Physical Significance of Count Bounds of the Fundamental Measures

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

Measurement Quantization (MQ) expresses physical measures as counts of fundamental units and investigates the physical significance of their upper and lower bounds. We examine these bounds using discrete gravitational relations together with expressions for the speed of light, escape velocity, and Heisenberg’s uncertainty principle. The analysis distinguishes the Internal Frame, the discrete count configuration domain of MQ, from the System Frame, its continuous relational encoding induced by the Frames mapping. This distinction connects count constraints with realized physical quantities and provides a framework for examining the fundamental measures and physical constants. Two applications illustrate the proposed role of measurement bounds: gravitational relations associated with the lower length-count bound and galactic stellar velocities associated with the upper mass-count bound. These applications explore how a common count-based description can connect quantum and cosmological measurement domains, while distinguishing internal mathematical consistency from independent empirical validation.

Keywords: Count Bounds, Bounds to Measure, Measure, Physical Significance, Discrete Measure

1. INTRODUCTION

The Planck frequency is defined as the inverse of the Planck unit for time [1]. The value describes an upper count bound, 1.8549218804(41) 10⁴³ discrete units of t_p per second, fundamental time t_f if we are using the more precise calculation which accounts for the contraction of length associated with discrete measure ([2], Appx. D). But are such bounds physically significant? And if so, are there physically significant upper and lower count bounds for each of the three measures?

While many are familiar with Planck’s unit expressions for length and time ([2], Appx. BN), our best interpretation has always been that they describe the smallest measures. Using a new approach to the quantization of measure we show that measure is discrete and countable ([2], Appx. I, J) with respect to the Internal Frame of the universe. We also show that a description of the same phenomena is non-discrete with respect to the System Frame of the universe ([2], Appx. K). Finally, when resolving the difference between these frames we resolve expressions and values for the physical constants and the laws of nature [3].

The fundamental length l_f and mass m_f are measured in metres and kilograms, respectively. Together with the fundamental time introduced above, they provide the measurement scales for the dimensionless counts n_L, n_M and n_T ([2], Appx. W). Multiplying a count by its corresponding fundamental measure produces a physical length, mass or elapsed time. This count nomenclature lets us express measurement bounds without identifying a count with its dimensional realization.

This paper establishes the physical correlation between a count nomenclature and measured phenomena. In Sec. 3.2 we consider better known phenomena, such as the speed of light, escape velocity and Heisenberg’s uncertainty principle [4]. Sec. 3.3 addresses the relation between the quantum and cosmological measurement domains. We then look at bounds as apply to observational domains ([2], Appx. BB), a concept not considered in modern theory. And finally, we look to CODATA publications of the physical constants to demonstrate support for discrete, bounded measure.

2. METHODS

This paper uses classical expressions in a way that account for new properties of measure resolved using discrete expressions describing gravitational curvature, the speed of light, escape velocity and Heisenberg’s uncertainty principle. We identify this approach with the name Measurement Quantization (MQ). Notably, several examples in the publication record exist [5-9] whereby MQ is applied to problems in modern theory.

While not essential to understanding this presentation, this paper rests on physical evidence supporting the discreteness and countability of measure. Derivations predicting these properties and supporting measurement data are presented in the introductory paper entitled, Measurement Quantization [2]. All referenced appendices, along with a more extensive list of term and keyword definitions are included. A pre-print can also be found in Online Resources.

https://www.researchgate.net/publication/361148724_OnlineResources

Also available are two spreadsheets. Suppl. 1 contains calculations of all expressions in the introductory paper, which include calculations for expressions listed in the referenced appendixes. Suppl. 2 contains calculations of all expressions in this and 49 other papers that implement the MQ approach. Importantly, MQ is predicated on the physical support established in these resources.

3. RESULTS

3.1. Establishing a Physically Correlated Nomenclature

Such that the fundamental measures are shown to be physically significant references in nature, we resolve that these references are discrete and countable ([2], Appx. I, J). This is the first time a physical substructure to classical theory has been shown to exhibit discrete properties. Moreover, we can now ask questions; what is the smallest count of a measure; what is the largest count; what are the lower and upper bound relations between each of the three measures: n_L/n_T, n_M/n_T, n_L/n_M?

We use the inverse fundamental time to express a time-count frequency. The length-to-time, mass-to-time and length-to-mass ratios are ratios of dimensionless counts. The labels length frequency, mass frequency and length-to-mass frequency are manuscript-local names for these count comparisons, not SI frequencies; their dimensional realization must still be specified.

An SI frequency measures counts per physical time interval. A length-to-time count ratio becomes a speed when multiplied by the fundamental length-to-time scale. These distinctions keep the count descriptions separate from metres, kilograms and seconds.

With this in mind, we can describe a count of one as the smallest count, at least with respect to length and time. Recall, the smallest mass count is n_M=1/2 ([2], Appx. I). And the largest count is described by the Planck frequency. We can also use the Planck frequency to describe each count in terms of their associated SI measurement bounds ([2], Appx. O).

Physically important, we have mapped all counts with respect to an elapsed time of one second. Why is this and is it physically appropriate? We present the significance of this approach with an investigation of the gravitational constant ([2], Appx. F).

Equation (4) expresses the upper-count gravitational constant as the product of the cube of the speed l_f/t_f and the inverse mass-rate factor t_f/m_f ([2], Appx. F). Their combined dimensions are cubic length divided by mass and squared time. This factorization preserves the role of fundamental time in both ratios. Equation (5) gives numerical identities at unity; those identities alone do not establish a unique physical realization.

We therefore use time as the common reference when comparing the count allocations in Eqs. (1)–(3). This choice specifies the comparison interval rather than following as a uniqueness theorem from Eq. (5).

3.2. Applications of Counts to Basic Phenomena

The speed of light c, measured in m s⁻¹, supplies the length-to-time conversion used in Eqs. (6) and (7). The first compares elementary counts and the second expresses their physical realization.

This describes the bound, a count n_L of l_f traveled in one second. Importantly, while numerically the same, this is more often expressed as,

Equation (7) gives the speed of light when the positive length and time counts are equal. The elementary case has n_L=n_T=1. For an arbitrary common count, the distance is n_Ll_f and the elapsed time is n_Tt_f; their ratio remains the same speed.

A count ratio is dimensionless, whereas the associated physical ratio acquires dimensions from the fundamental measures. In Eq. (7), the length-to-time count ratio is converted to a speed by l_f/t_f. A count per physical time interval is a frequency; a ratio of counts is not itself an SI frequency.

Likewise, using the expression for escape velocity ([2], Appx. H) also set equal to the speed bound c we resolve the lower count bound for length and mass.

Notice here that this is a 2-to-1 relation.

Finally, we consider the uncertainty principle ([2], Appx. B), which correlates counts associated with length, mass, and time. The terms take on a meaning which differs from their usual application. Here, n_Lr describes the distance between the observer and the particle (i.e., as applies to the often quoted conical pairing of the particle/observer relation), whereas n_L describes the change in position of that particle.

We can use this expression to resolve the lower count bound for all measures ([2], Appx. I). For instance, such that n_L=n_T at the bound c=n_Ll_f/n_Tt_f, then

And where 2n_M≤n_Lr also at the escape velocity bound c=(2GM/r)^1/2, then we make the substitution to resolve that,

And such that n_Lr and n_L are both counts of length and such that n_L=n_T, then all three counts are equal to one. This resolves n_M=1/2.

3.3. Correlating the Quantum and Cosmological Domains

Not all count bounds are straight-forward. For example, one might conclude that an upper count bound n_M of m_f describes the total observable mass. Not entirely.

This is several orders in magnitude short of the visible mass in the universe, 1.4794 10⁵³ kg. The discrepancy is multifactored. For one, what is visible per increment of elapsed time t_f differs from what is observable over elapsed time n_T>1. What is visible or observable is also a function of what has been measured Ω_vis and what will be measured Ω_uobs ([2], Appx. BB).

To compare the quantum and cosmological domains, we distinguish the invariant dimensionless coefficient θ_si from its momentum realization p_si, measured in kg m s⁻¹. The fundamental relation is l_fm_f=2p_sit_f ([2], Appx. V). The two representations retain the invariant numerical coefficient while carrying different physical dimensions. Measurements of the cosmic microwave background (CMB) provide the cosmological context for this comparison.

With MQ, we recognize that the early universe can be characterized as a quantum bubble less contained no discrete interior point able to reference a point externally ([2], Appx. BF). This constrained expansion to a quantum crawl lasting 363,312 years ([2], Appx. BG-BH). When the universe reached a radius of square root of three l_f, the quantum epoch ended ([2], Appx. BG), external referencing became possible, and the universe began to expand with a leading-edge velocity of c relative to that edge. This event marked the beginning of the expansionary epoch we observe today.

Notably, it is understood that the rate of universal expansion – the metric expansion of space – differs from the rate of stellar expansion – the rate at which galaxies move away from one another. While these rates are similar, their relation is not well formulated.

Equation (15) gives the mass accretion rate M_acr, measured in kg s⁻¹, for fixed fundamental measures and the invariant coefficient ([2], Appx. BE). Its first form uses the dimensionless coefficient with the mass-to-time ratio; its second uses the momentum realization divided by fundamental length. Both forms therefore have mass-rate dimensions.

We use this along with the elapsed time describing the quantum epoch to resolve the quantity of mass/energy accumulated, which by 2n_M≤n_Lr (Eq. 8) cannot be baryonic, thereby constituting the CMB. We can then resolve its present-day density and temperature and compare that with measurements of the CMB made with respect to the Internal Frame of the expansionary epoch.

The quantum-epoch interpretation provides the MQ context for the comparisons in Table 1 ([2], Appx. AY). These comparisons test selected predicted quantities against observational benchmarks. Agreement in an individual row does not by itself establish that every fundamental parameter has been independently measured or that the same agreement holds for all rows.

Table 1. Tests extending the fundamental relation across the measurement domain, reprinted from [2].

^a A consolodated study of CMB temperature measurements prior to 2009 [10].

^b Astrophysical constants [11].

^c 2018 Planck Collaboration Results [12].

Table 1 compares the reported quantum and cosmological calculations with their observational benchmarks. The diameter D_U, measured as a length, and the System Frame age A_U, measured as a time, are linked by Eq. (16) ([2], Appx. AZ). The speed c converts the age to a length so that the dimensionless coefficient can relate quantities with matching dimensions.

Equation (17) restates Eq. (16) as a dimensionless diameter–age ratio. The coefficient controls that ratio, while the quantum fundamental relation carries momentum dimensions through p_si. The cosmological ratio therefore does not determine the fundamental mass.

3.4. Observational Domains as Bounds to Measure

We can now turn our attention to observational domains ([2], Appx. BB). Or in other words, how do we use the three measures to describe what we observe cosmologically?

While classical mechanics is well understood, a parameter free description of the early universe is difficult to resolve with ΛCDM. We will focus our understanding of cosmological observations with respect to geometry and elapsed time. First, there is what is presently observed, the visible Ω_vis. There is what will be observed, the unobserved Ω_uobs. Summed together these domains describe all that is observable Ω_obs. There is what will never be observed due to the expansion of space, the dark Ω_dk. And finally, there is the fundamental Ω_f, a mathematical inflection point between the observable and the dark.

Less apparent is a mapping of observational domains to those commonly identified today as dark energy and dark matter. These popularized terms represent collections of one or more unexplained phenomena. They are organized according to their conjectured association. But on what physical basis have we assembled these phenomena.

We present as example the following question. Is there physical evidence that suggests that the considerable velocity of stars in a galaxy is related to the likeness of star velocities across a disk. We lump the two observations together as properties of the dark matter phenomenon, but under what physical grounds?

For this reason, we can demonstrate that the observational domains resolved here (Table 2) closely match those resolved using ΛCDM [12, 13], but we would not claim that calculating the relative size of a domain leads to a physical description of all aspects of these phenomena [14]. The specific properties of dark matter and dark energy are better described elsewhere ([2], Appx. AI, BA).

Table 2. Comparison of CMB Planck Survey Results with MQ Calculations, reprinted from [2].

^a 2018 Planck collaboration results using TT,TE,EE+lowE [12].

MQ describes observational domains such that their composition is equivalent, more universe, more mass.

3.5. Confirming the Lower Bound with Length Contraction

Bounds to measure are directly correlated with physical support for discrete measure. Using a discrete approach to the description of gravity we discover that discrete measure carries with it length contraction, specific to the measure of length only ([2], Appx. D). In MQ, the Informativity differential denotes this count-dependent, non-Lorentz contraction of realized separation under the Frames mapping, rather than gravitational curvature itself. The details of this effect can be described with the Pythagorean theorem and as such are a property of those dimensions squared or higher. This, we recognize as applicable only to space.

Equations (18) and (19) retain the finite-count dependence on Q_Ln_L ([2], Appx. L, M). Here n_L is the radial separation count and Q_L is its fractional length-count remainder. Their product is dimensionless. Equation (18) uses p_sil_f to supply action dimensions, while Eq. (19) uses c³t_f/m_f to supply the dimensions of gravitational coupling. The upper-count limit must be taken explicitly; neither finite-count expression is an upper-limit value merely by its typography.

Had Planck realized this effect, solutions to Planck unit expressions with respect to component terms would match with greater precision. We can demonstrate this by solving for the gravitational constant (Table 3).

Table 3. Calculations of G using Planck’s unit expressions, reprinted from [2].

^a Planck’s unit expressions are resolved using those values provided in CODATA publications for 2010, 2014 and 2018 [15-17].

For the comparisons in Table 4, the realization of each input must be specified consistently. The finite electromagnetic demarcation, the fixed electromagnetic interaction scale also described through the blackbody construction ([2], Appx. AB, AE) differs from the upper-count limit. Equation (20) takes the latter limit, in which the product Q_Ln_L tends to one half, and consequently the reduced Planck constant tends to twice p_sil_f. The upper-limit symbol is upright; finite-demarcation measures retain their distinct mathematical styling.

The upper-count value of the gravitational constant reported here is 6.6740779428(56) × 10⁻¹¹ m³ kg⁻¹ s⁻². It is an MQ calculated value, distinct from the CODATA 2018 recommendation of 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻² [17]. Their central values differ by about 1.48 times the CODATA standard uncertainty. This comparison does not establish the much smaller uncertainty quoted for the MQ calculation as an experimental uncertainty, nor does it show that all experimental determinations yield the same value.

Next, we compare this with measures offered by the CODATA collaboration with respect to the 2010, 2014 and 2018 publications. An analysis of selected experiments representing G and ħ show that the focus has been on G with Planck’s expressions being used to resolve the most consistent result for ħ. But why is there such significant variation in G? This we map out in the remaining four lines (Table 4) to demonstrate calculated solutions for all values as a function of values for G and ħ either at the blackbody demarcation ([2], Appx. AB) or the upper count limit.

Table 4. Matching CODATA values for ℏ, G and the Planck units with MQ calculated values, reprinted from [2].

^a CODATA publications for 2010, 2014 and 2018 [15-17].

^b Using precise G and ℏ, we account for the Informativity differential at the blackbody demarcation n_Lr= 84.6005496647(07) (i.e., italicized), the upper count limit (i.e., lim_nLr→∞) (not italicized), or a mix of the two.

The two reconstructions below use different inverse fine structure inputs. The Planck-like input α_p is reconstructed from the ground-state atomic orbital expression, whereas the classical input α_c is the electromagnetic value used in the CODATA comparison ([2], Appx. N). In Eqs. (21) and (22), f denotes the corresponding reconstruction of gravitational coupling. The fundamental-measure inputs in the two evaluations must be interpreted within their respective reconstructions; holding both the fundamental time and mass fixed would produce the same coupling.

And with this, we find that all results in these publications can be calculated to the same precision offered, thus demonstrating the physical significance of discrete measure by way of the length contraction effect we call the Informativity differential. And where measure is discrete and countable, bounds to those counts are also physically significant.

3.6. Confirming the Upper Bound with Mass

We now consider a phenomenon best described as an upper measurement bound. More detailed calculations are provided in the supplemental materials ([2], Appx. AF-AI). These calculations combine the effects of expansion with the difference between frames along with the upper count bound n_M of m_f per increment of elapsed time t_f to describe the velocities of stars in a galaxy. By example, looking at the first 84,000 lightyears of Milky Way data offered by McGaugh [18], the MQ approach resolves a velocity expression having a 1.394 km s^-1 standard deviation from McGaugh’s model data.

The calculation begins with a description of the bound velocity as defined with respect to the Internal Frame of the universe, v_bc ([2], Appx. AK). The metric differential denotes the representation-level mismatch between discrete counts and their continuous System Frame realization; it is distinct from the non-Lorentz length contraction described by the Informativity differential. We then apply the metric differential ([2], Appx. T) (i.e., multiplied by θ_si) to resolve the expression with respect to the System Frame of the universe. And finally, we use the same approach to resolve the upper count of fundamental units of mass (i.e., the bound mass M_b) per increment of elapsed time t_f, set with respect to the System Frame.

With the bound mass resolved, we can then resolve the effective mass M_e, by considering the relative difference between the modelled and bound velocities. This is not ideal, but physically correct. For a more familiar implementation one would replace the modelled velocity with Newton’s expressions for orbital velocity as a function of the bound mass M_b closest to the resultant orbital path. This is challenging but demonstrates the idealistic form. For simplicity, we proceed with the modeled velocity v_m.

We can now use Newton’s expression to resolve the effective velocity v_e. We replace the mass term with the effective mass M_e and reduce.

Thus, we enter the modeled velocity to resolve the effective velocity. The solution exists only where v_m=v_e and only where respective of the mass density profile of a system. This later quality is physically important. Only values v_m reflective of the closest mass beneath the bound along a path of lowest energy will produce a value v_e that does not cause diverge of the velocity curves. This approach, as noted, produces a precise match with McGaugh’s model data.

4. DISCUSSION

Of importance, is that the forces of nature are constrained to what is measurable in each increment of elapsed time, not what is observable over elapsed time. This represents a significant departure from modern interpretations. To consider a description of nature where this is not true also describes a universe where the relation between the three measures is without bound (i.e., c=n_Ll_f/n_Tt_f). This would be to say that objects could exceed the speed of light n_L>n_T, which we know to be incorrect.

5. APPENDICES

APPENDIX A: TERMS

Complete set of term and keyword definitions in the paper entitled, Measurement Quantization [2].

• The angular realization Θ_si carries radians and retains the invariant numerical coefficient. Shwartz and Harris calculate signal and idler wave-vector directions of 3.26239 rad in one frequency-degenerate diamond configuration [19], Table I. Those propagation directions are distinct from a measured polarization angle, and the table does not assign them the ±2 μrad measurement uncertainty previously quoted here. The separate down-conversion experiment [20] and X-ray–optical wave-mixing experiment [21] provide related nonlinear-optics context. The discussion of conserved quantities at null infinity in [22] addresses supertranslation invariance; it does not establish the MQ realization mapping.

• c is the speed of light which may also be written as c=n_Ll_f/n_Tt_f =299,792,458 m/s [17] such that n_L=n_T=1 is physically significant.

• G is the gravitational constant, 6.6740779428(56) 10^-11 m³ kg^-1 s^-2 such that its value considers the effects of length contraction associated with discrete measure at the upper count limit. Italicized G identifies a measure not at the limit (e.g., G=6.6738448362(53) 10^-11 m³ kg^-1 s^-2 at the blackbody demarcation).

• l_f, m_f and t_f are the fundamental measures, more precise expressions for Planck’s units – length, mass, and time – that consider the effects of length contraction associated with discrete measure.

• l_p, m_p and t_p are Planck’s units for length, mass, and time.

• n_L, n_M and n_T are physically significant discrete counts of l_f, m_f and t_f respectively.

• n_L describes the count of l_f representative of a change in position of an observable measured with respect to the observer’s frame of reference.

• n_Lr describes the count of l_f representative of the position of an observable with respect to the frame of a center of mass.

• n_Lc describes the count of l_f representative of a change in position of light measured with respect to the observer’s frame of reference.

• The notation reference for n_θ identifies the System Frame count of θ_si adjusted by the metric differential. The nearest-whole-unit example is ⌊84.60055/2⌉=42.

• The mixed left-floor and right-ceiling bracket pair denotes rounding to the nearest whole-unit value in this manuscript. No additional one-half term is inserted into its argument.

• Q_L is the fractional portion of a count of l_f when engaging in a more precise calculation.

• The notation reference 2Q_Ln_L identifies the Informativity differential. It is the dimensionless normalization modifier associated with length contraction from discrete measure; here the length count denotes radial separation as in Eqs. (18) and (19).

• ℏ is the reduced Planck constant, 1.054571817 10^-34 m² kg s^-1. When accounting for the Informativity differential at the upper count bound, this term is not italicized (i.e., ħ=1.0545349844(45) 10^-34 m² kg s^-1).

• m_e is the mass of an electron.

• a₀ is the ground state orbital of an atom.

• α_f^-1, α_p^-1, and α_c^-1 are the fundamental, Planck-like, and classical forms of the inverse fine structure constant. α_f^-1=42θ_si is defined with respect to the System Frame; the remaining two against the Internal Frame such that α_c^-1 differs from α_p^-1, accounting for the length contraction associated with discrete measure.

• γ is a collection of terms that describe the geometry between the System and Internal frames.

STATEMENTS AND DECLARATIONS

The author has no relevant financial or non-financial interests to disclose.

The author has no competing interests to declare that are relevant to the content of this article.

DATA AVAILABILITY

The manuscript has data included as electronic supplementary material.

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