*2.2. Observation*

The sequences addressed above lead to the following outcome.

• Any meaningful micro-act A either saves a state (*α* A *α*) or turns it into a conserved one (Ψ A *α*).

The two extremes do not contradict this fact. The first—maximally rough observations—is when all states are destroyed into a certain one: Ψ Ψ0 ("whatever and however we watch, all we see is one and the same"). In this, the state Ψ0 is not destroyed: Ψ0 Ψ0. Another extreme is when none of the states are destroyed: Ψ Ψ. This is the case of ideal (quantum) observation, but, due to the absence of any changes, it is indistinguishable from the case where observations are entirely absent.

Situated in between these extremes lies the simplest case with two distinctive states

$$
\underline{\mathfrak{A}}\_{1} \ - \stackrel{\underline{\mathfrak{a}}\_{1}^{\prime}}{\dashrightarrow} \underline{\mathfrak{a}}\_{1} \prime \qquad \qquad \underline{\mathfrak{C}}\_{2} \ - \stackrel{\underline{\mathfrak{A}}\_{2}^{\prime}}{\dashrightarrow} \underline{\mathfrak{A}}\_{2} \ \vdots \tag{7}
$$

Of course, these are prohibited from transitioning into each other. Because there is still the free admissibility of transitions Ψ A *α*1, Ψ A *α*2, we can turn the semantic sequence

> Aarbitrariness preservation distinctive *α*'sB

into the more rigorous scheme

$$\left[\mathfrak{L} = \{\underline{\mathtt{A}}, \underline{\mathtt{Q}}, \ldots\}\right] \; + \; \left[\mathfrak{A}\text{-observations}\right] \; \rightarrow \quad \left\{\mathfrak{g}\_{1}, \mathfrak{g}\_{2}, \ldots\right\} \coloneqq \mathfrak{T}\_{\mathsf{A}} \subset \mathfrak{T} \; , \tag{8}$$

which gives, even though partially, rise to the concept of a physical distinguishability ("distinguo"). It is formally defined only on the subset TA : the statement *α*1 ≈ *α*2 is equivalent to (7). To avoid overloading the further notation, we do not use symbols such as ≈A and ≈A ; the context is always obvious.

O By a *physical observation* A or, in short, *observation* we will mean such interventions A , in which the "never-ending" chaos (3) is replaced by chaos with the notion of preservation, i.e., "chaos with rule (6)":

$$
\underline{\Psi} \dashv \overline{-\,} \overline{\,} \overline{\,} \overline{\,} \overline{\,} \overline{\,} \begin{array}{c} \text{where} \\ \end{array} \begin{array}{c} \text{where} \\ \end{array} \begin{array}{c} \text{ $\underline{\Psi}$ } \xrightarrow{\underline{\text{ref}}} \underline{\text{\,}} \end{array} . \tag{9}
$$

The set of *α*-objects TA with the property

$$
\mathbb{Q}^1 \quad - \xrightarrow{\overline{\underline{\boldsymbol{m}}}} \mathbb{Q}^1 \land \qquad \qquad \mathbb{Q}^2 \quad - \xrightarrow{\overline{\underline{\boldsymbol{m}}}} \mathbb{Q}^2 \land \qquad \qquad \dots \tag{10}
$$

is discrete, and the *αs* themselves are termed *the eigen* (proper) *for observation* A . They define A and do not depend on S. No logical connection between Ψ (the left of (9)), family TA , and system S exists.

(The comprehensive terminology here is this: a micro-act of observation by instrument A . The zig-zag arrow is replaced with the straight one .) Expressed another way, the introduction of the concept "the eigen" is equivalent to the following informal, yet minimal, motivation: *at least some* certainty instead of *total* arbitrariness.

Two instruments A and B may have arbitrarily different eigen-states {*α*1, ..., *αn*} = {*β*1, ... , *βm*}. Accordingly, as regards observation B, the (distinctive) states {*αs*} do not differ, in general, from the "regular" Ψ's, i.e., from those chaotically destroyable into the B-eigen states: *αj* B *βk*. All kinds of instruments {A , B, . . . } are thus defined by aggregates {TA , TB, . . . }. The number |TA | of corresponding *α*-objects therein may be an arbitrary integer. There are also no (logical) grounds for restricting/prescribing the composition of TA . Any element of T may be the conserved one for a certain instrument. Parenthetically, the notion of an eigen-state—in different forms—is sometimes present in axiomatics of QM [18,72,122].

In a generic case, the chaos present in Rule (9) leaves open the problem of correlating the recognizability Ψ ≈ Φ (or Ψ ≈ Φ) with physics. Clearly, the issue is linked to the ambiguity of the term Ψ-state itself, which is used in pt. S—an important point—because we need to start with something since building the mathematical description without some sort of a set is impossible.

**Remark 2.** *Informally, metalinguistic semantics—the association of meanings with texts [58]—is in general as follows. Inasmuch as we are receiving* different *α-responses to each micro-act* A *, let us say that "on the other side from us there is something that can also be different", and all of that is to be described. This reflects our intuitive perception of reality, which, both at the micro-level and the macro-level, boils down to pt.* S *and to an ineradicable pair:* ,*something outside*- + ,*that which can be different for us*-*. If we give up either of these semantic premises—"something outside" (*Ψ*) or "can be different" (αj)—then, as above, we face a linguistic dead end, as the possibility for reasoning disappears. There must be two sides present. Because of this, the arrow* A *must be accompanied by "some things" to the left and right of it. The low-level set* T = {Ψ*,* Φ*,..., α*1*, α*2*, . . . } does arise. Then, the arbitrary elements* Ψ ∈ T *(unrestricted chaos) are assigned to the left of this act instead of "some thing" and the micro* ≈*-distinguishable α-objects (αj* ≈ *αs) to the right. Put another way,*

•*What is being abstracted is not "concrete things" ([13], p. 27) or behavior of things" ([123], p. 414) but a primitive element of perception—a micro-event—the α-click.  This is a kind of "Radical Empiricism . . . [by] William James" ([23], pp. 289, . . . ). The "click . . . and nothing more" ([16],*

*p. 42; C. Brukner) is a kind of experimental zero-principium of QT. Therefore, the initial math premises of QT should contain nothing but the* ≈A *-distinguishability and formalization from* (9) *and* (10)*.*

*Ideas of "a click (signal) in a counter" ([126], p. 758) have, time and again, already been expressed in the literature [2] (A. Peres), [123], and we draw attention to answers of C. Brukner ˇ on pp. 41–43 in [16], their work [127] (p. 98), and page 635 in [22]. "Having grown up collecting clicks . . . I would start with "clicks" as the only point of contact between observer and observed", wrote J. Summhammer in [23] (p. 261). It may be added that the micro-observation, as such, is terminated at the eigen elements; one and the same αj has always remained on the right.*

As a result, the minimal entity Ψ A *αj* constitutes, mathematically, an ordered pair (Ψ, *αj*) of elements of the set T, which are labeled by the symbol A , which is equivalent to the TA -family (8). Accordingly, the customary physical notion of the observation is substituted for a micro-event, an act. "Physics should forget" about processes or time of interaction when observing about the interaction itself and about anything but Ψ A *α*. This object represents a completed formalization of the empirical/laboratory notion of a quantum micro-event—a detector click. The click is sometimes considered from an information viewpoint as an information bit [22]. However, it cannot be such a (classical) bit with reified content because it is completely unpredictable. The next (different) click does entirely negate the previous one, and the information bit is in turn a concrete thing— the bit. For the same reason, there cannot be any information behind the single event. It is "too small and too momentary" to possess or to carry information about something inasmuch as even the "something" is composed from elementary clicks—see below.

#### *2.3. Numerical Realizations*

*ˇ*

Is there a possibility of relying exclusively on the inflexibility of the eigen-type elements (10) or of defining the sought-for ultimate distinguishability ≈ through the A - (micro)distinguishabilities *αj* ≈A *αs*? Let us formulate a thesis.

T There is no (linguistic) means of recognizing the system S to be different (pt. S) other than through the results of its destructions into the {*α*1, *α*2, . . . }-objects of observational instruments A .

Granted, the stringency of this linguistic taboo (T) must be accompanied by something constructive, and we will adopt the following program, which reflects the fact that the unequivocal description may only take the form of a quantitative mathematical theory.

**R** • Out of the primary ("proto")elements {Ψ, *α*, ...} ∈ T, one constructs a new set H, of which the elements

$$|\Xi\rangle := \oplus(\mathfrak{a}\_1, |\mathfrak{a}\_1\rangle; \mathfrak{a}\_2, |\mathfrak{a}\_2\rangle; \dots) \in \mathbb{H} \tag{11}$$

are said to be (number) *representations* in the "reference frame for instrument A ", and a*s* are the numerical objects. The distinguishability relation ≈A is carried over to H and admits an a-coordinate realization there—symbol ≈.

•• No preferential or preordained observational reference frame A {*α*1, *α*2, ...}—an absolute instrument—exists.

Identification (11) is always tied to a certain family TA . Accordingly, images of *αs*—symbols |*αs* —are present in Equation (11), and character *⊕* is also no more than a symbol here. Even though coordinates a*s* are declared to be numbers or aggregates of numbers, there is no arithmetic stipulated for them yet. The number is a name for a*s*. The distinguishability |**Ψ** ≈ |**Ψ**˜ of two representatives

$$\oplus(\mathsf{a}\_{1\prime} \, | \, \mathsf{a}\_{1} \rangle; \mathsf{a}\_{2\prime} \, | \, \mathsf{a}\_{2} \rangle; \dots \rangle =: \langle \, \mathsf{Y} \rangle, \qquad \oplus(\mathsf{a}\_{1\prime} \, | \, \mathsf{a}\_{1} \rangle; \mathsf{a}\_{2\prime} \, | \, \mathsf{a}\_{2} \rangle; \dots) =: \langle \, \mathsf{Y} \rangle.$$

by means of numbers a*k* = a˜ *k* and mathematical implementation of (11) and of the H-space, i.e., a "coordinatization" scheme have yet to be established. This will comprise the meaning of the word "constructs" (Sections 7–8), which may not be even linked to the mathematical

term mapping yet, since *no math of QM exists at the moment*. It immediately follows that the question about number entities—specifically, about (11)—is nontrivial in physics.

II To speak of an exact correspondence between experiment and mathematics (,obser- vation + measurement-) makes no sense until there is a detailed *mechanism for the emergence* of what is understood by number.

#### (*The second principium of quantum theory*)

In other words, we wonder what an empiricist/observer understands (semantics) by the word (syntax) "number". The underlying message here implies that the reliance upon the all-too-familiar arithmetic elucidates nothing. *There is no arithmetic in interferometers/colliders*—there are only clicks there—and the empirical nature arising from this construction (along with the measurement) must be scrutinized.

From pts. T, R, and II, it also follows that the search for a description through hidden variables, over which something is averaged, is indistinguishable from the utopian attempts to find out intrinsic content of boxes (5).

#### *2.4. Macro and Micro*

The task becomes more precise at this point. Instead of nonphysical identity/ noncoincidence ( Ψ = Φ or Ψ = Φ) of two abstract elements Ψ, Φ of the abstract set T, we need the concept of a physical ∼∼∼-equivalence (∼∼∼-distinguishability) of H-representatives {| **Ψ** , |**Φ** , . . . }. That is, there must hold either relation |**Ψ** ∼∼∼ |**Φ** or its negation |**Ψ** ∼∼∼ |**Φ** for all |**Ψ** , |**Φ** ∈ H. The primitive set T, initially required by point S, must disappear from the ultimate mathematics of symbols |**Ψ** ∈ H. Therefore, elements Ψ ∈ T are henceforth named *primitives*.

Let us sum up the fallaciousness of the metaphysical belief in the meaningfulness of the wording "there is a quantum state", i.e., the belief that the existence of a state has some math-numerical form.

•There is no a priori way to endow the term (quantum) state of system S with any meaning ([15], p. 419). It may not have a definition and any predefined semantics. This term should be created. Meanwhile, one cannot get around the concept of the (micro)observation A [127] (pp. 98–100), [113] (p. 646), [34,96]. Essentially, no one thing, including Ψ, *α*, or the T-set itself, can be the primary bearer of data about S. "There is an entirely new idea involved, . . . in terms of which one must proceed to build up an exact mathematical theory" (P. Dirac [26] (p. 12)).

There is no escape from quoting K. Popper: ". . . language for the theory; . . . it remains (like every language) to some extent vague and ambiguous. It cannot be made "precise": the meaning of concepts cannot, essentially, be laid down by any definition, whether formal, operational, or ostensive. Any attempt to make the meaning of the conceptual system "precise" by way of definitions must lead to an infinite regress, and to merely *apparent* precision, which is the worst form of imprecision because it is the most deceptive form. (This holds even for pure mathematics.)" ([108], p. 13).

The notions of a physical observable and of its observable values are also ambiguous at this point ([87], p. 5). Their ambiguity is even greater than that of state due to questions such as "what is being measured?" and even 'what is a measurement?'. Nonetheless, up until the end of this section, we will not discard the term state within the context of pt. S.

The irreproducibility of outcomes, i.e., the "turnability of Ψ-primitives into the various", leaves only one option: "to take a look at S again, once again, . . . "—in other words, to seek the source of description in repeatability. It is necessary, then, to move to the subject of macro- rather than micro-observation. This intention fits perfectly with the undefined verb "constructs" in pt. R, and the following paradigm should be understood as the macro.

M The only way of handling the uncontrollable micro-level changes is the treatment of the results of repeated destructions, accompanied by what we shall call the common physical macro-setting (experimental context):

> Ψ ··· Ψ Ψ ··· Ψ ······ A A *α*1 ··· *α*1 *α*2 ··· *α*2 ······ + ,common macro-environment **M**- . (12)

To be precise, we should have to (and we shall do) indicate the different {Ψ, Ψ, . . . } here because the same ingoing Ψ's in (12) is a preassumption, which we eschew throughout the work. This point will be very fully addressed further below (Sections 2.5, 2.6, and 3.1).

The importance of repetitions and distinguishability had long been noted (Bohr, von Neumann et al. [78]), and recently, it was particularly emphasized in the work [128]. The words "copy/repeat. . . /distin..." occur 90 times therein.

Thus, the empiricism of quantum statics forces us to operate exclusively with such formations of copies *α*,..., Ψ, and this is the maximum amount of data provided by the supra-mathematical problem setup. *All* further mathematical structures may come only from constructions such as (12) and from nothing else. Getting ahead of ourselves, let us once again turn our attention to the fact that the implementation of this idea *is not shortlength*—"the mathematization process (cor) is not simple" ([58], p. 24), and Sections 3–9 are devoted specifically to this—see, e.g., the chain (105).

One can once more repeat (Section 1.3) that much of what follows does not and cannot contain the mathematical definienda and proofs as they are usually present in the literature on quantum foundations. Instead, there appears a step-by-step *inference* of objects as they result themselves: numbers, operations, groups, algebras, etc. The only instrument that may be applicable here is the empirical inference.

The common macro-environment **M** in (12) is also viewed as a supra-mathematical notion [106], the mathematical implementation of which is yet to be created. The same considerations regarding qualitative adjectives are applicable as to the physical convention **M** as well as the transition acts in Section 2.1. Representations (11) will be the formalization of the meaning ,observation- + ,data on system S-, but now with no references to the elementary acts in (12). The physical distinguishability criteria |**Ψ** ∼∼∼ |**Φ** may not be formulated yet because the physical attributes are not yet available, but |*αs* -elements have already appeared in (11) as prototypes of explicitly distinguishable *αs*.

**Remark 3.** *The dual form of the typical quantum statements such as "*S *is a* micro*-system and* A *- instrument is a* macro*-object" (N. Bohr) is identical to the initial premise "observation does always destroy a system". It follows that there is actually little need for that terminology. Indeed, QM-micro has no internal structure and, hence, an oft-discussed issue about boundary (and limit (According to A. Zeilinger, ". . . no limit. The limit is only a question of . . . money and of experience" ([39],* 1309*).)) between micro and macro [8,30,33,90] is devoid of sense; "The notions of 'microscopic' and 'macroscopic' defy precise definition" ([28], p. 215). Therefore, this may be a matter only of "different macro", either "smaller/bigger", i.e., when they describe certain models.*

*As a (partially philosophical) note, what is understood by observational randomness does, in fact, boil down to distinguishability, and more specifically, to postulating the micro-chaos* (9)*. In considering the denial of* (9) *as an impossible proposition, we arrive at the* M*-paradigm and conclude that the only way to deal with that which is contemplated for the subject-matter of a physical description must be the treatment of micro-acts as assemblages ([129], Lect. 6). In other words, and in accordance with the outline of the clicks' analysis set out below, the determinism of micro-processes (micro-ontology)—much less the microscopic time-arrow—is meaningless as a concept since they are not processes but rather structureless acts that have not even any relationship to each other. Since there are no physical phenomena as of yet, the claim that "phenomenon-1 appears to be the cause that precedes phenomenon-2 as the effect" is no more than a collection of words. To attribute physical content and mathematical formulation at the micro-level to them is*

*impossible in principle—the "problem of boxes" noted above. Accordingly, the cause of (classical) macro-indeterminism is the absurdity of the notion of its twin concept—micro-determinism—and the unavoidable repetition of the arrows (*M*). N. Bohr puts the point very definitely: "there can be no question of causality in the ordinary sense of the word" ([78], p. 351), and Heisenberg adds that "l'indeterminismo, . . . ë necessario, e non solo consistentemente possibile" ([17], Section IX.4). See also ([129], p. 223).*

#### *2.5. Quantum Ensembles and Statistics*

Let us call the upper row in Scheme (12), as a collection of the Ψ-copies, a (quantum) homogeneous *ensemble* (Kollektiv, by von Mises [129]). We will designate it, simplifying when needed, by

{Ψ Ψ ··· Ψ C DE F *N* times}≡{Ψ ··· Ψ}*N* ≡ {Ψ}*N* ,

where *N* is understood to be an arbitrary large number. Scheme (12) also dictates considering the generic ensembles

$$\{\{\underline{\pi}\_1\cdots\underline{\pi}\_1\}\_{n\_1}\{\underline{\pi}\_2\cdots\underline{\pi}\_2\}\_{n\_2}\cdots\cdots\cdots\}, \qquad \{\cdots\underline{\Psi}\cdots\underline{\Psi}\,\underline{\Phi}\cdots\underline{\Phi}\,\underline{\Theta}\cdots\underline{\Theta}\cdots\}\tag{13}$$

as collections of homogeneous sub-ensembles. Ensembles are symbolized in the same manner as sets but, for typographical convenience, without the numerous commas and internal parentheses {} in Ensemble (13); for example,

$$\{ab\cdot\cdot\cdot b\{bca\}\cdot\cdot\cdot\} = \{a,b,\dots,b,b,c,a,\dots\} = \begin{array}{ccc}\cdot\cdot\cdot & = \vdots \end{array} = \text{: } \{ab\cdot\cdot\cdot bbca\}\cdot\cdot$$

Scheme (12) is the first point in which numbers emerge in theory, and conversion

$$\left[\underline{\alpha}\text{-ensemble (13)}\right] \quad \begin{array}{c} \longrightarrow \quad \left(n\_1, n\_2, \dots\right) \end{array}$$

into the integer collection anticipates a *numerical* A *-measurement of* S. Quantities *ns* ∈ Z+, however, should not be associated with such, as they are potentially infinite. The minimal way of creating the knowingly finite numbers out of independent and potential infinities *ns* (without loss of their independence) is to divide each of them by a greater infinity, which is a "constant" Σ for Ensemble (13). It is clear that one should put

$$\Sigma := n\_1 + n\_2 + \dotsb \quad \text{and} \quad \left\{ \mathfrak{k}\_1 := \frac{n\_1}{\Sigma}, \quad \mathfrak{k}\_2 := \frac{n\_2}{\Sigma}, \quad \dotsb \right\} \quad (\Sigma \leadsto \infty) \,, \tag{14}$$

and that Ensemble (13) does not provide any numerical data besides the relative frequencies (14). All the other data are functions of f*s*. An independence of the theory from the ensemble's Σ-constant, i.e., the scheme Σ ∞, is also implied to be a principle, and it can only be the semantic one. Without it—the Σ-postulate of infinity—there can be no question of a rational theory, i.e., empiricism will not turn into a mathematics (Sections 5.1 and 5.2). In turn, the concepts "closely, limit, the limiting frequencies", and the like will arise later when we obtain the state of space as a Hilbert one H and topology on it [130].

Thus, the M-paradigm in Scheme (12) does not only give birth to a concept of numerical data in the theory per se but also converts their Z+-discreteness into the R-continuum of real measurements. Namely, numbers f*s* ∈ R are the statistics (f1, f2, ...) of destructions A into the ensemble of primitives {{*α*1}*n*1 , {*α*2}*n*2 , . . . }.

#### *2.6. Distinguishability and Numbers*

The distinguishability of the two ensembles now turns out to be the R-numerical, i.e., it is determined by the difference between f-numbers. As a result, and according to pt. R, the two elements |**Ψ** ≈A |**Ψ**˜ of H will differ in the numbers a*s* and a˜*s* if the latter turn out to be the bearers of different statistics

$$\mathfrak{t}\_{/}(\mathfrak{a}\_{1}, \mathfrak{a}\_{2}, \dots) \not\equiv \mathfrak{t}\_{/}(\mathfrak{a}\_{1}, \mathfrak{a}\_{2}, \dots) \; . \tag{15}$$

As a consequence, distinguishability ≈ is carried over to H with an extension to the non-eigen objects, but it is inherently incomplete since it does not take into account the most significant fact—arbitrariness of transitions (6).

The collection (f1, ...), as a final result of transitions {Ψ *αs*}, actually "knows nothing" about their left-hand side, much less about its uniqueness Ψ. For instance, if under the equal *α*-statistics {f*s*} for the two families {**?** *αs*}*N* and {Ψ *αs*}*N* (collectivity of **?**'s), we would claim **?** = Ψ, which would mean a mass control over transitions (9). Instead of a "black box" above, we find that prior to acts A , all the undefined **?**'s were equal to Ψ. This, however, is the declaration of a property: "prior to observation the system S was/dwelled in . . . ". With any continuation of this sentence, it is pointless and prohibited if one theoretically accepts that, prior to observation, nothing exists, and there are no properties (Section 2.1). The words "initial state of S" thus make no sense. The indeterminacy of the ongoing **?**'s is therefore mandatory, and numbers (f1, f2, ...) required for recognition are manifestly insufficient. Considering that the micro-changeability of single primitives Ψ also means nothing [15] (p. 419 (!), left column), [33] (p. 493), [41], only a generic ensemble

$$\mathbf{x} \cdot (\mathbf{z} \cdot \cdots \mathbf{z}) \quad \longmapsto \quad \{\cdots \cdot \overline{\mathbf{z}} \cdot \cdots \cdot \overline{\mathbf{z}} \cdot \overline{\mathbf{Q}} \cdots \cdot \overline{\mathbf{Q}} \cdot \overline{\mathbf{z}} \cdots \cdot \overline{\mathbf{z}} \cdots \} =: \mathfrak{A} \tag{16}$$

can be an intermediary in the sought-for translation of Ψ's onto representations |**Ξ** ∈ H under Construction (11).

In the accustomed physical terminology, the above is expressed in the sequence

$$\left[ \begin{array}{c} \text{state} \\ \end{array} \right] \begin{array}{c} \left[ \begin{array}{c} \text{"} \\ \end{array} \right] \left[ \begin{array}{c} \text{state'} \\ \end{array} \right] \left[ \begin{array}{c} \text{state'} \\ \end{array} \right] \left[ \begin{array}{c} \text{measurement} \\ \end{array} \right] \end{array} \tag{17}$$

The removal of the intermediate component here, i.e., switch to the sequence

$$\begin{array}{c} \text{[state]} \quad \text{---} \stackrel{\text{ad}\_{\text{class}}}{\longrightarrow} \quad \text{---} \quad \text{---} \quad \text{[} \text{measurement} \text{]} \tag{18}$$

amounts to the rejection of micro-destructibility and of unpredictability. Even with the classical framework, this supposition is questionable since the notion of a "change when observed" disappears. The relationships between the dual concepts—(micro/macro)- scopicity, big/middle/small, etc.—do also get lost. That is the reason why, developing Heisenberg's question ". . . is it . . . I can only find in nature situations which can be described by quantum mechanics?"' ([78], p. 325), we conclude that, strictly speaking,

•*All observations*, regardless of (the envisioned physical) macro/meso/micro characteristics, do have the structure (17), i.e., are quantum. *No non-quantal observations exist*.

With their idealized "roughening", the classical description appends numerical fstatistics to (18), which is when the left/right sides of (18) become indistinguishable with respect to the arrow symbols. The arrows may then be replaced with the equivalence

$$\left[ \begin{array}{c} \text{state} \\ \end{array} \right] \quad \begin{array}{c} \text{\(\text{state}\)} \\ \text{\(\text{\(\(\text{\(\(\)}} \text{\(\(\)} \text{\(\(\)} \text{\(\(\)} \text{\(\(\)} \text{\(\(\)} \text{\(\)} \text{\(\(\)} \text{\(\)} \text{\(\)} \text{\)} \\ \text{\(\(\)} \text{\(\(\)} \text{\(\)} \text{\(\)} \text{\(\)} \text{\)} \end{array} \right] . \tag{19}$$

Supplementing the right-hand side here with the concept of numerical values {*αs*} for all of the observables A = *A*(*q*, *p*) (or for phase variables {*q*1, *q*2, ... ; *p*1, *p*2, ...}), this side will turn into an exhaustive numerical realization of the left-hand side. Criterion ≈, then, turns into the R-number equality = of all the A -statistics or into an equality of phase distributions (*q*1, ... ; *p*1, ...). This is a situation of the classical (statistical) physics (ClassPhys), i.e., when "the physics is initially identified" with quantities being numerical in character: the particle coordinates/numbers, the number values of field functions, etc. The ill-posedness of such a paradigm—the core motive of QT—is discussed further below at greater length in Sections 6.4, 6.5, and 7. Consequently, "classicality" is not and cannot be regarded as a primitive in the logical construct. In both these cases, distinguishability ≈ depends on the concept of *α*-states.

**Remark 4.** *From this point onward, by state we will strictly mean representations* (11)*. Thus, it makes* no sense to speak of transitions between states*, much less of "transition from possible to actual" ([107], p. 189; Everett), [117–119]. The writing* |**Ξ** |*α and its typical wave-function collapse interpretation are not correct. Indeed, in treating transition* |**Ξ** |*α*1 *as a state-tostate destruction, its left-hand side cannot carry any information about* f(A )*-frequencies for other events* |**Ξ** |*αs ,* much less *about the amount of destruction from envisioned* B*-observations* |**Ξ** B |*β . Such "*f(B)*-amounts" are always present at the experimental interpretation of the* |**Ξ** *-symbol. For this reason, the concept of a state should not be used as a correct term at all [58]; the terminology, however, has been settled.*

The motivation given above—S (system, primitives), O (observation), R (representations), T (taboo), the semantic principia I (QM-statics), and II (numbers) complemented below with the principium III—is sufficient for further creating the basis of the mathematical formalism of QM. These tenets should hardly be regarded as postulates, at least in the common meaning of the phrase "postulates of a physical theory", since they are a natural language and are, as we believe, the points of departure for reasoning whatever the approach to the micro-world. It is clear that they are directly concerned with the familiar dialogs, which reflect, in the words by Bohr, "[Einstein's] feeling of disquietude as regards the apparent lack of firmly laid down principles . . . , in which all could agree" ([131], p. 228).

The underpinning of QT must thus begin, at least to a large extent, with a simplification/reducing the terminology in use and putting the language and the semantics of observations/numbers in order, rather than giving the "improved" postulates or definitions.

"The task is not to make sense of the quantum axioms by heaping more structure, more definitions, . . . , but to throw them away wholesale"

> C. Fuchs ([50], p. 989)

"Simplicity is implicit in the basic goals of scientific inquiry. . . . only simple theories can attain a rich explanatory depth. . . . the Basic Propert[ies] should indeed be very simple"

#### N. Chomsky ([132], pp. 4–5)

As was underscored above, these (organizing) principles do not stipulate for predetermined mathematics and physics, with the exception of a linguistic/metamathematical understanding [105,106] of how to look at the mathematical axioms, structures, rational theories, and their interpretations altogether. See also Remarks 7 and 10 and Sections 5 and 10.

#### **3. Ensemble Formations**

Your acquaintance with reality grows literally by buds or drops of perceptions. . . . they come totally or not at all—W. James (1911)

Are billions upon billions of acts of observer- participancy the foundation of everything?—J. Wheeler ([62], p. 199)

The key corollary of Macro-paradigm (12) is not merely the appearance of numerical data in the theory but also the fact that the further construct cannot rely on isolated primitives but rather on their aggregates being considered as an integrated whole, i.e., as a set. This causes a choice for the ensemble notation.

#### *3.1. Mixtures of Ensembles*

Returning to the analysis of transitions **?** *αs*, one obtains that the lower row in (12) actually comes from indeterminacy

$$\{\underline{\underline{\alpha}}\_1\}\_{n\_1}\{\underline{\underline{\alpha}}\_2\}\_{n\_2}\cdots\cdots\cdots\cdots$$

$$\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\downarrow\cdots$$

$$\underline{\underline{\underline{2}}}\cdots\cdots\underline{\underline{2}}\cdots\cdots\cdots\cdots\cdots$$

and thus (12), by virtue of (16), should be replaced with the scheme

$$\{\cdots\underbrace{\Psi}\ \cdots\ \underbrace{\Psi}\ \cdots\ \underbrace{\Psi}\ \cdots\ \underbrace{\Psi}\ \cdots\ \}$$

$$\{\cdots\ \underbrace{\alpha\_1}\ \cdots\ \underbrace{\alpha\_2}\ \cdots\ \alpha\_n\}$$

wherein the composition of the upper ingoing row may not be predetermined. Fundamentally, according to (17), it may not be withdrawn from (20), yet at the same time, the meaning of the row can in no way be aligned with the adjective "observable" via typical empirical/physical words: properties, readings, quantities/amounts, and other "observable" characteristics. Such non-detectability is the equivalent of a box that may be prepended to Scheme (6):

$$\overbrace{\boxed{\cdots \cdots \cdots \cdots}}^{} - \stackrel{\cdots}{\cdots} \stackrel{\cdots}{R} - \stackrel{\cdots}{\stackrel{\cdots}{\cdots} \stackrel{\cdots}{\cdots}} \stackrel{\cdots}{\mathbb{R}} \tag{21}$$

If *β*'s serve as Ψ in (21), then we have the schemes of precedence and of continuation:

$$\begin{array}{ccccc}\hline\hline\cdots\cdots\cdots\cdots\\\hline\end{array}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\qquad\quad\text{or}\qquad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\cdots\\\hline\begin{array}{ccccc}\hline\cdots\cdots\cdots\\\hline\end{array}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel{\scriptstyle\neg\infty}{\longrightarrow}\mathbb{R}\quad\stackrel$$

Let an observer capture the fact of any distinguishability in the penultimate A . Section 2.1 tells us that this may only be the distinguishability of objects {*α*1, *α*2, ...}; hence, this very A turns into an observation (pt. O). The M-paradigm then gives rise to the numbers of *α*-events (*n*1, *n*2, ...) and, thereupon, their relative frequencies (1, 2, ...) by the rule (14). If subsequently micro-observations B are to follow, then a composite macro-observation B ◦ A has been formed, and frequencies {*j*} cannot impact statistical characteristics of these later B-observation's micro-events. However, being an ongoing ensemble for B, each homogeneous {*αs* ··· *αs*}*ns* is indistinguishable from an indefinite ensemble {· · · Ψ ··· Φ ···}*ns* since the concept of "≈A -sameness" is unknown for B. Instrument B is "aware of only its own ≈B and cannot know *what* it destroys", or that the source-object consists of one and the same *αs*. Rejecting this point brings us once again (p. 9) to attempts at "penetrating the black box" of transitions (5), i.e., to attempts at creating the physics of a more primary level. According to pts. O and M, an instrument produces nothing more than its own "destruction list"; in this case, ({*β*1}*m*1 , {*β*2}*m*2 , ...). This list is completely independent of the preceding one since, according to pt. R••, there cannot be restrictions on TA and TB. In case the set {*αs* ··· *αs*}*ns* transits into collection {*βk* ··· *βk*}*ns* , this means that *αs* has always transited into one and the same *βk* every time (under the convention Σ ∞), and merely a coincidence *αs* = *βk* of eigen-primitives in the lists TA and TB takes place.

If B ◦ A is proceeded with a third observation C , the preceding analysis is repeated recursively with the same result; only the values {*j*} will be changed. As a consequence, only the following two ongoing types for macro-scheme (20) are conceivable:

$$\{\cdot,\cdot\:\mathbb{X}\cdot\cdots\mathbb{A}\cdot\cdots\mathbb{A}\cdot\cdots\}\quad\stackrel{\text{indefinite ensemble}}{\lceil \text{no statistics} \rceil}\,,\tag{22}$$

$$\left\{ \left\{ \cdots \overleftarrow{\Psi} \cdots \overleftarrow{\Phi} \cdots \overleftarrow{\Phi} \cdots \right\}^{(\mathfrak{e}\_{\mathbb{J}})} \left\{ \cdots \overleftarrow{\Psi} \cdots \overleftarrow{\Phi} \cdots \overleftarrow{\Phi} \cdots \right\}^{(\mathfrak{e}\_{\mathbb{Z}})} \cdots \right\} \quad \underset{\left[ \text{with statistics} \ (\varrho\_{1}, \varrho\_{2}, \dots) \right]}{\text{ensimtle}} \quad \text{(23)}$$

It is reasonable to regard Case (23) as a "non-interfering" mixture of the system's A -preparations

$$\{\mathcal{S}^{(\mathfrak{e}\_1)}, \mathcal{S}^{(\mathfrak{e}\_2)}, \dots\} \quad \Longleftrightarrow \quad \{\underline{\mathfrak{e}}\_1^{(\mathfrak{e}\_1)}, \underline{\mathfrak{e}}\_2^{(\mathfrak{e}\_2)}, \dots\} \dots$$

to each of which one assigns the positive number *s* < 1 referred to as its statistical weight. These weights—"an element of reality" ([113], p. 649)—are all that is inherited from the preparation A , and subsequent micro-observation acts B are performed again on indefinite ensembles (22).

It is clear that in the view of transitions in scheme (20), this situation is a derivative of (22) and this very type (22) is crucial ([34], p. 53). In other words, if the preparation is regarded as a concept as essential as observation (pt. O), we still remain within the framework of the binate essence of the transition:

> Ψ B *β*, *α* B *β* .

Its left-hand side should always be seen as an undetermined primitive, even though we treat/call it the preparatory (micro)observation. See also "preparation-measurement reciprocity" in [133].

#### *3.2. Ensemble Brace*

According to pts. R and M, the representations in (11) must reflect all information about the physics of the problem: primitives/incomes, transitions ("arrows" ), and outgoing statistics. All the data are contained in Scheme (20), which is why the maximum that the model of a future mathematical object—it characterizes everything we obtain while watching the S—can rely on is the *ensemble brace*:

$$\{\Xi\} := \left\{ \{\cdots\underbrace{\Phi}\_{\bullet} \cdots \underbrace{\Phi}\_{\bullet} \cdots \underbrace{\Phi}\_{\bullet} \cdots \underbrace{\Phi}\_{\bullet} \cdots \right\} \tag{24}$$

(or a couple of ensemble bunches).

It is immediately seen that (24) carries the radical difference between situation (17) and its "roughening" (19) because of the upper row. The enormous arbitrariness within the brace and arrows A is "programmed" to give birth to the different processing rules of statistics and to effects that are typical for QM. Thanks to the maximality of (24), it is only this row that encodes all the sought-for cases of distinguishability ≈. In particular, by varying the upper row while the lower one remains unchanged, we get into a situation when *α*-statistics (f1, f2, ...) are found to be the same for ⎧⎩Ξ⎫⎭ and ⎧⎩Ξ˜ ⎫⎭, and meanwhile, ⎧⎩Ξ⎫⎭ ≈ ⎧⎩Ξ˜ ⎫⎭.

The problem is thus as follows. With the indefinite A-ensemble (16) in hand, i.e., with the upper row of (24), is it possible, based on the principles described above, to bring the still incomplete relation ≈ to the maximal quantum-physical distinguishability of states?

#### **4. Why Does Domain** C **Come into Being?**

. . . quod ideo sint imaginariae, . . . quod ideo sint . . . tum certe forent reales ideoque non imaginariae—L. Euler (1736)

(. . . this is why they are imaginary. Were they ..., they would certainly be real and therefore not imaginary.)

. . . denn die imaginären Größen existierten doch nicht?—D. Hilbert (1926)

The first priority in the ≈-distinguishability of objects (24) is to separate the closest and unconditional criterion—the outgoing *α*-statistics. To do this, let us split the lower row into families {*α*1}∞1{*α*2}∞2··· , where

$$
\infty\_1 + \infty\_2 + \dots = \infty \,\, , \tag{25}
$$

and, subsequently (rather than the reverse, otherwise (23)), taking into account the "arbitrariness of arrows", we also split the upper row:

$$(\underline{\mathbf{T}}) = \left\{ \begin{array}{c} \{ \{ \cdots \underline{\Psi} \cdots \underline{\Phi} \cdots \underline{\Phi} \cdots \}\_{\approx\_1} & \{ \cdots \underline{\Psi} \cdots \underline{\Phi} \cdots \underline{\Phi} \cdots \}\_{\approx\_1} \\ \{ \quad \{ \underline{a}\_1 \cdots \cdots \cdots \underline{a}\_1 \}\_{\approx\_1} & \{ \underline{a}\_2 \cdot \cdots \cdots \cdot \underline{a}\_2 \}\_{\approx\_2} & \cdots \} \end{array} \right\} \tag{26}$$

(the indication of observation A is omitted further below since it has been mirrored in primitives *α*). Hereafter, infinities ∞*j* stand for cardinal numbers (a number of elements, possibly finite) of their own ensembles. Therefore, the extension of distinguishability (15) should be produced by comparing the sub-objects such as

$$\{\cdots\underbrace{\Phi}\ldots\underbrace{\Phi}\ldots\underbrace{\Phi}\ldots\underbrace{\Phi}\ldots\}^{\omega\_1}\_{\omega\_1},\tag{27}$$

that differ from each other in the upper-row composition.

#### *4.1. Continuum of Quantum Phases*

The cardinality of the T-set cannot be finite. This would finitely entail many *α*primitives for all kinds of instruments. However, the finiteness of this number *n*T would mean an exclusivity of its value that does not follow from anywhere. At the same time, all the A-ensembles (16) are subsets of the set T (boolean 2T); any finite portion of it is ruled out. Hence, the endless variety of upper rows in (27) is uncountable.

Aside from the number of f-statistics, program R does also require an association of the numerical objects with each row

$$\mathbf{v} = \{\cdots \cdot \mathbb{Z} \cdot \cdots \mathbb{Z} \cdot \mathbb{Q} \cdot \cdots \mathbb{Q} \: \mathbb{Q} \cdot \cdots \mathbb{Q} \cdot \cdots \}^{\omega} \quad \Longleftrightarrow \quad \cdots \cdot \mathbb{Z}$$

because primitive's symbols must disappear in the ultimate description. To avoid introducing the structures ad hoc, we will produce numbers here—the upper row—in the same manner, in which statistics were producing in Section 2.5—the lower row. Indeed, the genesis of the concept of the number must be single in theory. That is, we should again take into account the presence of copies of primitives and write

$$\cdots \quad \iff \quad \left\{ \overbrace{\{\overline{\mathbb{K}}\}^{\simeq\_{\mathbb{K}}} \{\overline{\mathbb{K}}\_{n}\}^{\simeq\_{\mathbb{K}}} \cdots}^{\times\_{\mathbb{K}}} \right\} \; , \tag{28}$$

and numbers per se will come into being by the Σ-convention, such as (14), i.e., through the cardinal ratios

$$\aleph\_{\prime} := \frac{\infty}{\infty^{\prime}}, \qquad \aleph\_{\prime} := \frac{\infty}{\infty^{\prime}}, \qquad \dots \tag{29}$$

Now, the discreteness of micro-transition acts is embodied in (28) with the sequence (Ψ, Ψ, . . . ), and the uncountability of micro-arbitrariness is inherited by attaching the symbolic "quantities"—"countless" characters (∞, ∞, . . . )—to elements of this sequence. The global discreteness says that there are no grounds to assume a more than countable infinity ℵo for the set T, i.e., |T| = ℵo. The infinity of the family (28), hence, has the type

$$
\mathcal{Z}^{\ast\_{\bullet}} = \aleph\_{\prime}
$$

i.e., it is continual [134]. Parenthetically, the 2ℵo is the only known way of introducing the continual (more than discrete) mathematical infinity. Which possibilities exist for the form of row (28)?

The trivial case A = {Ψ}∞, i.e., *K* = 1 in (28) drops out at once since element Ψ would always go into the same primitive:

$$\{\underline{\Psi}\}\cdot\underbrace{\cdots\cdot\cdots\cdot\underline{\Psi}}\_{\{\underline{\Psi}\_{1}\cdot\cdots\cdot\cdots\cdot\underline{\Psi}\_{l}\}\_{\omega\_{1}}}=\underset{\{\underline{\Psi}\_{1}'\}\_{\omega\_{1}}\,\underline{\omega}\_{l}}{\{\underline{\Psi}\_{1}'\}\_{\omega\_{1}}}.\tag{30}$$

However, this is tantamount to the identity Ψ ≡ *α*1, which robs of any meaning the concept of the transition Ψ A *α*. We obtain a single number here—the number of *α*1-clicks—and arrive thereby at classical statistics, the physics of which is inadequate with respect to the interference patterns. Hence, the following options are admissible for the formations (28):

$$\begin{array}{c} \left( \overbrace{\{\Psi'\}}^{\alpha'} \overbrace{\{\Psi'\}}^{\alpha'} \right), \quad \dots, \quad \left( \overbrace{\{\Psi'\}}^{\alpha'} \overbrace{\{\Psi'\}}^{\alpha'} \overbrace{\{\Psi'\}}^{\alpha'} \right), \quad \dots \\\end{array} \quad \dots, \quad \left( \underbrace{\{\Psi'\}}^{\alpha'} \overbrace{\{\Psi'\}}^{\alpha'} \overbrace{\{\Psi''\}}^{\alpha'} \dots \dots \right), \quad \dots \quad \dots \quad \text{(31)}$$

with minimal *K* = 2. If some of the infinities (∞, ∞, ...) are finite here or countable, this does not change the total continuality ℵ. The extreme case *K* = ∞—a countable infinity of continuums—also changes this count because of ℵ + ℵ + ··· = ℵ [135]. All of these infinities may be even countably duplicated without augmenting the continuum since ℵ·ℵ··· = ℵℵo = ℵ.

What can one say about relationship of cases (31) between each other? Do we have to deal with their total arbitrariness or with only one of these schemes? The latter case—the sameness/indistinguishability of upper rows in (27)—would correspond to the structural staticity of theory. Otherwise, whether one (unrecognizable upper) row should differ (why?) from another in the number (what?) of defining primitives {Ψ, Ψ, ...} (which ones?)?

Suppose the variability of *K*. That is, consider the simultaneous existence of, say, the *K* = {2, 3} rows

$$\{\{\overline{\Psi}\}^{\omega\_{\prime}}\{\overline{\Psi}\_{\bullet}\}^{\omega\_{\ast}}\}, \qquad \{\{\{\overline{\Psi}\}^{\omega\_{\ast}}\{\overline{\Psi}\_{\bullet}\}^{\omega\_{\ast}}\{\overline{\Psi}\_{\bullet}\}^{\omega\_{\ast}}\}.$$

However, each of the 2-row is a particular case of the 3-row with a cardinal number ∞ = 0:

$$\left\{ \{ \mathbf{\tilde{K}}' \}^{\omega\_{\mathbf{\tilde{\cdot}}}} \{ \mathbf{\tilde{K}}\_{\mathbf{\tilde{\cdot}}} \}^{\omega\_{\mathbf{\tilde{\cdot}}}} \right\} = \left\{ \{ \mathbf{\tilde{K}}' \}^{\omega\_{\mathbf{\tilde{\cdot}}}} \{ \mathbf{\tilde{K}}\_{\mathbf{\tilde{\cdot}}} \}^{\omega\_{\mathbf{\tilde{\cdot}}}} \{ \mathbf{\tilde{K}}\_{\mathbf{\tilde{\cdot}}} \}^{\{ \omega\_{\mathbf{\tilde{\cdot}}} = 0 \}} \right\} \subset \left\{ \{ \mathbf{\tilde{K}}' \}^{\omega\_{\mathbf{\tilde{\cdot}}}} \{ \mathbf{\tilde{K}}\_{\mathbf{\tilde{\cdot}}} \}^{\omega\_{\mathbf{\tilde{\cdot}}}} \{ \mathbf{\tilde{K}}\_{\mathbf{\tilde{\cdot}}} \}^{\omega\_{\mathbf{\tilde{\cdot}}}} \right\}.$$

(the case in point is sets). Therefore, these situations are structurally indistinguishable from each other, and the *K* = 2 theory is a *sub*theory for *K* = 3. So, the cases *K* = {2, 3} are actually not mutually exclusive; rather, they form an embedding. We thus have arrived at the one cumbersome and common construct akin to the Russian dolls 2 ⊃ 3 ⊃ 4 ⊃···. Hence, the minimal 2-theory will always be present inside all the higher orders *K* > 2 as an "independent (sub)world". For this reason, the *K* = 2 theory must be created in any way; incidentally, it will enclose the *K* = 1 case.

In the other part, we have no criteria to terminate the sequence 2 3 ··· at some intermediate *K* < ∞. Such a cut-off does immediately raise an issue of the questionable empirical exclusivity of a certain "world integer *K* 3" that defines the number of "physically inaccessible" Ψ-objects. Moreover, these options would be related to a certain topological dimension *K* 3 that has an unmotivated origin. We thus conclude that the non-minimal options *K* = 3, *K* = 4, . . . in (31) should be dismissed.

**Remark 5.** *A few remarks can be made in connection with the case when K* = ∞*. It is related to a conglomerate of infinities, which has the form of a discretely infinite family of continual infinities* {κ*,* κ*, . . . }, and things would have been even "worse" had the staticity of the schemes* (31) *been changed to variability. Such formations would need to be equipped with topology and with associated concepts of convergence and of limit. However, all this touches on principally unobservable numerical entities, for which it is not clear how to motivate the further reductions to "finite mathematics" as required: dimensions, finite approximations, finite numbers (which ones?), and the like. More to the point, all of that would pertain to the global structural parameters of the theory prior to constructing it per se, not to mention the physical models. To put it plainly, such an assumption would result not in a theory but in a theory of theories, and so on ad infinitum, which should be somewhere terminated in some way. For these reasons, we leave the case K* = ∞ *aside, though it might be worth elaborating on it. However, in Section 7.6, we will give a further justification that the number domain of the theory is what it has already been known in QM.*

As a result, one has a choice: the structural staticity *K* = 2 or entirely non-structured/ undetermined set of outgoing primitives {Ψ*j*, Ψ*k*, ...}, i.e., extremely complex case *K* = ∞. We do choose *K* = 2. This option might have been adopted even before on the ground that the most minimalistic construction, which set-theoretically gives rise, as a minimum, to the minimal numerical object—a single number—corresponds to the minimal *K* = 2 in (31). The maximal case is problematic, while the mid-ones are ruled out. That is to say, all possible assumptions regarding the upper row structure in (27) are indistinguishable from a case just as if the row contained two primitives only {Ψ, Ψ} =: {Ψ, Φ}. The functionality of the symbol ∪, with regards to the inclusion of the {Ψ, Φ}'s copies, is unchanged (see Section 5.1 further below).

We establish in the following writing of Scheme (27) that

$$\{\underline{\Psi}\cdots\underline{\Psi}\}\_{\simeq^{\mathbb{N}}\_{1}}\cap\{\underline{\Phi}\cdots\underline{\Phi}\}\_{\simeq^{\mathbb{N}}\_{1}}\cdots\underset{\cdots\cdots\in^{\mathbb{N}}\_{l}}\{\underline{\Psi}\cdot\cdots\cdot\underline{\Psi}\}\_{\simeq^{\mathbb{N}}\_{1}}$$

none of the primitives {Ψ, Φ} coincide with *α*1. Otherwise, the unrestricted adjunction of identical transitions *α*1 *α*1 to (27) would mean indeterminacy of both the number κ1 and the actual statistics (f1, f2,...).

Let us take into account that numbers (29) are mathematically generated by the standard scheme: ,(ordered) integers- ,(ordered) rationals- ,(ordered) continuum -. The natural ordering < is always present here and, as is well known ([136], p. 52), can be isomorphically represented by the set-theoretic inclusion ⊂ on a certain system of sets. That inclusion (= "to be contained in"), in turn, is directly concerned with the semantics of Section 2. The natural-language term "accumulating"—"the old is being nested into the new"—is formalized to create sets by the cumulative ensembles (see Section 5.1).

We now conclude that all kinds of schemes (27) form an ℵ-continuum, for which there is no reasonable rationale for equipping it with a topology other than the standard order topology of the one-dimensional real R-axis or its equivalents. Call the quantity κ ∈ R *quantum phase*.

It should be added that in considering some two upper rows in (27) as infinite sets

$$\{\underline{\Psi}\cdots\underline{\Psi}\}\_{\bowtie\prime}\cup\{\underline{\Phi}\cdots\underline{\Phi}\}\_{\bowtie\prime} \quad \text{and} \quad \{\underline{\Psi}\cdots\underline{\Psi}\}\_{\bowtie\prime}\cup\{\underline{\Phi}\cdots\underline{\Phi}\}\_{\bowtie\prime}$$

one can always establish their formal identity. However, physics requires distinguishing the rows, which is what the numerical part of pt. R and comparison of cardinals (∞, ∞) do "serve".

#### *4.2. Statistics* + *Phases*

Thus, the closest reconciliation of Scheme (26) with the R•-postulate is an ensemble brace of the form

$$\mathbf{(\underline{\nabla})} = \left\{ \begin{array}{c} \{\{\underline{\Psi}\}\_{\simeq\_1^1}\{\underline{\Phi}\}\_{\simeq\_1^1}\} & \{\{\underline{\Psi}\}\_{\simeq\_1^1}\{\underline{\Phi}\}\_{\simeq\_1^1}\} & \dots & \dots \\ \vdots & \downarrow & \downarrow & \downarrow & \downarrow \\ \{\underline{\Psi}\_1 : \cdots : \cdots \underline{\Psi}\_1\}\_{\simeq\_1} & \{\underline{\Psi}\_2 : \cdots : \cdots \underline{\Psi}\_2\}\_{\simeq\_2} & \dots & \dots \end{array} \right\} \tag{32}$$

followed by the (upper) continual numeration through R-numbers

$$\mathsf{x}^{\circ} := \frac{\mathsf{os}\_{\circ}^{\prime}}{\mathsf{os}\_{\circ}} \qquad \left(\mathsf{os}\_{\circ} := \mathsf{os}\_{\circ}^{\prime} + \mathsf{os}\_{\circ}^{\prime}\right) \,. \tag{33}$$

In other words, the quantitative description in the theory is created on the basis of the minimal building bricks

$$\left\{ \begin{array}{c} \{\{\underline{\Psi}\}\}\_{\curvearrowleft} \{\underline{\Phi}\}\_{\curvearrowright} \\ \vdots \\ \{\underline{\Psi}\}\ldots\end{array} \right\} \qquad (unitary \,\hspace{1cm} \times \underbrace{\begin{array}{c} \{\underline{\Psi}\}\\_{\scalebox{0.0}{\curvearrowleft}\{\underline{\Psi}\}\\_{\scalebox{0.0}{\curvearrowleft}\{\underline{\Psi}\}\\_{\textit{op}}\\_{\textit{op}} \\ \downarrow \\ \downarrow \end{array}} \right\} \qquad (unitary \,\hspace{1cm} \times \underbrace{\begin{array}{c} \{\underline{\Psi}\}\\_{\textit{op}} \\ \underline{\Psi}\end{array} \right) \tag{34}$$

with *two* abstract ongoing primitives.

Now, we have had cardinals connected by Relation (25) and Structures (32) and (33). In the above-described context, parentheses { } and symbols Ψ, Φ, no longer carry meaning at this point. Therefore, we may omit them as "extraneous" and write (32) as

$$(\underline{\mathbf{S}}) \quad \iff \quad \left\{ \begin{array}{c} \approx\_1 \ \mid \quad \approx\_2 \ \mid \quad \dots \\ \approx\_1 \ \mid \quad \approx\_2 \ \mid \quad \dots \end{array} \right\} = \cdots \ \mid \quad \underline{\mathbf{S}}$$

where *αs* are well represented by a subscripted numerals; observation A has been fixed so far. Let us now introduce a statistics from the "embracing infinity" (25):

$$\begin{aligned} \boldsymbol{\pi} &:= \left\{ \begin{array}{c} \varkappa\_{1} \\ \mathtt{f}\_{1} \cdot \infty \end{array} \; \bigg| \; \mathtt{f}\_{2} \cdot \infty \; \bigg| \; \frac{\cdots}{\cdots} \right\} &= \left\{ \begin{array}{c} \varkappa\_{1} \\ \mathtt{f}\_{1} \end{array} \; \bigg| \; \mathtt{f}\_{2} \; \bigg| \; \frac{\cdots}{\cdots} \right\} \cdot \boldsymbol{\infty}^{\prime} \end{aligned} \qquad \begin{aligned} \mathtt{f}\_{s} &:= \frac{\infty\_{s}}{\infty} \; \mathtt{f}\_{s} \end{aligned}$$

Then, by Σ-postulate, one arrives at a continually numeral labeling of objects (32):

$$\left(\Xi\right) \quad \Longleftrightarrow \quad \left\{ \left( \begin{smallmatrix} \varkappa\_1\\ \mathbf{f}\_1 \end{smallmatrix} \right) , \left( \begin{smallmatrix} \varkappa\_2\\ \mathbf{f}\_2 \end{smallmatrix} \right) , \dots \right\} \dots$$

Recall that the arithmetical operations on the emergent pairs (f,κ) are still out of the question, and Σ-limit does not care the "innards" of ⎧⎩Ξ⎫⎭. Only one of all the potentially infinite quantities tends to the ∞-infinity—the total cardinality (25) of brace (32). What remains "non-extraneous" in (32) is *α*'s, and we return them to their place. Hence, from the viewpoint of observation A , the aggregate of the possible brace (24) is indistinguishable from an order-indifferent two-parametric family of data

$$(\boldsymbol{\Xi}) = \left\{ \begin{pmatrix} \mathbb{X}\_1 \\ \mathbf{f}\_1 \end{pmatrix} \underline{\mathbf{a}}\_{1'} \begin{pmatrix} \mathbb{X}\_2 \\ \mathbf{f}\_2 \end{pmatrix} \underline{\mathbf{a}}\_{2'} \; \; \; \; \right\}. \tag{35}$$

We drop a lower bar in the symbolic designation ⎧⎩Ξ⎫⎭, highlighting the fact that the meaning of the ⎧⎩Ξ⎫⎭-object becomes increasingly divorced from primitives in pt. S and gets into the number domain to match program R.

As an outcome, despite the freedom of ingoing collection in (26) and quantum microarbitrariness, the distinguishability ⎧⎩Ξ⎫⎭ ≈ ⎧⎩Ξ˜ ⎫⎭ is *indeed determinable*, it is determinable not only by statistics, and is the (R × R)-numerical:

$$(\Xi) \not\approx (\Xi)' \quad \text{if} \quad (\mathfrak{k}^\*, \aleph^\*) \not\equiv (\mathfrak{k}^\*, \aleph^\*) \,. \tag{36}$$

What is more, the preliminary (classical) ≈-criterium (15) fits in (35)–(36) as a particular case by omitting the κ-numbers and middle link from (17). That is to say, ignoring quantum "κ-effects" is only possible via the (3 %→ 2) reduction of (17) into (18), with an automatic imposition of the ClassPhys description. A simplified and hypothetical version of QM over R1 is also ruled out. It would mean a reduction in the two numbers (f,κ) to a single one. However, they have fundamentally different origins. The construct and reasoning in Section 2.1 also tell us that the attempt at a greater "quantum specification" to (5) and (17) is impossible by virtue of the two-row structure—ingoing/outgoing—of the object ⎧⎩Ξ⎫⎭, and distinguishability by numerical pairs (36) is the highest possible.

The ⎧⎩Ξ⎫⎭-objects (35) remain, and they, as a family, exhaustively inherit the problem's physics. The quantities f*s* are the really observable (unitless) numbers—the percentage quantity of events—which are declared by instrument/observer to be the distinguishable *α*objects. The quantities κ*s* are the internal and unremovable degrees of freedom. Figuratively speaking, the κ's may be speculatively referred to as phases, but they may not be associated with an actual quantity of something. Not only is any material or the classical treatment of these "amounts" impossible, but it is fundamentally *prohibited* since the converse would have meant endowing the nonexistent boxes (5) and (6) with a notional content or asserting the nature of their origin. Justification is only allowed here for the fact of their existence,

which is mirrored by the presence of the left-hand side in the concept of the transition of Ψ A *α* (Remark 2).

In view of numerous ongoing discussions of the meaning to the quantum state [21], note that, for the same reason, any (even merely similar) classical/ontological and causal "visualization mechanisms" ([5], p. 137) as the wave function of a certain real matter, of a hypothetical observable, of an "objective knowledge', or of the classical data (whatever this all means) are—and this we stress with emphasis—pointless. This is why, strictly speaking, without further theoretical conventions,

•It is impossible ([12], p. 13) to make/prepare, observe/read-off, transmit or measure/approximate a state, or to endow it with the property of being known/unknown, or physically recognize/compare/distinguish it from the other.

We will be repeatedly turning back to this matter in Sections 6.3.1, 6.4, 6.5, and 10.2. The present thesis has not undergone a change even with regard to the word "statistics" in the Born rule [6], if only because the rule is a substantial—two-to-one—reduction in the (f,κ)-data. The state will itself, when created as a mathematical object, determine the meaning of all of these words (see Section 5.3) with an appropriate concept of the physical distinguishability (Section 2.4). Cf. the works [53,54] and the "methods to directly measure general quantum states . . . by weak measurements" in [137] and, on the other hand, the statements in Section 15.5 of the book [33].

All the κ*s* and f*s* are independent of each other, except for relation f1 + f2 + ··· = 1. Taking into account the admissible renormalization of both R-numbers, the pair (f,κ) can be topologically identified with a point on the complex plane:

$$\left(\begin{smallmatrix}\mathbf{x}\end{smallmatrix}\right) \rightleftharpoons \left(\begin{smallmatrix}\lambda\end{smallmatrix}\mu\right) \in \mathbb{R}^2 =: \mathbb{C}\text{ .} $$

That is, the domain C is at the moment just a two-dimensional numeric continuum without algebra of complex numbers. Notice that the pairs of R-numbers is a starting point—different from ours—to the QT in ref. [138]. More than that, the impossibility of the real-number QM became a subject of the direct experimental test to distinguish between the complex-number and real-number representations of QT: on photonic systems [139] and the superconducting qubits [140].

The issue of the numerical domain over which the quantum description is being conducted—the real R, the complex C, the quaternions Q, or whatever—is non-trivial and continues to be the subject of study [57,93,138,141,142]. The complexity C is often motivated by quantum dynamics (Schrödinger's equation) ([36], p. 132; Stueckelberg), [143]; however, such a motivation is inconsistent, and as we have seen, there is no need for it. The rigidity of the C-domain points to the fact that, in particular, the quaternion QM also has no place to originate from ([33], Section 10.1), although it was the object of theoretical constructs in the 1960–1970s [144]. Note that even the most comprehensive works [36] (p. 131), [72] (p. 234), [93] (p. 217), [96,138], and [142] (!) observe a difficulty in the full substantiation of the Cdomain in QT. Within the last decade, this theme had also attracted the particular attention of the information-theoretic approach to QT [138,145,146].

The above-outlined emergence of the numerical quantities in theory is a draft at the moment and will be refined further below in Section 7.

#### **5. Empiricism and Mathematics**

Set theory does not seem today to have . . . organic interrelationship with physics— P. Cohen and R. Hersh ([147], p. 116)

. . . physics has . . . to say about the foundations of mathematics . . . "if we believe in ZF there is nothing for physics to say" is not right—P. Benioff ([2], p. 31)

Up to this point, we have dealt, roughly speaking, with a single abstract aggregate ⎧ ⎩Ξ ⎫ ⎭ isolated from the others. However, the constructional nature of the ensemble brace (32) entails the following closedness relation between them. Every brace ⎧ ⎩Ξ ⎫ ⎭ is composed of

some others in infinitely many ways (for remote analogies, see ([40], Section 11.2)), i.e., it is a union

$$\{\Xi\} = \{\Xi'\} \cap \{\Xi'\_\*\} \,\,\,\tag{37}$$

and, to put it in reverse, any union of two braces is a third object-brace. In assemblages (37), the operation ∪, which generates them, is commutative and associative:

$$A \cup B = B \cup A,\qquad \left(A \cup B\right) \cup C = A \cup \left(B \cup C\right),\tag{38}$$

and these two- and three-term relations not only are not a formal supplement, but should be read as the *structural* properties in general. Let us address the matter more closely.

#### *5.1. Union of Ensembles*

Consider the lower *α*-rows of brace (26) and experimentally forming the new real *α*-ensembles from them. Let the procedure of that forming be denoted by U(*A*, *B*, ...), where (*A*, *B*, ...) are the ensembles per se. Its essence is such that it is comprehensively determined by the following minimum. A rule that involves the *fewest* (i.e., two) number of arguments U(*A*, *B*) = **?** and a rule of the *repeated* applying U to itself: U(U(...), ...) = **?**. Obviously, we should write

$$\mathbb{U}(A,B) = \mathbb{U}(B,A), \qquad \mathbb{U}(\mathbb{U}(A,B), \mathbb{C}) = \mathbb{U}(A, \mathbb{U}(B, \mathbb{C})) \,, \tag{39}$$

which is of course merely the empirical rephrasing the standard properties (38) of operation ∪. However, the converse is logically preferable: Empiricism (39) is formalized into the abstract properties (38). If we now attach the upper "quantum" primitives to the low *α*-rows—a requirement of Section 2.1—then the operationality of actions with the resulting ⎧⎩Ξ⎫⎭-braces would be just like that of U, i.e., (39). In other words, we carry over (and had already used everywhere) properties (39) to the general operation on ⎧⎩Ξ⎫⎭-brace, without distinguishing between the essences of symbols ∪ and U. "Micro-operationality' of empiricism and its formalization are confined, at most, by the rules (38) and (39).

Let us temporarily discontinue using the numerical terminology as applied to ⎧⎩Ξ⎫⎭- objects. They differ from each other due to relationships between their "innards", rather than because of our assignment of differed symbols (*λ*, *μ*) to them. The brace is comprised of elements that are combined into sets and are added to them. In the language of the abstract logic, we are dealing with the fact that transitions *x* form the brace *A*, *B*,..., i.e., they are in the membership relationships *x* ∈ *A*, *x* ∈ *B*, . . . or, when accumulated as microacts, "get belonged to them". That is to say, the braces themselves and their formation (accumulation of statistics for the Σ-limit) are equivalent to a huge number of propositional "micro-sentences *x* ∈ *A* or *x* ∈ *B* or . . . ". However, again, this is nothing but a logically formal equivalent of the union operation ∪:

$$A \cup B = \{ \mathbf{x} \mid (\mathbf{x} \in A) \lor (\mathbf{x} \in B) \}\tag{40}$$

.

which is already being constantly exploited above.

**Remark 6.** *As is well known [134,136], due to properties of logical atoms* ∈ *(*membership*) and* ∨ *(*or*), the properties of sentences such as* (40) *are determined precisely by rules* (38) *for* ∪*. Technically, we should also take an idempotence A* ∪ *A* = *A into account, however. At the same time, the need to have a number requires that the duplicates in ensembles have to be taken into consideration. Nevertheless, this situation is easily simulated by the set theory itself. Indeed, look first at the lower row in* (24) *as a strictly abstract set* {*α*, *α*, ...} ⊂ T*. Then, instrument* A *"asserts" the distinguishable elements* {*α*1*, α*2, ...} *and those that should be thought of as their equivalents:*

$$
\underline{\mathfrak{a}}'\_1 \approx \underline{\mathfrak{a}}''\_1 \approx \dots = \colon \underline{\mathfrak{a}}\_1 \,, \qquad \underline{\mathfrak{a}}'\_2 \approx \underline{\mathfrak{a}}''\_2 \approx \dots = \colon \underline{\mathfrak{a}}\_2 \,, \qquad \dots \, \underline{\mathfrak{a}}\_n
$$

*This equivalence can be characterized, say, by words "a detector click at one and the same place α*1*". Upon such a formalization, one obtains the formation* {*α*1*α*1 ···}{*α*2*α*2 ···}··· ≈ {*α*1 ···}{*α*2 ···} ···*, i.e., the very lower row in* (26)*. It is within this context that we think of the union operation without running into inconsistencies. Accordingly,* ⎧⎩Ξ⎫⎭ ∪ ⎧⎩Ξ⎫⎭ = ⎧⎩Ξ⎫⎭*, but the standard symbol* ∪ *continues to be used for simplicity.*

Therefore if we get back to the numeral labels (35) but ignore the "inner composition" of ⎧⎩Ξ⎫⎭, i.e., the M-paradigm, thus excluding ∪ and (38) from the reasoning, then all possible ⎧⎩Ξ⎫⎭-objects would turn into the semantically "segregated ideograms". Micro-transitions, their mass nature, arbitrariness, ≈-distinguishability, and the "quantumness" of the task simply disappear. To illustrate, the obvious statement

> the brace {Ψ A *α*} =: ⎧⎩Ξ⎫⎭ has an empirical "kindred" with its duplication {Ψ A *α*, Ψ A *α*} =: ⎧⎩Ξ ⎫⎭

becomes pointless because the property ⎧⎩Ξ⎫⎭ ∪ ⎧⎩Ξ⎫⎭ = ⎧⎩Ξ ⎫⎭ is missing. Furthermore, this is despite the fact that creating the transition copies in ⎧⎩Ξ ⎫⎭ is a primary operation for generating the objects and reasoning at all. The construction of the theory would then become possible only with the interpretative introduction of the vanished concepts anew. Therefore, macro-empiricism necessitates that the relationships (38) be operative rules, and with that, the quantumness or classicality of consideration is of no significance.

**Remark 7.** *Let us take a closer look at the situation on the opposite—mathematical—side. The union of sets* ∪ *is already a fundamental operation at the level of the set-theoretic formalization, e.g., the Zermelo–Fraenkel (ZF) axioms [134]. This is one of the first ways to create sets—the axiom of union. Thus, if we believe in the set-theoretic mode of explaining/creating the quantum rudiments, the quantitative description will inevitably invoke the operationality of the mathematical primitive* ∪ *through rules* (38)*. This would be suffice to declare,*

•*Inasmuch as* we have nothing but ∪ and ⎧⎩Ξ⎫⎭ (taboo T)*, commutativity/associativity of theory is then postulated from the outset by* (38)*, with the subsequent carrying these structures over to numerical representations, i.e., to* R *or* C*.*

*It is preferable, however, to adhere to the sequence order in ideology more stringently—*,*observation*- ,*mathematics*-*,* ,*empiricism*- ,*numerical representation*-*—without substituting it for the opposite. At least, if we rely upon the comprehension of the empiricism as a formalization of the zero-principium of QT (Remark 2):*

• *Our primordial perceptions are formalized only into sets and set-theoretic* ∪*-abstraction* (40)*.*

*See also [2] (p. 178), [58] (Ch. 3), [78] (p. 323), [96,104], [148] (pp. 12, 86, Ch. 4), [149] (Section V.9), and Section 11.1.*

Summing up, we detect a kind of junction point: the physical and mathematical fundamentality of operation ∪ for describing the elementary acts. That is to say, the mathematics of ⎧⎩Ξ⎫⎭-brace (32) and of objects (35) may not inherently be exhausted by them as "bare" sets without structures.

Recalling now pr. II, we draw a conclusion regarding the very construction of the theory.

• The reconciliation of the R-paradigm with empiricism must transform itself into *rewriting* the primary ensemble ∪-constructions (26), (32), (34), and relationships between them into the language of numerical symbols.

More formally, we have the following continuation of pt. R•.

**R**+ *Homomorphism of the ensemble-brace properties "onto numbers"*: mutual ∪-relationships (38) between the ⎧⎩Ξ⎫⎭-brace should be carried over to relations between their numerical ⎧⎩Ξ⎫⎭-representations (35).

Thereby, we once again fix the maximum that is available for the building up of quantum mathematics. One may only handle the ∪-aggregates of transitions—constructions (32), (35)—and the minimal modules (34).