (a ⊕ b).

#### 6.3.1. "Physics" of Superposition

Besides the essentially unphysical nature of the ( <sup>+</sup>)-superpositions, i.e., "we cannot recognize them" ([12], p. 13), the primary and salient property of quantum addition is in the fact that, due to the group subtraction, it is possible to experimentally obtain a "quantum zero" in statistics from "non-zeroes'. With that, these "seem to be" positive, but there are "negative non-zeroes", i.e., negative numbers (Section 9.2). Subtraction manifests by the typical obscurations in interference pictures. S. Aaronson adds to this: "We have go<sup>t</sup> minus signs, and so we have go<sup>t</sup> interference" ([20], p. 220). No classical composition

$$w\varrho\_1 + (1 - w)\varrho\_2\tag{60}$$

of non-zero statistics 1, 2 can provide a zero value since the zero will never be obtained via the ∪-unions. The same is true for the pre-superposition in isolated brace ⎧⎩Ξ⎫⎭A , i.e., when one instrument is in question.

**Remark 12.** *One cannot help but mention yet another counterexample to the superposition's "physicality": the (in)famous "quantum cat". Any combination of the dead and living animal is meaningless as a statement about new/nonclassical entity such as a "(half-)dead/alive cat" or such statements about particles as "their being neither here nor there but everywhere", especially with the stress on "at the same point in time" (see pr.* I*). It makes absolutely no sense to add (allegedly in accord with the character* +*) to each other the nature's phenomena and notions that have not yet been created and are dynamical ("alive") at that.*

• **What is being added is states, not their denominations** or verbal descriptions of envisioned *("fantasized", "fantastic phantoms" ([12], p. 15))* physical properties *such as spin up/down or dead/alive. Cf. [151] (pp. 134 (!), 135).*

*The "cat-box open" is a click, not state, without a notion of "a cat". Accordingly, the word combination "the quantum objects exist in "superpositions" of different possibilities" (a representative excerpt from the literature) is at most an interpretative allegory (Section 10) without physical and mathematical content. That is to say, strictly speaking,*

• *No quantum (micro)system has ever been/dwelled in any state, much less in a superposition one, and much less at an instant t. Ludwig, on pp. 16 and 78 of the book [58], insists that it is a "myth" and "a fairy tale, . . . the very widespread idea that each microsystem has a real state . . . represented by a vector in a Hilbert space", and M. Nielsen remarks in [21] that "Saying* 0.6|0 + 0.8|1 *is simultaneously* 0 *and* 1 *makes about as much sense as Lewis Carroll's nonsense poem* Jabberwocky*: . . . ". K. Svozil does also underscore that "'coherent superpositions' just correspond to improper, misleading representations of nonexisting aspects of physical reality. They are delusive because they confuse ontology with epistemology" ([152], p. 26).*

*The meaning of the word "add" is still being created, including an implementation at objects to be thought of as the "atomic irreducible" entities—the numbers (Section 7).*

*T. Maudlin notes on p. 133 of the work [151]: "Our job . . . is to invent mathematical representations ..., rather than merely linguistic terms such as "z-up." . . . we are in some danger of* confusing physical items with mathematical items*" (italics supplied). Here is an example of confusion. If we are going to measure the z-spin in one of the* (*x* )*-beams in a Stern–Gerlach device, then why and when does this "*observable*" certainty—say* |→*x —get turned into a* (↑↓*z* )*- uncertainty* |↑ + |↓ *? (see [153] (p. 232)). However, what if we are about not to do this? We come up against the question:*

• *What does one mean by an equal-sign* = *in the orthodox notation* |→ = |↑ + |↓ *?*

*Which state does the system "intend" to fall into: the z-uncertainty or the x-determinacy? Which of the states is it in, after all? Examples to the "physicality of states" may be continued endlessly [154].*

A statement about QM-superposition (without C-numbers) as a non-independent axiom can be found in the book ([36], p. 108) but arguments given there are circular: ,Hilbert space- ,quantum logic of propositions- ,superposition principle-. Similarly, in the works [122] and [72] (p. 164), all of that is "derived" from modular lattices [155]. However, the lattices are known to enter QM from the Hilbert space structure and, on the other hand, the purging quantum rudiments of such a space' axiomatics constitutes Birkhoff's 110-th problem ([155], p. 286). Note also that, in connection with the formal logic approaches to the theory construction [9,36,72,111,122,156,157], the issue of vindicating the *matters* that this logic deals *with* (logic of what?) [58,93,158] should not be neglected. What we mean here is the questions on logic: of propositions? [105,106] of relations? of (mathlogic) classes/sets? [120] of phenomena/properties? (which ones?) of quantum/classical events? . . . ? "For example, would one have to develop a quantum set theory?" ([110], p. 17). "If by "logic" we mean something like "correct reasoning," then it would make no sense to think of logic as "just another theory."" ([73], p. 258). The more abstract micro-events and Boolean logic we have used in metamathematical reasoning at the moment ([87], pp. 189, 193) contain nothing that depends on classical physics. That is, quantum foundations do not require [58] a different quantum/non-classical logic. See also [74] (p. 29).

#### 6.3.2. When and What Is Non-Commutativity?

Yet another fact that results from the above constructs is that the availability of a superposition math-structure (58) reflects the presence of at least *two* A , B with *noncoinciding* families of eigen-primitives {*<sup>α</sup>s*}, {*βk*}. This consequence of pt. R•• should be particularly emphasized since it will manifest in the *non-commutativity* of operators A and B in the future. Although the present work does not ge<sup>t</sup> to operators as a mathematical structure, it is clear that the emergen<sup>t</sup> eigen-states and spectra have a direct bearing on them. In this context, the "commuting instruments" {|*<sup>α</sup>*1 ⎫⎭, |*α*2 ⎫⎭, ...} = {|*β*1 ⎫⎭, |*β*2 ⎫⎭, ...} can be treated, roughly speaking, as coinciding because this fact is independent of the specific spectra {1*<sup>α</sup>*1, 1*<sup>α</sup>*2, ...}, {1*β*1, 1*β*2, ...} assigned to them. If they differ, this is merely a different (numerical) graduation of the spectrum scale. It is the same for all instruments, and its length is the parameter D.

Notice that the definition of an A -observation is not different from the formal assignment of the family TA (pt. O and (8)), which is why the non-coinciding sets TA , TB do always exist. This provides a kind of abstractly deductive existence's proof for the non-commutativity, QM-interference—see Section 6.5 further below—and for the utmost low-level finality of QM altogether [13,33,93]. The whys and wherefores of theory do not require invoking the physical conceptions; cf. [10] (p. 2).

Of no small importance is that this point entails an independence of the (existence/ presence of) classical physics or of its formal deformation, which are ye<sup>t</sup> to be created from the quantum one (cf. a selected thesis on page 15). In particular, no use is required of the notion of a certain pretty small—again the classical/physical term—quantity, i.e., the Plank constant *h*¯ ([123], Section 6.5). (Parenthetically, no numerical value of this constant matters here; it is not dimensionless and its zero limit is not meaningful.) What is more, the quantum paradigm (17)–(19) tells us that the classical description begins,i.e., we do create/introduce, with the notions of a micro-event's average and of time, whereas these conceptions are still absent at the moment and in the present work. Similarly for the notions of locality, causality, the classical event, and the classical object.

#### *6.4. Physical Properties*

Now, the "general physics" S, **<sup>M</sup>**,... is mathematized into representations (50) of states |**Ξ** of system S. There is, however, an ambiguity, the source of which is the fact that the natural/classical language also lays claims to a similar formulation. This refers to the belief in the existence of mathematics ("bad habit" [3]; see also [38], [58] (p. 122), and [159]) that describes S as an individual object with properties regardless of observation; an observation that is not a *functioning* attribute of the mathematics itself. In classical description, it is specified by definitions: point P of a phase space, (*q*, *p*)-coordinatization of the point (manifold), and statistical distribution (*q*, *p*).

On the other hand, quantum empiricism provides nothing more to us besides the ensemble brace and |**Ξ** -states (pt. T). Preordained definienda with physical contents are unacceptable, i.e., S should not be conceived as "something with *physical* properties" or as an "*individual* object"[93,94], [113] (p. 645). However, since the observational data (in the broadest sense of the word) may not originate from anywhere but a certain |**Ξ** -object, there should subsequently create:


This is habitually referred to as elements/images of reality [27] (p. 194), [40] (Section 10.2), [94] (Section XIII.4.8)—Bell's "beables" [28]—or what we have been calling attributes of a physical system.

•"The very notion of 'phenomenon' or of 'the appearance of things,' . . . is a cognitive and perceptual act of abstraction"

> M. Wartofsky ([160], p. 220)

That is to say, the physical phenomena per se do not exist [92] (p. 310), [127].

Indeed, the primary ideology of Sections 1.3 and 2.1 tells us that an invasion of physically self-apparent images into the theory should be avoided ([87], p. 69) because "quantum theory not only does not use—it does not even dare to mention—the notion of a "real physical situation"" ([27], p. 198; E. Jaynes). Continuing a quotation from R. Haag on page 5, one requires "the renunciation of the absolute significance of conventional physical attributes of objects" ([131], p. 238; N. Bohr) and of concomitant and accustomed logic in reasoning. In fact, we are led to (re)build the language of the classical description. Therefore, everything, with no exceptions, should be created mathematically: coordinates, momenta, energies, optical spectra, device readings, lengths/distances and time, extension and lifetime of objects, the language of particles, their number/numeration (Fock space), (in)discernibility/individuality (bosons/fermions), the notions of a subsystem of system S (see (23)), and even a notion of the physical rigor (in reasoning), etc.

Degrees of freedom, the concepts of the field/body/mass/inertia/interaction, the numerical labeling the space-time continuum, Newtonian mechanics with its equations and the concepts of the force, interaction, and the causality of classical events, thermodynamics, the very term "the classical state", the numerical labels of the space-time continuum and numerical forms of what is known as the classical reference frames—coordinates on manifolds—need to also be created. Once more to underscore, the numerical forms of the classical space coordinates and the time (e.g., the metric tensor *<sup>g</sup>αβ*(*x*)*dxαdx<sup>β</sup>*) have a quantum empirical origin. The latter fact is required for carefully posing the questions of quantum gravity, and it should be noted in passing that the simultaneity is an ill-defined term not only in the (general) relativity theory; in QTit is even worse. In common with the simultaneous measurability, this term appears to have come from the classical framework, which is why it is illegal as a quantum-theoretical primitive (pr. I and [87]).
