What Defines the Fundamental Measures?

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60607

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We develop a Measurement Quantization (MQ) description in which observable length, mass, and time emerge through the Frames mapping from discrete Internal Frame count structure to non-discrete System Frame realizations. Taking the MQ count structure as a theoretical premise, we combine its gravitational expression with Heisenberg’s uncertainty principle, escape velocity, and the speed of light to derive relations among invariant fundamental measures and the model’s radial rate of universal expansion. An invariant dimensionless coefficient θ_si parametrizes these relations. Its momentum realization corresponds to one-half of the fundamental momentum, while its angular realization is examined in the phase-matching geometry of polarization-entangled X-rays; their common coefficient does not imply dimensional identity or establish the proposed angular identification experimentally. The count 42 yields the fundamental inverse fine structure form 42θ_si, with System Frame realizations incorporating the metric differential and the length-contraction correction described by the Informativity differential. Because the coefficient is inferred from the measured inverse fine structure constant, this construction provides a consistency relation rather than an independent prediction. A cosmic microwave background comparison likewise establishes numerical compatibility, while independent validation requires documented input dependencies and compatible epoch definitions.

Keywords: Fundamental Measures, Planck Units, Constraints to Measure, Measure, Frames of Reference, Physical Significance

1. INTRODUCTION

We propose that measurement references are physically significant and that observable measure emerges from the relation between the discrete Internal Frame and the realized System Frame. The Internal Frame is the discrete, count-based configuration domain in which the fundamental measures are defined, whereas the System Frame is the physical measurement domain in which corresponding relational observables are realized through the Frames mapping. The fundamental measures therefore provide the dimensional references through which count structure is physically realized rather than independently constituting observable physical geometry. The broader proposition that physically meaningful descriptions may involve a fundamental scale or discrete structure is well represented in quantum-gravity research, although neither discreteness nor a minimum measurable length is established experimentally and different approaches implement such structure differently [21, 22].

To establish the MQ distinction, we begin with the relation between the two frames ([1], Appx. K). A discrete Internal Frame acquires physical dimensional significance through quantities realized in the System Frame. A single solution set to counts of the fundamental measures provides the discrete basis for this relation ([1], Appx. I, J). The difference between the frames can then be used within MQ to resolve expressions for physical constants and other measurable quantities [1, 2].

Several prerequisites are essential to this discussion. Let l_f, m_f, and t_f denote the fundamental measures of length, mass, and time, respectively, and let n_L, n_M, and n_T denote their corresponding discrete counts. The invariant scalar θ_si relates the fundamental measures and admits different representations according to the frame and physical realization under consideration. The whole count n_θ identifies the nearest-whole-count realization associated with one-half of the relevant length count. The count relation between n_θ and n_L contributes to the metric differential between the Internal Frame and System Frame ([1], Appx. T). Using the single paired floor/ceiling operator ⌊ · ⌉ to denote nearest-whole-unit realization, this relation is

n_{θ} = ⌊ n_{L} / 2 ⌉ = 42 .

(1)

Using this relation together with the discrete gravitational expression ([1], Appx. C), the speed of light, escape velocity, and Heisenberg's uncertainty principle [3], MQ resolves the simplest relation among the fundamental measures. We call this the fundamental relation ([1], Appx. V).

l_{f} m_{f} = 2 θ_{s i} t_{f} .

(2)

These relations are assessed against physical phenomena spanning quantum, macroscopic, and cosmological scales. The CMB is used later in the paper to test consequences of the resulting relations rather than to establish the relations themselves. This separation is important because the theoretical derivation and its empirical assessment constitute distinct claims. The present section therefore establishes only the framework needed to distinguish the properties of measure in the Internal Frame and System Frame ([1], Appx. K) and to separate the discrete counts n_L, n_M, and n_T from the dimensional fundamental measures l_f, m_f, and t_f.

An additional prerequisite is careful treatment of θ_si. Its physical interpretation depends on the representation in which it appears. In the count structure of the Internal Frame, θ_si is an invariant scalar. Its momentum realization is represented dimensionally through the fundamental measures and corresponds within MQ to one-half of the fundamental momentum ([1], Appx. Q). The MQ development also considers an angular realization in connection with polarization-entangled X-rays at their degenerate frequency ([1], Appx. A). Shwartz and Harris established that polarization-entangled X-ray photon pairs can be generated by spontaneous parametric down-conversion and that Bell-state production depends on the incidence angle and pump polarization [14]. That experimental result supports the physical setting used by the angular construction, but it does not independently establish the MQ identification of θ_si.

These representations must not be interpreted as asserting dimensional identity between angle and momentum. They are distinct MQ representations associated with the same underlying invariant scalar. Likewise, the treatment of conserved quantities at null infinity by Chen et al. ([4], Sec. 4.4.2) provides mathematical context for conserved asymptotic quantities in general relativity but does not independently establish an equivalence between angular measure and momentum. The MQ identification developed later must therefore stand on its own derivation rather than on that external literature.

The objective of this paper is to determine how the fundamental measures are related to physically measurable phenomena. First, we establish the relation between the fundamental combination l_fm_f/(2t_f) and the invariant scalar θ_si, thereby connecting the fundamental length, mass, and time measures through the fundamental relation. Second, we examine the momentum realization of this scalar and its relation to the angular construction used in the quantum analysis while maintaining the dimensional distinction between those representations. Third, we examine how the resulting count structure enters the inverse fine structure constant and thereby characterizes the metric differential between the Internal Frame and System Frame. Together, these results test an MQ description in which dimensional length, mass, and time are realized from discrete count structure rather than assumed as independently continuous primitives.

2. METHODS. METHODS

This paper uses established physical expressions together with the discrete count relations developed in Measurement Quantization. The derivation draws on gravitational curvature, the speed of light, escape velocity, and Heisenberg's uncertainty principle to investigate the relation between discrete count structure and dimensional physical measure. We identify this approach as Measurement Quantization (MQ). Several examples in the publication record apply MQ to problems in modern theory [1, 2, 5–9].

The present analysis depends on the discreteness and countability developed in the introductory paper, Measurement Quantization [1]. That work presents the derivations and measurement comparisons underlying the count structure used here. This premise should be distinguished from established experimental physics. The possibility of a fundamental resolution scale is extensively studied in quantum-gravity phenomenology, but the literature does not establish that physical length, mass, and time are discrete in the particular MQ sense adopted here [21, 22]. The MQ count structure is therefore treated in this paper as a theoretical premise whose physical consequences are subjected to the comparisons developed in later sections.

The referenced appendices in [1] contain the supporting definitions and derivations required by the present analysis. A preprint and supporting material are available in Online Resources.

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

Two supporting spreadsheets accompany that material. Suppl. 1 contains calculations for the expressions developed in the introductory paper, including those appearing in its referenced appendices. Suppl. 2 contains calculations associated with this paper and the broader set of MQ analyses. These resources provide numerical implementations of the expressions used in the present investigation but do not substitute for the analytic derivations presented in the paper.

The analysis proceeds by keeping discrete count structure distinct from dimensional realization. Counts such as n_L, n_M, and n_T belong to the Internal Frame. Dimensional quantities constructed using l_f, m_f, and t_f are realized through the Frames mapping in the System Frame. This distinction also prevents the fundamental measures from being identified without qualification with experimentally established minimum measurable intervals. The broader literature itself distinguishes a fundamental or minimal scale from the stronger assertion of a minimum measurable length [21].

Where a quantity is evaluated at a finite count, the corresponding MQ notation retains the finite-count form. Where the upper count limit is explicitly taken, the notation identifies that limiting realization. This distinction becomes particularly important for the gravitational constant G and the reduced Planck constant ħ in the analysis that follows. Neither symbol is otherwise introduced or used mathematically in the present sections. Their finite-count and upper-count-limit forms are distinguished when they first enter the subsequent derivation.

3. RESULTS

The results distinguish finite-count measurements from upper-count-limit values and develop the relation between fundamental measure and cosmological expansion, including the quantum epoch, the cosmic microwave background, and the unity expression. Dimensional interpretation and experimental assessment are addressed in subsequent subsections.

1. Nomenclature

The MQ treatment of length contraction distinguishes finite-count measurements from their upper-count-limit values ([1], Appx. D), using the discrete count structure and fundamental measures established in [1] (Appx. I, J). For a realized measurement, the length count n_L specifies the number of fundamental length measures l_f associated with the relevant separation. The Informativity differential describes non-Lorentz length contraction arising from normalization of conserved discrete count structure under the Frames mapping. It differs from the metric differential, which characterizes the discrete–continuous mismatch between the Internal Frame and System Frame.

Italic G denotes finite-count gravitational coupling, whereas upright G denotes its upper-count-limit value. The same distinction applies to the reduced Planck constant ħ when its upper count limit is taken. It is relevant to the electromagnetic demarcation, where a fixed finite interaction count remains significant, and to macroscopic measurements, where the upper-count-limit form may be numerically adequate.

Let Q_L denote the mapping-induced length-count residual entering geometric realization. At the electromagnetic demarcation, the fixed interaction count is n_L=84.6005496647(07). The finite-count gravitational expression is

G = {2 Q_{L} n_{L} c^{3} \frac{t_{f}}{m_{f}} |}_{n_{L} = 84.6005496647 (07)} = 6.6738448362 (53) \times 1 0^{- 11} m^{3} k g^{- 1} s^{- 2} .

(3)

Here c denotes the speed of light. At the upper count limit, the corresponding expression is

G = \lim_{n_{L} \to \infty} 2 Q_{L} n_{L} c^{3} \frac{t_{f}}{m_{f}} = c^{3} \frac{t_{f}}{m_{f}} = 6.6740779428 (56) \times 1 0^{- 11} m^{3} k g^{- 1} s^{- 2} .

(4)

These expressions provide two MQ predictions for gravitational coupling. They can be compared with CODATA publications, including the 2010 and 2014 adjustments [10, 11]. Numerical proximity to individual measurements does not establish correspondence to electromagnetic or gravitational realization without examining measurement methods and systematic uncertainties.

2. Relation of Measure with Respect to Expansion

Special relativity describes the relation between relative motion and measured length [12]. MQ examines this relation through a count-based Pythagorean construction, distinguishing discrete counts from continuous geometry.

Let n_Li denote the initial reference length count, n_Lo the observed length count, and n_Lc the length count associated with the realized change in position of light. The count n_L describes the change in position of the observable relative to the observer’s reference. These quantities enter the length-count relation ([1], Appx. AU).

n_{L o} = n_{L l} {(1 - \frac{n_{L}^{2}}{n_{L c}^{2}})}^{1 / 2},

(5)

\frac{n_{L o}}{n_{L l}} = {(1 - \frac{n_{L}^{2}}{n_{L c}^{2}})}^{1 / 2} .

(6)

The ratios

n_{L}^{2} / n_{L c}^{2}

and n_Lo/n_Li are identified with the squared velocity ratio v²/c² and length ratio l/l_i, respectively, where v denotes relative velocity, l observed length, and l_i initial reference length. The count relation reproduces the algebraic form of relativistic length contraction, but its physical interpretation must follow from the Frames mapping.

During the quantum epoch, the absence of an accessible external spatial reference constrains initial geometric realization. The quantum-epoch radius is bounded by

\sqrt{3}

, corresponding to the three-component count construction ([1], Appx. BF). Let n_a and n_b denote its orthogonal component counts. The geometric relations establish the first admissible radial reference.

\sqrt{n_{a}^{2} + n_{b n}^{2}} = \sqrt{n_{a}^{2} + {(\sqrt{n_{a}^{2} + n_{b}^{2}})}^{2}} =,

(7)

\sqrt{n_{a}^{2} + n_{a}^{2} + n_{b}^{2}} = \sqrt{1^{2} + 1^{2} + 1^{2}} = \sqrt{3} .

(8)

The radial scale R_qe and non-dilated elapsed temporal measure A_qe characterize the quantum epoch. The MQ radial expansion relation connects them to θ_si, the invariant scalar introduced in Section 1 ([1], Appx. BG).

R_{q e} = \frac{2 l_{f}}{θ_{s i}^{3}} \int_{}^{} (\frac{1}{A_{q e}}) d t = \frac{2 l_{f}}{θ_{s i}^{3}} \ln (A_{q e}),

(9)

A_{q e} = e^{\sqrt{3} θ_{s i}^{3} / 2} = 1.14652655409 (32) 1 0^{13} s =

(10)

363,312.341272(21) y.

MQ characterizes the quantum epoch by constrained radial expansion, termed the quantum crawl. Let v_i denote its concluding radial velocity and H the fundamental expansion rate quantity. The elapsed temporal measure and radial expansion expression determine this model-dependent kinematic quantity, not a directly measured early-universe velocity.

v_{i} = \frac{1}{H_{f}^{2}} \frac{2 l_{f}}{θ_{s i}^{3} A_{q e}^{3}} = A_{q e}^{2} \frac{2 l_{f}}{θ_{s i}^{3} A_{q e}^{3}} = \frac{2 l_{f}}{θ_{s i}^{3} A_{q e}} = 9.309311 0^{- 37} m s^{- 1} .

(11)

The MQ mass relation ([1], Appx. BE, BH) determines the mass associated with the cosmic microwave background (CMB). Let n_Tu denote the fundamental-time count corresponding to the age of the universe and M_CMB the mass assigned to CMB realization.

M_{C M B} = n_{T u} m_{f} \frac{θ_{s i}^{3}}{2} = 1.50161 0^{50} kg .

(12)

This mass belongs to the specified cosmological count construction and must be distinguished from directly observed radiation energy density. The model considers additional contributions during the transition to the expansionary epoch [1, 9].

For present-epoch radiation quantities, let ρ_CMB denote CMB energy density and V_U the cosmological volume used in MQ. The density follows from the mass-energy relation ([1], Appx. BH).

ρ = \frac{M_{C M B} c^{2}}{V_{U}} = 4.17501 0^{- 14} J m^{- 3},

(13)

a = \frac{4 σ}{c} = \frac{8 π^{5} k_{B}^{4}}{15 c^{3} h^{3}} = 7.5657332501 0^{- 16} J m^{- 3} K^{- 4},

(14)

T_{C M B} = {(\frac{ρ}{a})}^{1 / 4} = 2.7255 K .

(15)

The resulting temperature can be compared with CMB observations. Agreement tests the MQ construction numerically but does not independently establish its assumptions about initial cosmological mass or volume.

MQ does not introduce a separate inflationary epoch in this construction [1, 13]. Instead, restricted external referencing during the quantum epoch precedes the expansionary epoch. This differs from standard inflationary cosmology and requires independent empirical discrimination ([1], Appx. BF).

The derivation connects the epochs through geometric volumes. Let D_qe and D_ee denote the diameters associated with the quantum and expansionary epochs, respectively. Their volume relation incorporates the radial expansion factor θ_si ([1], Appx. V).

\frac{(4 / 3) π (D_{e e} θ_{s i} c)^{3}}{(4 / 3) π (D_{q e} θ_{s i} c)^{3}} = 2 θ_{s i},

(16)

D_{e e} = D_{q e} {(2 θ_{s i})}^{1 / 3} .

(17)

The diameter relation is expressed through fundamental length counts. The count-based speed-of-light relation connects length and temporal counts, while the initial count condition relates the descriptions of the epoch boundary.

n_{L l} = n_{L o} {(2 θ_{s i})}^{1 / 3},

(18)

\frac{n_{L o}}{n_{L l}} = \frac{1}{{(2 θ_{s i})}^{1 / 3}} .

(19)

Substituting the resulting length-count ratio into the earlier contraction relation gives a dimensionless expression for expansion geometry.

\frac{1}{{(2 θ_{s i})}^{1 / 3}} = {(1 - \frac{n_{L}^{2}}{n_{L c}^{2}})}^{1 / 2},

(20)

\frac{1}{{(2 θ_{s i})}^{2 / 3}} = \frac{n_{L}^{2}}{n_{L c}^{2}} = 1,

(21)

The fundamental relation established in Eq. (2) then expresses the geometric term through fundamental measures.

{({(\frac{t_{f}}{l_{f} m_{f}})}^{1 / 3})}^{2} + {(\frac{n_{L}}{n_{L c}})}^{2} = 1 .

(22)

This MQ unity expression relates the fundamental measures to the expansion and light-propagation count ratio through a normalized Pythagorean identity. Its first term depends on the fundamental-measure combination and its second on the count ratio.

The construction applies to the quantum-to-expansionary transition, where the initial temporal count is n_T=1. It does not assert that expansion velocity remains constant; subsequent evolution follows the model’s expansion relations.

The cubic dependence of the fundamental length-to-time ratio can also be examined through the MQ gravitational expression ([1], Appx. C, F).

G = {(\frac{l_{f}}{t_{f}})}^{3} (\frac{t_{f}}{m_{f}}) .

(23)

This separates the cubic length-to-time contribution from the time-to-mass contribution. Dimensional consistency supports the form of the MQ gravitational construction but does not independently derive three spatial dimensions from the cubic exponent.

Finally, MQ distinguishes its radial rate of universal expansion from the conventional Hubble parameter. The invariant scalar θ_si enters the fundamental momentum and expansion relations ([1], Appx. AD), whereas H describes the present expansion rate through recession velocity and distance. The MQ radial expansion quantity must be interpreted within its own definitions rather than identified directly with H.

3. Relation of Dimensions with Respect to the Internal Frame

We now consider the angular and momentum realizations of the invariant dimensionless coefficient θ_si, arising in the System Frame through the Frames mapping of discrete Internal Frame structure. We begin with the phase-matching geometry discussed by Shwartz and Harris ([1], Appx. A) [14, 15]. In Eq. (24), n_sx, n_sy, n_ix, n_iy, n_px, and n_py are count coefficients for the signal, idler, and pump terms. The subscripts s, i, and p identify those contributions, while x and y label planar components in the System Frame. The angle θ describes the propagation geometry, and G⃗ denotes the reciprocal-lattice contribution, distinguished from the scalar gravitational constant G. Physical phase matching requires the reciprocal-lattice contribution and count terms to share a normalization, which remains unspecified.

(n_{s x} \cos (θ), n_{s y} \sin (θ)) + (n_{i x} \cos (θ), n_{i y} \sin (θ)) = (n_{p x} \cos (2 π - θ), n_{p y} \sin (2 π - θ)) + \vec{G} .

(24)

At degenerate frequency, the signal and idler frequencies are equal, but their polarization correlations depend on the phase-matching geometry and selection rules. Equal frequency alone does not require identical polarization angles relative to their wave vectors. The cited work establishes conventional wave-vector phase matching without independently establishing the MQ count normalization.

The expression in ([1], Appx. S) supplies a momentum scale using the fundamental length l_f and the upper-count reduced Planck constant ħ. In current MQKB notation, p_si denotes the momentum realization of θ_si, with units kg m s⁻¹.

p_{s i} = \frac{ħ}{2 l_{f}} .

(25)

The Informativity differential ([1], Appx. D) is the non-Lorentz length contraction associated with realizing discrete Internal Frame count structure in the System Frame. It increases at smaller count separations and diminishes toward the upper count limit. Electromagnetic interactions in this construction are evaluated at the electromagnetic demarcation, about 84.60055 l_f, where the correction remains significant ([1], Appx. AB, AE, AN).

With consistent upper-count realizations of G and ħ, Eq. (26) expresses the same scale as the momentum p obtained from the fundamental-length identity, where c denotes the speed of light.

p_{s i} = \frac{l_{f}}{2} (\frac{c^{3}}{G}) = \frac{l_{f}}{2} (\frac{ħ}{l_{f}^{2}}) = \frac{ħ}{2 l_{f}},

(26)

The angular realization is Θ_si=θ_si rad. The conversion factor C_pθ=p_si/Θ_si, with units kg m s⁻¹ rad⁻¹, relates it to the momentum realization.

p_{s i} = C_{p θ} Θ_{s i} .

(27)

The radian has dimension one in SI, but an angle is not a momentum. Equation (27) converts defined realizations; identifying the phase-matching angle θ with Θ_si requires a separate physical derivation. The conventional relation between Planck's constant and the reduced Planck constant does not supply it.

The momentum realization follows from the dimensionally explicit fundamental relation, l_fm_f=2p_sit_f, using the fundamental mass m_f and fundamental time t_f([1], Appx. V).

p_{s i} = \frac{l_{f} m_{f}}{2 t_{f}} .

(28)

4. Relation of the Internal and System Frames

In the fine-structure construction, n_θ denotes the System Frame count associated with θ_si, obtained from the electromagnetic demarcation value n_L, expressed in units of l_f. This noninteger interaction-scale value follows MQKB's demarcation definition and must be distinguished from primitive integer Internal Frame configurations. The floor expression rounds to the nearest integer, with half-integers rounded upward.

n_{θ} = ⌊ n_{L} / 2 ⌉ = ⌊ 84.6005496647 (07) / 2 ⌉ = 42 .

(29)

This operation assigns the count without replacing the full Frames mapping. The fundamental, Planck-like, and classical inverse fine structure forms are

α_{f}^{- 1}, α_{p}^{- 1}, and α_{c}^{- 1}

, respectively ([1], Appx. AA). Following MQKB Terms, the fundamental form is associated with the Internal Frame construction and the others with its System Frame realization.

α_{f}^{- 1} = 42 θ_{s i} = 137.020392764 (84) .

(30)

α_{p}^{- 1} = 84 θ_{s i} - ⌊ 42 θ_{s i} ⌉ = 137.041 .

(31)

In the right-triangle construction, Q_L is the residual count contribution, and 2Q_Ln_L supplies the Informativity differential used in the classical realization.

α_{c}^{- 1} = 2 Q_{L} n_{L} (84 θ_{s i} - ⌊ 42 θ_{s i} ⌉) = 137.036 .

(32)

With the MQKB central coefficient θ_si=3.262390305, these forms round to 137.020, 137.041, and 137.036 at three decimal places. The former printed value 137.20392764(84) for the original Eq. (30) does not follow from that coefficient and has been removed. Exact symbolic expressions are retained without unsupported numerical uncertainties.

MQKB records that θ_si is inferred from the measured inverse fine structure constant. Evaluating Eq. (32) with that coefficient therefore checks the count-realization construction rather than independently predicting the constant.

5. Experimental Support

Relations between constants do not alone establish their fundamental status. The CMB comparison below examines MQ cosmological consistency. It constitutes an independent test only if its prediction inputs were fixed without the CMB observable being tested.

We use D. J. Fixsen's CMB temperature determination [16], compiled from measurements available before 2009. The 2013 and 2018 Planck Collaboration analyses provide cosmological context; their separate results are not entered in Table 1. Planck 2018 adopts the measured present-day CMB temperature rather than independently measuring it.

In current MQKB usage, the quantum epoch is the count regime preceding a valid realized System Frame. Ordinary elapsed System Frame time applies in the subsequent expansionary epoch, so the historical age comparison requires a conversion of the quantum-epoch scale to time units.

Table 1 retains the historical labels Q_dk and A_CMB. The MQ entry for Q_dk is 214.767; MQKB variously describes it as a dark count or dark monopole without a verified physical definition or observational provenance. The label A_CMB denotes the compared epoch-age entries, ρ_CMB denotes present-day CMB radiation energy density, and T_CMB denotes CMB temperature. WMAP's age refers to photon decoupling, while the MQ entry is assigned to quantum-epoch termination. Their identification requires an additional physical argument.

Table 1. Historical comparison of reported MQ CMB values with observational quantities, adapted from [1]. The WMAP age uncertainty is restored; the interpretation and attribution of Q_dk remain unresolved.

	Q_dk	A_CMB	ρ_CMB	T_CMB
MQ	214.767	363,312.34127(22) y	4.1750 10^-14Jm^-3	2.7255 K
Measurements	216.4 ± 1.3^b	379,000 y^b	4.17 10^-14Jm^-3^c	2.72548 ± 0.00057 K^a

^a A consolidated study of CMB temperature measurements prior to 2009 [16].

^b Microwave (WMAP) All-Sky Survey [17].

^c Corresponds to about 411 photons/cm³ [18].

The quoted MQ temperature differs from Fixsen's central value by 0.00002 K, about 0.035 of the observational standard uncertainty. This establishes compatibility at the displayed precision, without a prediction uncertainty or independent-input audit. MQ's energy density and temperature are linked by its blackbody conversion and are not two independent tests.

The historical MQ age entry is about 4.14% below WMAP's decoupling-age central value. Interpreting this difference requires a common event definition, compatible time conventions, and uncertainties for both constructions; it cannot be attributed to observational modelling alone. The Q_dk comparison likewise requires a defined observable and source.

The supplied calculation is presented as having no free or fitted variables, but the absence of adjustable parameters does not establish input independence. MQKB distinguishes a forward CMB test using independently measured cosmic age from an internal consistency test using an MQ-derived age or measured CMB temperature. Classifying the present comparison as the former requires documented dependencies.

The early-universe construction [9] reports 9.30931×10⁻³⁷ m s⁻¹ as the initial quantum configuration's expansion speed ([1], Appx. BG). A physical speed requires realized length and time assignments. Its original expression remains dimensionally unresolved under the current θ_si definition, so the value remains a source-reported model quantity.

The construction assigns M_acr, reported as 7.00888×10³⁶ kg s⁻¹([1], Appx. BE). MQKB distinguishes pre-realization count allocation from its temporal interpretation as mass accretion after realization, meaning the global realized mass-equivalent accumulation rate. The dimensionally explicit form

M_{a c r} = (θ_{s i}^{3} / 2) (m_{f} / t_{f})

requires realized elapsed time when applied as a physical rate. The shorthand

θ_{s i}^{4} / l_{f}

is not used as a physical mass rate with dimensionless θ_si.

Using the fundamental relation, the historical chronology expresses the quantum-epoch scale as 363,312 y([1], Appx. BG, BH), terminating at the referencing threshold represented by

\sqrt{3} l_{f}

([1], Appx. BF). Conflicting MQKB age and dimensionless-scale records leave its elapsed-duration interpretation conditional on a conversion to realized time. At termination, the construction admits the relational references associated with the expansionary epoch. Quantum-epoch mass allocation denotes count-derived mass–energy allocation before classical thermalization. Assigning it to the CMB and mapping it to present-day radiation energy density and temperature are model assignments; Table 1 does not establish the formation mechanism.

4. DISCUSSION

Encoded count relations and observable quantities occupy the distinct Internal Frame and System Frame descriptions. Fundamental measures, expansion, and invariance claims therefore require specified coefficients, normalizations, and realizations.

The right-triangle construction describes realized System Frame separations without assigning primitive continuous geometry to the Internal Frame. The Pythagorean theorem alone establishes neither the Frames mapping nor gravitational dynamics or a unique physical realization. Likewise, Eq. (27) relates defined representations without deriving a measured phase-matching angle.

The count 42 characterizes the fine-structure construction ([1], Appx. AA). The metric differential is the mismatch between discrete count structure and continuous realization under the Frames mapping ([1], Appx. T), distinct from the realized length contraction described by the Informativity differential. The count also occurs in the supplied atomic ground-state construction ([1], Appx. BL); recurrence within related derivations is not independent empirical evidence. Inferring how another count assignment or expansion rate would affect fundamental measures or physical properties requires a specified mapping, normalization, and atomic-scale derivation.

5. APPENDICES

APPENDIX A: TERMS

Complete set of term and keyword definitions in the paper entitled, Measurement Quantization [1].

θ_si can be measured as the polarization angle of quantum entangled X-rays at the degenerate frequency of a maximal Bell state. As an angle θ_si=3.26239 rad ± 2 μrad [14, 15, 19]; as a momentum θ_si=3.26239030392(48)kg m s^-1; and with respect to the Target Frame, θ_si has no units. Notably, the relation of angle and momentum are clarified as conserved quantities supertranslation invariant, by Po-Ning Chen, et. al. ([4], Sec. 4.4.2) consistent with our own solution of equivalence at the lower count bound.

c is the speed of light which may also be written as c=n_Ll_f/n_Tt_f =299,792,458 m/s [20] 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_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.

n_θ is the Target Frame count of θ_si resolved as a function of the Measurement Frame demarcation adjusted for the metric differential (i.e., RND(84.60055/2)=42).

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

Q_Ln_Lr, also known as the Informativity differential, describes the contraction of length associated with discrete measure.

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 Target Frame; the remaining two against the Measurement Frame such that α_c^-1 differs from α_p^-1, accounting for the length contraction associated with discrete measure.

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.

REFERENCES

[1] J.A. Geiger: Measurement Quantization, Int. J. Geom. Methods Mod. Phys., CC BY 4.0, doi:10.1142/S021988782350069X.

[2] J.A. Geiger: Measurement Quantization Describes the Physical Constants, International Journal of Theoretical and Mathematical Physics, 11(1), pp. 29-59 (Jan 2021), doi:10.4236/jhepgc.2020.62015.

[3] Heisenberg, W., Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik, 43 (3–4): 172–198 (1927), Bibcode:1927ZPhy.43.172H, doi:10.1007/BF01397280, S2CID 122763326.

[4] Po-Ning Chen, Mu-Tao Wang, Ye-Kai Wang, Shing-Tung Yau, Conserved quantities in general relativity -- the view from null infinity (Apr. 8, 2022), doi:10.48550/arXiv.2204.04010.

[5] J.A. Geiger: Measurement Quantization Unites Classical and Quantum Physics, Journal of High Energy Physics Gravitation and Cosmology. Vol. 4, No. 2, 262–311 (Apr 2018), doi:10.4236/jhepgc.2018.42019.

[6] J.A. Geiger: Quantum Model of Gravity Unifies Relativistic Effects, Describes Inflation/Expansion Transition, Matches CMB Data, Journal of High Energy Physics Gravitation and Cosmology. Vol. 4, No. 4 (Oct 2018), doi:10.4236/jhepgc.2018.44038.

[7] J.A. Geiger: Measurement Quantization Describes Galactic Rotational Velocities, Obviates Dark Matter Conjecture, Journal of High Energy Physics Gravitation and Cosmology. Vol. 5, No. 2 (Apr 2019), doi:10.4236/jhepgc.2019.52028.

[8] J.A. Geiger: Physical Significance of Measure, Journal of High Energy Physics Gravitation and Cosmology. Vol. 6, No. 1 (Jan 2020), doi:10.4236/jhepgc.2020.61008.

[9] J.A. Geiger: Measurement Quantization Describes History of Universe – Quantum Inflation, Transition to Expansion, CMB Power Spectrum, Journal of High Energy Physics Gravitation and Cosmology. Vol. 6, No. 2 (Apr 2020), doi:10.4236/jhepgc.2020.62015.

[10] P. Mohr, B. Taylor, and D. Newell, CODATA Recommended Values of the Fundamental Physical Constants: 2010, p. 73 (2012), arXiv:1203.5425v1.

[11] P. Mohr, B. Taylor, D. Newell, CODATA Recommended Values of the Fundamental Physical Constants: 2014, p. 3, (2015), arXiv: 1507.07956v1.

[12] Albert Einstein, http://einsteinpapers.press.princeton.edu/vol6-trans/42, Vol. 6: The Berlin Years: Writings, 1914-1917 (English translation supplement), p. 30 (1914)

[13] Alan H. Guth, The Inflationary Universe: The Quest for a New Theory of Cosmic Origins (1997), Basic Books. pp. 233–234. ISBN 978-0201328400, doi:10.1119/1.18814.

[14] S. Shwartz and S. E. Harris, Polarization Entangled Photons at X-Ray Energies, Phys. Rev. Lett. 106, 080501 (2011), arXiv: 1012.3499, doi:10.1103/PhysRevLett.106.080501.

[15] T. E. Glover et al., X-ray and optical wave mixing, Nature, 488, 7413, 603–608 (2012), doi:10.1038/nature11340.

[16] D.J. Fixsen, The Temperature of the Cosmic Microwave Background, (2009), arXiv: 0911.1955, doi:10.1088/0004-637X/707/2/916.

[17] Abbott, B. Microwave (WMAP) All-Sky Survey, Hayden Planetarium (2007), doi:10.1086/513698.

[18] 29. Cosmic Microwave Background: Particle Data Group P.A. Zyla (LBL, Berkeley) et al.

[19] S. Shwartz, R. N. Coffee, J. M. Feldkamp, et. al., X-ray Parametric Down-Conversion in the Langevin Regime, Physical Review Letters, 109, 013602 (July 6, 2012), doi:10.1103/PhysRevLett.109.013602.

[20] NIST: CODATA Recommended Values of the Fundamental Physical Constants: 2018, (May 2019), https://physics.nist.gov/cuu/pdf/wall_2018.pdf, doi:10.1103/RevModPhys.93.025010.

[21] S. Hossenfelder, “Minimal Length Scale Scenarios for Quantum Gravity,” Living Reviews in Relativity 16, 2 (2013), DOI 10.12942/lrr-2013-2.

[22] G. Amelino-Camelia, “Quantum-Spacetime Phenomenology,” Living Reviews in Relativity 16, 5 (2013).

[23] Planck Collaboration, “Planck 2013 results. XVI. Cosmological parameters,” Astronomy & Astrophysics 571, A16 (2014).

[24] Planck Collaboration, “Planck 2018 results. VI. Cosmological parameters,” Astronomy & Astrophysics 641, A6 (2020).

[25] P. J. Mohr, D. B. Newell, B. N. Taylor, and E. Tiesinga, “CODATA recommended values of the fundamental physical constants: 2022,” Reviews of Modern Physics 97, 025002 (2025).

[26] BIPM, The International System of Units, ninth edition, for the SI treatment of plane angle and quantity dimensions.

AIP Advances
8: Resolving the Hubble tension 68.261 and 73.510 km s Mpc

International Journal of Geometric Methods in Modern Physics
7: Measurement Quantization

Journal of High Energy Physics, Gravitation and Cosmology
6: Physically Significant Units of Measure
5: Measurement Quantization Unites Classical and Quantum Physics
4: Quantum Model of Gravity Unifies Relativistic Effects, Describes Inflation/Expansion Transition, Matches CMB Data
3: Measurement Quantization Accounts for Galactic Rotational Velocities and Obviates Dark Matter
2: Measurement Quantization Describes History of Universe -
Quantum Inflation, Transition to Expansion, CMB Power Spectrum

International Journal of Theoretical and Mathematical Physics
1: Measurement Quantization Describes the Physical Constants