It may seem strange that before starting the experiment you need to list all the base quantities used and the total number of variables considered in the model. Moreover, this important point is completely ignored in the canonical CODATA method for calculating the target value of the physical constant and its relative uncertainty. The need for this requirement is explained as follows.

The fact is that any measurement of a variable by itself implies the presence of an already formulated model. As mentioned in [11], a measurement model constitutes a relationship between the output quantities or measurands (the quantities intended to be measured) and the input quantities known to be involved in the measurement. In this case, the researcher, based on his own knowledge, experience, and intuition, uses, as a rule, dimensional and dimensionless variables from the International System of Units (SI). This means, on the one hand, that accounting for one variable or another is equally likely: to describe the phenomenon being studied, the scientist or engineer selects a qualitative and quantitative set of variables as he wishes. The most famous example of such a situation is the possibility of studying an electron as both a particle and a wave. Although two qualitatively different sets of variables are used to describe the motion of an electron, as it turned out, both have the right to life, which led to the concept of electron dualism. On the other hand, by choosing specific variables, the scientist thereby declares the type of process by which she or he intends to measure the physical constant, for example, mechanical, thermal, electromagnetic, combined heat–mass–electromagnetic, etc. Thus, the model contains variables with different dimensions determined by the seven (ξ = 7) base quantities in SI: L is the length, M is mass, Т is time, I is electric current, θ is thermodynamic temperature, J is luminous intensity, F is the amount of substance. These seven main quantities can be combined into different groups, the structure of which depends on the qualitative set of variables considered in the model. Each group is called a class of phenomenon (CoP). In other words, CoP is a combination of physical phenomena and processes described by a finite number of base and derived quantities (defined as products of powers of the base units) that characterize the features of the process of measuring the physical constant in terms of qualitative and quantitative aspects [12]. For example, during the gravitational constant measurements with a torsion balance, the following base quantities are typically used: L, M, Т (CoP_SI ≡ LMT). Measuring the Boltzmann constant is usually realized by CoP_SI ≡ LMТθF or CoP_SI ≡ LMТθI. It should be noted that SI is a product of human thinking and does not exist in nature. At the same time, SI is used in science and technology in accordance with the developed consensus [6] [13].

Refining the process of formulating the model from the perspective of choosing a specific CoP may offer a new interpretation of the results of measurements of physical constants, which will be discussed later in the article.

In addition, it should be noted that a researcher, choosing a specific CoP, in practice discards possible potential hidden relationships between the variables considered in the model and the ignored variables. Thus, of course, this can affect the accuracy of the proposed model and even lead to an increase in its uncertainty. This is explained by the fact that, although in the opinion of a significant part of the scientific community, the model error can be reduced by using a large number of variables thanks to improved algorithms and supercomputers, each variable introduces its own uncertainty into the total integral error that affects the desired result. However, as the dimension of the model increases, only the reliability of the model results improves [14].

To assess the magnitude of the threshold mismatch [1] between the model and the measurement process under study, due to the choice of CoP, we will give the following reasoning and calculations.

In science and technology, a wide variety of unit systems can be used that are most suitable for a particular application, for example, Imperial and US customary units or Natural units [15]. However, the most widely used system is the international standard metric system—the International System of Units (SI). Therefore, further reasoning and calculations are given as applied to SI, especially since SI units are also used in the CODATA methodology. However, since SI is an Abelian group [16] [17], like any other system of units, the final conclusions do not depend on the choice of a specific system of units.

It can be proved using the concepts and mathematical apparatus of the theory of similarity [12] that SI includes a large but finite number of dimensionless variables [16]:

μ SI = 38265 . (1)

μ_SI cannot be simultaneously taken into account in the model. Typically, a researcher uses 10, 20, or even 130 variables with CoP_SI ≡ LMTθF [18] to describe the process being studied.

For further reasoning, we indicate that information entropy [19] is manifested through the interaction of the studied physical system and the formulated model. This model is an information channel between the physical system and the observer. As a result, information entropy is subjective, depending on the consciousness of the researcher with his preferences in choosing a quantitative and qualitative set of variables taken into account.

We will use an analogy with a theory of signals transmission. Imagine that the observed measurement process has a huge number of properties (quantities, criteria) that characterize its content and interaction with the environment. Then, we assume that each dimensionless complex represents the original readout (reading [20] [21]), through which some information on the dimensionless researched field u (researched process) can be obtained by the observer. In other words, the researcher observing a physical phenomenon, analyzing the process or designing the device, selects—according to his experience, knowledge and intuition—certain characteristics of the object. With this selecting of the object, connections of the actual object with the environment enveloping it are destroyed. In addition, the modeler takes into account the relatively smaller number of quantities than the current reality due to constraints of time, and technical and financial resources. Therefore, the “image” of the object being studied is shown in the model with a certain uncertainty, which depends primarily on the number of quantities taken into account. In addition, the object can be addressed by different groups of researchers, who use different approaches for solving specific problems and, accordingly, different groups of variables, which differ from each other in quality and quantity. Thus, for any physical or technical problem, the occurrence of a particular variable in the model can be considered as a random process.

Then, let there be a situation where all µ_SI of SI values can be taken into account, provided that the choice of these quantities is a priori considered equally probable. In this case, we are guided by the idea of Brillouin connecting amount of information obtained in the simulation (observation) without making any disturbance in the measurement process, and the uncertainty inherent in the selected model [20].

Comparing the number of variables in SI with the selected number of variables in a particular model, it turns out that you can calculate the amount of information ΔA_e contained in it [16]

Δ A e = κ ⋅ ln [ μ SI / ( z ″ − β ″ ) ] (2)

where ΔA_e is expressed in units of entropy, μ_SI includes dimensionless criteria/variables that are considered equally probable when selected by the researcher in the model, z ″ and β ″ are the number of all and base quantities registered in the chosen model, respectively, γ = z ″ − β ″ , k is the Boltzmann constant.

Obviously, in practice, researchers can use dimensionless criteria that are not included in μ_SI. It is easy to show that a value of μ₂ (number of dimensionless criteria and numbers in an extended system of units numbered “2”) does not dramatically influence on a final result. Let us suppose that 2 μ SI = μ 2 . Taking into account that ln μ SI ≫ ln ( z ″ − β ″ ) SI , ln μ 2 ≫ ln ( z ″ − β ″ ) 2 , and ln μ SI ≫ ln 2 , we can obtain the following relation

Δ A eSI / Δ A e 2 = [ ln μ SI − ln ( z ″ − β ″ ) SI ] / [ ln μ 2 − ln ( z ″ − β ″ ) 2 ] = ln μ SI / [ ln 2 + ln μ SI ] ≈ 1. (3)

The physical content of Equation (2) is very important. For example, two research groups analyze the process of measuring a physical constant. The results are different from each other. Who presented the most respectable option? Obviously, the choice of the class of the phenomenon and the number of variables considered will affect the information content of the model and will cause a different amount of information contained in it [22]:

Δ A b 1 γ = ( ln ( γ CoP / γ 1 ) ) / ln 2   ( bits ) , (4)

Δ A b 2 γ = ( ln ( γ CoP / γ 2 ) ) / ln 2   ( bits ) , (5)

where ΔA_b1γ and ΔA_b2γ are an amount of information of the model formulated by the first research team and the second team, respectively, compared with the model that takes into account the optimal number of dimensionless criteria γ_CoP inherent to a particular CoP; γ₁ and γ₂ are the number of dimensionless criteria in the first and second models, respectively.

Let us suppose that γ 1 < γ CoP < γ 2 and | γ 1 − γ CoP | < | γ CoP − γ 2 | . By analyzing Equations (4) and (5), some readers may suggest that it is preferable to use a model with a large number of variables when modeling a physical process. However, this is a wrong conclusion, and here is why. By comparing ΔA_b1γ and ΔA_b2γ in absolute terms, the researcher can “instantly” determine which one is smaller. This means that the number of dimensionless criteria considered is closer to the optimal one γ_CoP corresponding to the minimum comparative uncertainty [20] (for its detailed calculation, see below). Thus, a project with a lower absolute value of | γ i − γ CoP | is more informative. Therefore, the information approach will significantly reduce the time spent by researchers on the analysis of publications.

It is also advisable to emphasize the importance of introducing the concept of “information content” of the model, ΔA, from the point of view of choosing a specific model of the measurement process.

First, information content can provide a natural explanation for the preferred choice of a particular measurement method. Until now, it was almost impossible to recommend scientists to focus their efforts on a specific method. However, with the introduction of the concept of information content (3)-(5), it is quite possible to state which of the models describing the same method of measuring the physical constant is most preferable.

Secondly, the content of the information may mean that some models of measuring the physical constant are less preferable. Specific examples and detailed explanations are presented in Section 3.

Third, the information content implies that many models are unsuitable for measuring a particular physical constant. In these models, the number of variables does not correspond to the recommended number inherent in the selected class of phenomena. The accuracy of the model, usually associated with the number of variables considered, is seen in a different light when implementing the information approach (Section 3).

Fourth, an accurate description of the experimental setup in terms of an information approach requires some knowledge of the future. We know very well that the experiment itself never allows the experimenter to look into the future, but if we try to interpret what is happening, some expectation of the future experiment seems necessary. We suspect that this approach may allow us to reflect a state where some hidden variables that can influence the result are not considered by the researcher’s conscious decision (Section 3).

The amount of information contained in the model (Equation (2)) is only a sufficient condition for choosing the preferred option. In addition to this, we can formulate a necessary condition. Using Equation (2), we can get an expression for calculating the absolute uncertainty of the model Δ_pmm [16], due to the choice of CoP and the number of variables considered in the model:

Δ pmm / S = ( z ′ − β ′ ) / μ SI + ( z ″ − β ″ ) / ( z ′ − β ′ ) , (6)

where S is the interval in which the dimensionless quantity u is located, z¢ and β¢ are the total number of dimensional quantities and the number of base quantities in the CoP, respectively, ε = Δ_pmm/S is the comparative uncertainty [20].

Four features of Equation (6), called the µ-rule, should be noted. First of all, this equation is applicable both to models with dimensional variables and with dimensionless variables, due to the following relations:

Δ U / S * = ( Δ U / a ) / ( S * / a ) = Δ u / S r / R = ( Δ U / U ) / ( Δ u / u ) = ( Δ U / U ) / [ ( Δ U / a ) / ( U / a ) ] = 1 (7)

where S and Δu are the dimensionless quantities, respectively, the range of variations and the total absolute uncertainty in determining the dimensionless quantity u; S^* and ΔU are the dimensional quantities, respectively, the range of variations and the total absolute uncertainty in determining the dimensional quantity U; a is the dimensional scale parameter with the same dimension as that of U and S^*; r is the relative uncertainty of the dimensional quantity U; and R is the relative uncertainty of the dimensionless quantity u.

Secondly, Equation (6) is a kind of correspondence principle for the model development process and can be related to the Heisenberg principle. During measuring a physical constant, the model must satisfy Equation (6). In other words, changing the level of detailed description of the test bench by choosing the class of the phenomenon ( z ′ − β ′ ) and the specific number of variables to be taken into account ( z ″ − β ″ ) causes a change in the smallest value of the comparative uncertainty Δ_pmm/S of the main studied function (main variable). Thus, the correspondence principle uniquely determines the achievable accuracy limit (for a given class of phenomena), while simultaneously revealing a pair of quantities observed by a conscious researcher, in particular, absolute uncertainty Δ_pmm in the measurement of the studied quantity and the interval of its change S.

Third, Equation (6) has the property of equivalence. This means that it is true for other measurement systems. Models formulated in other systems of units of measure, for example, in yards and pounds or centimeter-gram-second (CGS), will also have to comply with Equation (6) to maintain the basic relationships between physical variables. Equivalence ensures that physical models of reality remain consistent, regardless of units.

Fourth, the development of measuring equipment, improving the accuracy of measuring instruments, and improving existing and newly created measurement methods in the aggregate lead to an increase in knowledge about the object under study and, therefore, the value of achievable relative uncertainty decreases. However, this process is not infinite and limited by Equation (6). The reader should keep in mind that this principle is not a lack of measuring equipment or an engineering device, but the way the human brain works. Predicting the behavior of any physical process, physicists actually predict a tangible yield of instrumentation. It is true that, according to the µ-rule, observation is not a measurement but a process that creates a unique physical world in relation to each specific observer.

In addition, using Equation (6), one can find the necessary conditions for approaching the smallest relative uncertainty of each CoP, r_CoP, the fulfillment of which can confirm the legitimacy of the declared measured value of the physical constant. For this, it is necessary to take the derivative of Δ_pmm/S with respect to z ′ − β ′ and equate it to zero:

( z ″ − β ″ ) = ( z ′ − β ′ ) 2 / μ SI (8)

For example, for the thermal–electromechanical process (CoP_SI ≡ LMТθI), which is used in measuring the Boltzmann constant, it is necessary to consider the following statement. The dimension of any derived quantity q can be expressed as a unique combination of dimensions of the main base quantities to the different powers [23]:

q ∋ L l ⋅ M m ⋅ T t ⋅ I i ⋅ Θ Θ ⋅ J j ⋅ F f , (9)

− 3 ≤ l ≤ + 3 ,   − 1 ≤ m ≤ + 1 ,   − 4 ≤ t ≤ + 4 ,   − 2 ≤ i ≤ + 2 , − 4 ≤ θ ≤ + 4 ,   − 1 ≤ j ≤ + 1 ,   − 1 ≤ f ≤ + 1. (10)

( z ′ − β ′ ) L M T θ I = ( е l ⋅ е m ⋅ е t ⋅ е θ ⋅ е i − 1 ) / 2 − 5 = 4247 , (11)

γ L M T θ I = ( z ″ − β ″ ) L M T θ I = ( z ′ − β ′ ) L M T θ I 2 / μ SI = 4247 2 / 38265 ≈ 471 , (12)

where l , m , ⋯ , f are the exponents of the base quantities, taking only integer values and which vary in certain intervals, Equation (10); γ_LMTθI is an optimal number of criteria in a model inherent in CoP_SI ≡ LMTθI; “−1” corresponds “to the case where the exponents of all the base quantities are zero in Equation (9); 5 corresponds to the five base quantities L, M, T, θ, and I; and division by 2 indicates that there are direct and inverse quantities, e.g., L¹ is the length and L⁻¹ is the run length. The object can be judged based on the knowledge of only one of its symmetrical parts, while the other parts that structurally duplicate this one may be regarded as information empty. Therefore, the number of options of dimensions is reduced by a factor of two.

Then, one can calculate the minimum achievable comparative uncertainty ε_LMTθI:

ε L M T θ I = ( Δ u / S ) L M T θ I = 4247 / 38265 + 471 / 4247 = 0.222. (13)

Using calculations similar to (10)-(13), it is possible to calculate achievable comparative uncertainties ε_CoP and the recommended number of quantities γ_CoP corresponding to different classes of phenomena (Table 1).

Thus, there is provided an amazing opportunity to calculate r_CoP by two methodologies in the framework of the information-based approach.

The first, dictated by the μ-rule, is to analyze the data on the value of the achievable relative uncertainty at the moment, considering the latest measurement results. In this case, the possible interval of placement of the physical constant S is selected as the difference between its maximum and minimum values

measured by various scientific groups over a certain period of time. Thus, using the achievable comparative uncertainty inherent in the selected class of phenomena when measuring the physical constant, we can calculate the recommended minimum relative uncertainty, which is compared with the relative uncertainty of each published study. Moreover, the apparent randomness of the choice of the interval value S, depending on the dataset, does not ultimately affect the final result: an extended range of variation of the value of S only indicates the imperfection of the measuring instruments, which leads to a significant increase in relative uncertainty. This can be illustrated by Equation (14), which indicates that the value of the relative uncertainty is finite and not equal to zero.

r 1 / r 2 = ( Δ 1 / A 1 ) / ( Δ 2 / A 2 ) = ( ( ε 1 ⋅ S 1 ) / A 1 ) / ( ( ε 2 ⋅ S 2 ) / A 2 ) = ( ε 1 ⋅ S 1 ⋅ A 2 ) / ( ε 2 ⋅ S 2 ⋅ A 1 ) = ( ( z ′ − β ′ ) 1 / μ + γ CoP / ( z ′ − β ′ ) 1 ) ⋅ S 1         ⋅ A 2 / ( ( ( z ′ − β ′ ) 2 / μ + γ CoP / ( z ′ − β ′ ) 2 ) ⋅ S 2 ⋅ A 1 ) ≡ finitevalue (14)

where r₁, r₂, Δ₁, Δ₂, ε₁, ε₂, S₁, S₂, A₁, A₂ are the relative, absolute, and comparative uncertainties, intervals of placement of the physical constant, and magnitudes of physical constants, respectively; index 1 corresponds to a larger interval, index 2 corresponds to a shorter interval, S₂ < S₁.

Assuming that ( z ′ − β ′ ) 1 = ( z ′ − β ′ ) 2 and γ CoP ⋅ μ ≫ 1 (look at Table 1 and (1)), then

r 1 / r 2 ≈ S 1 ⋅ A 2 / ( S 2 ⋅ A 1 ) (15)

Equation (15) indicates that the ratio r₁/r₂ does not tend to infinity or to zero, its value is finite and reflects an increase in the accuracy of instruments when measuring a physical constant. An important advantage of this approach is the independence of the real instability of the results of experimental measurements.

Although the goal of our work is to obtain the main restriction on the accuracy of measuring physical constants, we may also ask whether it is possible to achieve this limit in a physically correctly formulated model. Because our estimate is given by optimization in comparison with the achieved comparative uncertainty and observation interval, it is clear that in the practical case the limit cannot be reached. This is because there is an inevitable primordial uncertainty of the model, depending on the preferences of the researcher, based on his intuition, knowledge, and experience. The magnitude of this uncertainty is an indication of how likely it is that your personal philosophical inclinations will influence the outcome of this process. When a person mentally builds a model, at each stage of its construction there is some probability that the model will not correspond to this phenomenon with a high degree of accuracy.

In what follows, this method is denoted as IARU and is represented by the below-mentioned procedure [24].

1) From the published data of each experiment, the value α, relative uncertainty r_α, and standard uncertainty u_α (possible interval of placing) of the physical constant are chosen;

2) The experimental absolute uncertainty Δ_α is calculated by multiplying the physical constant value z and its relative uncertainty r_α attained during the experiment, Δ α = α ⋅ r α ;

3) The maximum α_max and minimum α_min values of the measured physical constant are selected from the list of measured values α_i of the physical constant mentioned in different studies;

4) As a possible interval for placing the observed constant S_α, the difference between the maximum and minimum values is calculated, S α = α max − α min ;

5) The selected comparative uncertainty ε_CoP (Table 1) inherent in the model describing the measurement of the constant is multiplied by the possible interval of placement of the observed constant S_α to obtain the absolute experimental uncertainty value Δ_IARU in accordance with the IARU, Δ I A R U = ε CoP ⋅ S α ;

6) To calculate the relative uncertainty r_IARU in accordance with the IARU, this absolute uncertainty Δ_IARU is divided by the arithmetic mean of the selected maximum and minimum values, r I A R U = Δ I A R U / ( ( α max + α min ) / 2 ) ;

7) The relative uncertainty obtained, r_IARU, is compared with the experimental relative uncertainties r_i achieved in various studies;

8) According to IARU, a comparative experimental uncertainty of each study ε_IARUi is calculated by dividing the experimental absolute uncertainty of each study Δ_α by the difference between the maximum and minimum values of the measured constant S_α, ε_IARUi = Δ_α/S_α. These calculated comparative uncertainties are also compared with the selected comparative uncertainty ε_CoP (Table 1).

As follows from the presented description of the step-by-step procedure, the results do not depend on the complex, difficult to fulfill requirements inherent in statistical-expert methods (SEM), such as, for example, the CODATA method [6]. Moreover, the physical meaning of IARU is to assess the suitability of a method for measuring a specific physical constant. IARU can also be used to compare achieved measurement accuracy with various methods for different constants (Section 3.2).

In the second technique, S is determined by the limits of the used measuring instruments [20] in each particular experiment. This is confirmed by the fact that in experimental physics, unlike other areas of technology (for example, when studying the processes of heat and mass transfer in refrigeration equipment [22]), the researchers present measurement data with the obligatory indication of the standard uncertainty. At the same time, it is obvious that this uncertainty of a particular measurement is subjective because the observer is simply not able to consider all the uncertainties. The standard uncertainty is calculated considering the uncertainties observed by the experimenters.

Then, the ratio between the absolute uncertainty achieved in the experiment and the standard uncertainty, which acts as a possible interval for the placement of the physical constant, is calculated. Thus, in the framework of the information approach, the comparative uncertainties achieved in the studies are calculated, which, in turn, are compared with the theoretically achievable comparative uncertainty inherent in the chosen class of phenomena. This method is hereinafter referred to as IACU and includes the following steps:

1) From the published data of each experiment, the value α, relative uncertainty r_α, and standard uncertainty u_α (possible interval of placing) of the physical constant are chosen;

2) The experimental absolute uncertainty Δ_α is calculated by multiplying the physical constant value α and its relative uncertainty r_α attained during the experiment, Δ α = α ⋅ r α ;

3) The achieved experimental comparative uncertainty of each published study ε_IACUi is calculated by dividing the experimental absolute uncertainty Δ_α by the standard uncertainty u_α, ε_IACUi = Δ_α/u_α;

4) The experimental calculated comparative uncertainty of each published study ε_IACUi is compared with the selected comparative uncertainty ε_CoP inherent in the model (Table 1), which describes the measurement of the physical constant.

It should be noted that this methodology also does not require consistent experimental results. From the point of view of its physical content, the IACU reflects the situation, how thoroughly all possible sources of uncertainties for a certain class of phenomena were identified and considered in calculations using different methods of measuring a specific physical constant (Section 3.2).

In the next section, we will present the results of applying the information approach to analyze the measurement data of various physical constants using different methods. In the proposed analysis, only publications are considered that contain data on the value of a physical constant, its relative, and standard uncertainties.

As an example of the visual step-by-step application of the information approach, we consider the results of measuring the Boltzmann constant using the method of an acoustic gas thermometer (CoP_SI ≡ LMTθF). One of the many datasets can be found in [25], which consists of measurements taken in seven laboratories (Table 2) from 2009 to 2017.

We will apply IARU and IACU to calculate the estimated observation interval of k, S_k, according to IARU, its values obtained in two projects were selected: k_max = 1.3806508 × 10⁻²³ m²·kg·s⁻²·K⁻¹ [29] and k_min = 1.3806484 × 10⁻²³ m²·kg·s⁻²·K⁻¹ [32]. Then

S k = k max − k min = 2.4 × 10 − 29 m 2 ⋅ kg / ( s 2 ⋅ K ) . (16)

One can calculate the comparative uncertainty ε_LMTθF and the lowest relative uncertainty r_LMTθF taking into account Equations (1), (6), (8), (10), and (16):

( z ′ − β ′ ) L M T θ F = ( е l ⋅ е m ⋅ е t ⋅ е θ ⋅ е f − 1 ) / 2 − 5 = ( 7 × 3 × 9 × 9 × 3 − 1 ) / 2 − 5 = 2546 , (17)

γ L M T θ F = ( z ″ − β ″ ) L M T θ F = ( z ′ − β ′ ) 2 / μ SI = 2546 2 / 38265 ≈ 169 , (18)

ε L M T θ F = ( Δ u / S ) L M T θ F = 2546 / 38265 + 169.4 / 2546 = 0.1331. (19)

Δ L M T θ F = ε L M T θ F ⋅ S k = 0.1331 × 2.4 × 10 − 29 = 3.2 × 10 − 30 m 2 ⋅ kg / ( s 2 ⋅ K ) . (20)

r L M T θ F = Δ L M T θ F / ( ( k max + k min ) / 2 ) = 3.2 × 10 − 30 / 1.38064961 × 10 − 23 = 2.3 × 10 − 7 . (21)

where “−1” corresponds to the case where all the exponents of the base quantities are zero in Equation (9); 5 corresponds to the five base quantities L, M, T, Q, and F; Δ_LMTθF is the absolute uncertainty.

The value of r_LMTθF = 2.3 × 10⁻⁷ calculated by IARU is in sufficient agreement with 6.0 × 10⁻⁷ [30] and is much closer to 3.7 × 10⁻⁷ [13]. This, first, confirms the legitimacy and appropriateness of using (6), and second, confirms the

*Data are introduced in [6] [13] [33].

μ-rule, when the experimentally achieved relative uncertainty is always greater than that calculated by the information approach (Section 2.4). Furthermore, data introduced in Table 2 allows formulation of the following conclusions:

1) Although the authors of publications declared that they considered all the possible sources of uncertainty, the values of absolute and relative uncertainties can still differ by more than a factor of two. A similar situation exists in the spread of the values of comparative uncertainties (IARU). This reflects the existence of hidden uncertainties that have eluded the attention of researchers.

2) The results from the use of IACU indicate a relative agreement between the magnitude of the experimental comparative uncertainties and their significant discrepancy (more than 3 - 4 times) compared with the recommended one (0.1331). This situation is explained by the fact that, on the one hand, research teams learn from each other in the search and elimination of undetected or unaccounted for uncertainties, thereby ensuring the relative uniformity of the magnitude of experimental comparative uncertainty. On the other hand, it should be considered that the idea of an acoustic gas thermometer method is based on the concept of an ideal gas, although the interaction between gas particles is not well understood. An additional difficulty is associated with measuring the molar concentration of gas per unit volume and volume itself with a competitive degree of accuracy. It should also be noted that the total volume includes the volume of the connecting pipes to the pressure gauges. Therefore, there may be significant unaccounted uncertainties due to both the formulation of the experimental model and the achievable accuracy of the values considered in the calculation. Moreover, the proximity of the acoustic mode to shell resonance leads to an unacceptably large degree of data violation for this mode. In addition, experimenters consider a much smaller number of variables compared with the recommended ones (see Table 1). These reasons lead to a large difference between the theoretically calculated comparative uncertainty and the experimental values of the comparative uncertainties achieved in measuring k.

Because the step-by-step procedure for applying the information approach was described in detail in Section 3.1, generalized information on the data sets of measurements of the Planck constant, Boltzmann constant, Hubble constant, and gravitational constant is presented below (Table 3).

Looking closer at the data entered, we can make the following comments.

1) In measuring the Planck constant, h, when moving from a model (LMТF) to CoP_SI with a large number of dimensionless criteria (LMТI), the comparative uncertainty increases. This change is due to the potential effects of the interaction between an increased number of variables that may or may not be considered by the researcher. At the same time, the r_exp/r_SI ratio for CoP_SI ≡ LMТI is much smaller, which indicates the advantage of the Kibble balance method for measuring the Planck constant. This is also confirmed by the significant difference in the

¹KB—Kibble balance. Data include results of measurements taken in seven laboratories from 2014 to 2017. ²XRCD—X-ray crystal density. Data include results of measurements taken in seven laboratories from 2011 to 2018. ³AGT—acoustic gas thermometer. Data include results of measurements taken in seven laboratories from 2009 to 2017. ⁴DCGT – dielectric constant gas thermometer. Data include results of measurements taken in six laboratories from 2012 to 2018. ⁵JNT—Johnson noise thermometer. Data include results of measurements taken in six laboratories from 2011 to 2017. ⁶DBT—Doppler broadening thermometer. Data include results of measurements taken in six laboratories from 2007 to 2015. ⁷BDL—brightness of distance ladder. Data include results of measurements taken in seven laboratories from 2011 to 2019. ⁸CMB—cosmic microwave background. Data include results of measurements taken in six laboratories from 2009 to 2018. ⁹BAO—baryonic acoustic oscillations. Data include results of measurements taken in four laboratories from 2014 to 2018. ¹⁰Data include results of measurements taken in seven laboratories from 2000 to 2014. ¹¹Data include results of measurements taken in five laboratories from 2001 to 2018.

value of the comparative uncertainty CoP_SI ≡ LMТF (XRCD - 0.0146) compared with CoP_SI ≡ LMTI (KB – 0.0245) with almost equal experimental relative uncertainties achieved (1.3 × 10⁻⁸ ≈ 1.2 × 10⁻⁸).

As stated in [46], the implementation of the measurement of h using the Kibble balance or the XRCD methods allowed us to achieve a consistent reliable value for the latest results. In addition, the calculated relative uncertainty does not exceed the uncertainty due to the current implementations of primary and secondary units of mass. However, given the r_exp/r_SI ratio (LMТI: 2.9, LMТF: 9.1), there is an urgent need to reduce the influence of sources of uncertainty for XRCD.

It should be noted that, in the framework of the information approach, the statement that “after the Planck constant is constant (an exact number with zero uncertainty...)” [47], is unacceptable because the relative uncertainty of measurement of the Planck constant always varies depending on the selected CoP inherent in the selected model.

2) The data from Table 3 clearly show that the minimum achievable relative uncertainties, r_SI, calculated in accordance with the information approach, differ by two orders of magnitude for different methods of measuring the Boltzmann constant k! That is why, in the framework of the information approach, in contrast to the concept approved by CODATA, it is not recommended to determine and declare only one value of relative uncertainty when measuring the Boltzmann constant (and other constants) by various methods.

Using an information-oriented approach, both a respected scientist and a simple engineer can easily identify the advantages or disadvantages of a particular measurement method. Thus, analyzing the data of Table 3, it is obvious that the greatest success in achieving high accuracy of measurement of k in recent years was achieved using JNT and DBT, considering the smallest values of the ratio r_exp/r_SI (1.9) and (1.1). At the same time, the achieved experimental least relative uncertainty of 3.7 × 10⁻⁷, realized using DCGT, is doubtful. This is explained by the requirement of the μ-rule, according to which the theoretically calculated relative uncertainty (4.3 × 10⁻⁷) is always less than the experimental one (3.7 × 10⁻⁷). Therefore, researchers of [13] [38] should reanalyze all possible sources of uncertainty.

3) From Table 3, it is obvious that in measuring H₀ using BDL and BAO (CoP_SI ≡ LMТ), the experimental relative uncertainties (0.01 [41] and 0.01 [43]) calculated according to IARU are many times greater than the recommended 0.00023 and 0.00018, respectively. This situation indicates that hidden variables are not considered and CoP_SI ≡ LMT cannot be used in the future. Therefore, the conviction of scientists in accounting for all possible sources of uncertainties is far from providing a guarantee of achieving the true value of Н₀ by these two methods.

Following the logic of the information approach, it is again necessary to recognize that the method of measuring H₀ using the cosmic microwave background is the most promising, theoretically justified, and implements the most reliable experimental data. This conclusion can be confirmed by calculating the ratio ε_SI/r_exp considering the data in Table 3

( ε SI / r exp ) BAO = 0.0048 / 0.01 = 0.05 , ( ε SI / r exp ) BDL = 0.0048 / 0.01 = 0.05 , ( ε SI / r exp ) CMB = 0.0442 / 0.007 = 6.3 , ( ε SI / r exp ) BAO = ( ε SI / r exp ) BDL < ( ε SI / r exp ) CMB . (22)

Relation (22) reflects the fact that the best accuracy in measuring the Hubble constant can be achieved for the class of phenomena with a large number of base quantities.

Data of Table 3 show that the experimental minimum relative uncertainties r_exp exceed the recommended r_SI by 43 and 56 times for BDL and BAO, although when measuring H₀ with CMB, r_exp/r_SI = 2.4. Because consistency is one of the basic requirements for analyzing results, the current situation needs to be explained. The information approach declares that the inevitable primordial absolute uncertainty of the model already exists in the process of developing a method for measuring a physical constant. That is why, when making predictions to increase the accuracy of the Hubble constant, great caution should be exercised. The fact is that with an increase in the number of observed space objects, according to most astronomers using various methods of calculating H₀, absolute (ideal) statistical stability of the observed parameters and characteristics of any physical phenomena (real events, processes, and fields) is achieved. However, as was proved [48], the nonideal character of statistical stability (statistical predictability), which manifests itself in the absence of convergence (mismatch) of statistical estimates, plays a key role in limiting accuracy. At small temporal, spatial, or spatiotemporal intervals of observation, an increase in the amount of statistical data leads to a decrease in the level of fluctuations in statistical estimates, which creates the illusion of ideal statistical stability. However, starting with a certain critical amount of data, the decrease in the level of fluctuations stops. A further increase in the amount of data either practically does not affect the level of fluctuations in the estimates, or even leads to their growth.

4) The huge difference between the achieved experimental relative uncertainty in measuring the gravitational constant by means of mechanical methods and the theoretically recommended (r_exp/r_SI = 12.7) confirms the thesis of the information approach about the inappropriateness of their use in determining the true value of the gravitational constant. At the same time, a higher measurement accuracy of G was achieved using electromechanical methods: r_exp/r_SI = 1.9. From the point of view of the information approach, further clarification of the true value of the gravitational constant and a decrease in the experimental relative uncertainty is possible when using models and measurement methods with a large number of base quantities, for example, CoP_SI ≡ LMТθI.

In addition to the comments made in 1) - 4), one can make the following comments and conclusions, which are sometimes not obvious and do not coincide with the provisions of the generally accepted CODATA methodology.

a) The values of the minimum attainable comparative and relative uncertainties calculated according to the information approach depend on the choice of class of phenomena. Theory can predict their value. It is important to note that during the transition from the mechanistic model (LMТ) to CoP_SI with a large number of base variables, the uncertainty increases. This is explained by a change in the number of potential interaction effects between an increased number of quantities that may or may not be considered by the researcher.

b) You may notice large differences in the level of consistency between ε_SI and ε_exp calculated according to IACU. This level can be called a “coefficient of consistency” for a physical constant measured by various methods. In particular, when measuring Н₀, the ratio ε_exp/ε_SI is 710 (BDL) and 104 (BAO), while using CMB this ratio is only 4.1. A similar situation exists for measuring the gravitational constant: ε_exp/ε_SI = 100 when implementing mechanical methods, and ε_exp/ε_SI = 7.9 using electromechanical methods. At the same time, when measuring the Planck constant with KB and XRCD and using AGT and DCGT to calculate the Boltzmann constant, the values of the ε_exp/ε_SI ratios are very close to each other. As part of the information approach, this situation indicates that the BDL, BAO, and mechanical methods for G have limited use, and it can even be argued that they are not recommended for use. Moreover, using simple relationships calculated in accordance with a theoretically sound approach, we can draw very serious and far-reaching conclusions. It is important to emphasize once again that, using the IACU, researchers can find out for which method of measuring the physical constant it is necessary to continue the search for all possible sources of uncertainties. Thus, the ratio ε_exp/ε_SI is an objective criterion for assessing the achieved accuracy when comparing different methods of measuring one specific physical constant.

c) The introduction of comparative uncertainty, through the IARU, to evaluate the accuracy of measurements of physical constants allows the calculation of the r_exp/r_SI ratio. From the data in Table 3, a very obvious trend is clearly traced: models of measurements of physical constants with a small number of base quantities (LMT) and (LMTF) have clearly overestimated values of this ratio: 9.1, 12.7, 44, and 56. This is due to insufficient consideration of the effect of unaccounted base quantities and possible relationships between variables in calculating the value of the physical constant. At the same time, for models with a large number of base quantities, for example, LMTI or LMTθF, the r_exp/r_SI ratio varies from 0.9 to 2.9. Thus, in the framework of the information approach, we can consider the r_exp/r_SI ratio as a universal indicator of the achievements of scientists in measuring any physical constant using a variety of methods.