Angular and Momentum Realizations at the Fundamental Measurement Bound

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We examine the dimensionless MQ coefficient quoted as 3.262390305(36), its angular realization in radians, and its momentum realization in kg m s⁻¹. In Measurement Quantization (MQ), the Internal Frame is the discrete count domain of fundamental measures. The System Frame is the continuous relational representation generated from discrete count configurations through the Frames mapping. Conditional on these definitions and count assumptions, the equations distinguish the coefficient from its realized quantities and give consistent momentum expressions in terms of fundamental length, mass and time, gravitational coupling, and the reduced Planck quantity. An arc-length construction describes the angular realization separately. We compare this realization with selected propagation angles calculated for polarization-entangled X-rays. The comparison establishes neither a measured angle–momentum law nor an experimental determination of the MQ coefficient. Its quoted uncertainty is an input to the presentation, rather than a demonstrated uncertainty budget for a physical observable.

Keywords: Angular Measure, Momentum, Angular Momentum, Planck Momentum

1. INTRODUCTION

The physical significance of Planck’s units [1] motivates the MQ investigations [2-8], which use count-resolved gravitational relations, the speed of light, escape velocity and quantum uncertainty [9]. We denote the fundamental measures of length, mass and time by l_f, m_f and t_f, with units m, kg and s. The dimensionless counts n_L, n_M and n_T refer to these measures. Fundamental time is fundamental length divided by c, the speed of light, whose value is 299,792,458 m s⁻¹ [10]. We write the dimensionless coefficient introduced in the abstract as θ_si, its angular realization as Θ_si in rad, and its momentum realization as p_si in kg m s⁻¹. The realization maps ℛ_L and ℛ_p assign the angular and momentum representations, respectively. Equation (1) fixes this notation before their physical roles are examined.

θ_{s i} = 3.262390305 (36), Θ_{s i} = ℛ_{L} (θ_{s i}), p_{s i} = ℛ_{p} (θ_{s i}) .

(1)

Polarization-entangled X-rays provide a context for comparing the angular realization with the geometry of a specified Bell state, a maximally entangled two-photon polarization state [11, 12]. The comparison uses propagation angles calculated by Shwartz and Harris [13]. Table 1 separates the MQ coefficient and its two realized representations, retaining their distinct units. The expansion-rate construction in [2], Appx. AX, supplies additional model context; it is not an independent observational determination of this coefficient.

Table 1. MQ coefficient and realized quantities at the fundamental measurement bound. Updated from the representation comparison in [2].

Dimension	Expression	Results	Units
Angle	$Θ_{s i} = ℛ_{L} (θ_{s i}) = θ_{s i} rad$	3.262390305(36) a	radians
Momentum	$p_{s i} = l_{f} m_{f} / 2 t_{f}$	3.262390305(36) b	kg m/s
Non-dimensional	$θ_{s i} = 3.262390305 (36)$	3.262390305(36) c	-

a Angular realization assigned by the map in Eq. (1). Its comparison with selected calculated propagation angles in [13] is discussed in Sec. 3.2. All Table 1 uncertainties are quoted MQ inputs, not uncertainty estimates for measurements made in this paper.

b Momentum realization calculated from fundamental measures. The earlier calibration based on 2018 inputs [10] is historical.

c Dimensionless coefficient in Eq. (1); its MQ expansion-rate interpretation is discussed in [2], Appx. AX.

These quantities retain distinct symbols throughout the paper. Sharing a numerical coefficient under the realization maps does not remove their physical dimensions. The purpose of the calculations is to establish the internal mapping consistently and identify what the angular comparison can test.

2. METHODS

Measurement Quantization (MQ) expresses classical relations through counts of fundamental measures and proposes physical interpretations of the resulting discrete relations. The present calculation adopts that framework; it does not independently establish discreteness of spacetime or a modification of gravitational or quantum dynamics. Published MQ applications [2-8] provide the model context.

The count assumptions underlying this presentation, their derivations and their proposed physical support are presented in Measurement Quantization [2]. Its published symbol and term context is available in [2], Appx. BP and BQ. A preprint can also be found in Online Resources.

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

The MQ publication [2] describes two supplementary spreadsheets for its calculations and applications. Those descriptions establish the intended computational provenance, rather than independent physical evidence. The current coefficient in Eq. (1) is an MQ parameter determined from the 2022 recommended inverse fine-structure input of the Committee on Data of the International Science Council (CODATA) [14] under the specified count and nearest-whole-unit assumptions, not an independent prediction of that input. Its central value is used here as a quoted calibration. The quoted parenthetical uncertainty is retained, but its complete propagation is not established by this presentation. Precision-dependent physical conclusions require an explicit uncertainty budget and covariance treatment.

3. RESULTS

The finite-count and upper-count realizations must be kept distinct. The Informativity differential describes non-Lorentz contraction in realized separations under the Frames mapping ([2], Appx. D, L, M). The dimensionless right-triangle residual is

Q_{L} = \sqrt{1 + n_{L}^{2}} - n_{L},

for positive n_L. The dimensionless modifier 2Q_Ln_L tends to unity as the count increases. In Eq. (2), italic ℏ denotes the finite-count reduced Planck quantity, with units J s. In Eq. (3), italic G denotes the finite-count gravitational coupling, with units m³ kg⁻¹ s⁻². Upright ħ and G denote their upper-count limits. Equations (2) and (3) use the same residual normalization and the momentum realization defined above. These finite and upper-count quantities are MQ model realizations. They must be distinguished from the reduced Planck constant of the International System of Units (SI), whose value is fixed by the exact SI Planck constant [10]. The count dependence here is not evidence that the SI constant changes with separation.

ℏ = \frac{p_{s i} l_{f}}{Q_{L} n_{L}},

(2)

G = 2 Q_{L} n_{L} c^{3} \frac{t_{f}}{m_{f}} .

(3)

Equation (4) evaluates the coupling at the historical finite count supplied with this calculation. The electromagnetic demarcation is a fixed interaction count close to 84.60055 fundamental lengths ([2], Appx. AB, AE, AN). Equation (5) instead takes the upper-count limit, where twice the residual-count product tends to unity. These are different evaluation regimes; the finite value must not be assigned the upper-count normalization.

{G |}_{n_{L} = 84.6005496647 (07)} = {2 Q_{L} n_{L} c^{3} \frac{t_{f}}{m_{f}} |}_{n_{L} = 84.6005496647 (07)},

(4)

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}} = 6.67407794 (10) \times 1 0^{- 11} m^{3} k g^{- 1} s^{- 2} .

(5)

CODATA publications [10, 15, 16] provide historical reference values for comparison. Equations (4) and (5) are internal MQ evaluations, and numerical proximity to a recommended constant alone does not identify the measurement regime of the experiments entering a CODATA adjustment. The finite-count calibration in Eq. (4) and the upper-count limit in Eq. (5) must therefore be distinguished in any comparison.

Within MQ, count-based expressions originate in the Internal Frame, while physical representations are assigned in the System Frame through the Frames mapping ([2], Appx. K and W; [8], for published framework context). The angular and momentum realizations in Eq. (1) are distinct assigned representations of the coefficient. The following derivations preserve that distinction. Identifying an assigned representation with an experimentally observed quantity requires additional physical evidence.

1. Calculations

Arc length construction at the fundamental radius — **Fig. 1.** Arc length at radius l_f for angular realization Θ_si. Eq. (7); adapted from [3], Fig. 2.

We first separate the momentum calculation from its angular representation. Let r denote a radial separation, in m, containing n_L fundamental lengths. Rearranging the finite-count gravitational coupling ([2], Appx. C and M) gives the momentum expression in Eq. (6). Figure 1 illustrates the separate angular construction at radius l_f. The arc length L_si, measured in m using radian angular measure, is given by Eq. (7). This construction assigns Θ_si to the angle without identifying the arc length with an action.

p_{s i} = \frac{Q_{L} r c^{3}}{G} = \frac{l_{f} m_{f}}{2 t_{f}} = \frac{ħ}{2 l_{f}} .

(6)

L_{s i} = l_{f} Θ_{s i} .

(7)

Equation (6) has momentum dimensions throughout. Equation (7) has length dimensions and provides the geometric interpretation of the angular realization. The upper-count normalization used in Eq. (8) belongs to a limit; its value is not assigned exactly to a finite residual-count product. The terminal momentum identity follows from the upper-count gravitational and reduced Planck quantities.

p_{s i} = \lim_{n_{L} \to \infty} (Q_{L} n_{L} l_{f} \frac{c^{3}}{G}) = \frac{l_{f} c^{3}}{2 G} = \frac{ħ}{2 l_{f}} .

(8)

Equation (9) substitutes the upper-count gravitational expression from Eq. (5) into Eq. (8). Cancellation of the speed-of-light factors gives half of the fundamental momentum expressed through length, mass and time ([2], Appx. F). This is the same momentum realization as Eq. (6).

p_{s i} = \frac{l_{f} c^{3}}{2 G} = \frac{l_{f} c^{3}}{2} \frac{m_{f}}{c^{3} t_{f}} = \frac{l_{f} m_{f}}{2 t_{f}} .

(9)

The angular and momentum realizations are related by the dimensional conversion in Eq. (10). The factor C_pθ has units kg m s⁻¹ rad⁻¹ and is defined as their ratio. This equation states the representation mapping; it supplies no independent prediction of a beam momentum.

p_{s i} = C_{p θ} Θ_{s i}, C_{p θ} = \frac{p_{s i}}{Θ_{s i}},

(10)

The selected calculated X-ray propagation angle of 3.26239 rad [13] is numerically close to the angular realization. Its tabulated digits are not a measured uncertainty or a precision test of the MQ coefficient. The experimental study [17] concerns down-conversion coincidence rates in a different configuration; it does not measure the bound-related MQ momentum assigned in Table 1.

The Informativity differential ([2], Appx. D) enters the finite-count normalization in Eq. (2). To express the upper-count reduced Planck quantity consistently, Eq. (11) takes that limit with the momentum realization and fundamental length held fixed. This recovers the same bound-related momentum normalization discussed in [2], Appx. Q.

ħ = \lim_{n_{L} \to \infty} \frac{p_{s i} l_{f}}{Q_{L} n_{L}} = 2 p_{s i} l_{f} .

(11)

Equation (11) is the rearrangement of the terminal momentum identity in Eq. (6). The agreement of these expressions is an internal algebraic consistency check. Finite-count factors remain relevant to intermediate evaluations of gravitational coupling, but their limiting cancellation does not establish agreement with an independently measured angle or momentum.

2. Comparison with Published X-Ray Calculations

Table 2 preserves the historical numerical comparison from [2], Table AB.1, with the 2010 CODATA values [16]. Within the MQ framework [2], the two model rows select different reduced Planck realizations while using upper-count gravitational coupling. Their differing Planck-form results illustrate sensitivity to that choice. They do not establish that the CODATA gravitational coupling was measured at an electromagnetic demarcation.

The MQ entries in Table 2 are retained historical values, including their reported parenthetical uncertainties; they are not recalculated from Eq. (1), and a complete covariance model for them is not supplied here. The quotient column uses the fixed fundamental length of the historical upper-count calibration, rather than the Planck-form length in the same row. The quotient has momentum units and is not an angular measurement. Its representation as an angle uses Eq. (10).

Table 2. Historical Planck-form calculations using finite-count and upper-count reduced Planck quantities adapted from [2], Table AB.1, and compared with 2010 CODATA [16]. a

Phenomenon	$n_{L}$	$ℏ / (2 l_{f}) (kg m s^{- 1})$	Length (10^-35 m)	Time (10^-44 s)	Mass (10^-8 kg)
$ℏ$	84.6005496647(07)	3.26250425394(01)	1.6162281374(12)	5.3911567628(97)	2.1764705491(26)
$ħ$	∞	3.26239030392(48)	1.6161999120(35)	5.3910626131(72)	2.1764325398(24)
CODATA^b			1.616199(97)	5.39106(32)	2.17651(13)

a The historical MQ rows use upper-count gravitational coupling with the indicated reduced Planck realization. Reported values and uncertainties are retained for source context, rather than presented as new metrological estimates.

^b 2010 CODATA publication [16].

The angular comparison uses selected numerical solutions from Shwartz and Harris [13]. Their 2011 paper treats phase matching, the wave-vector balance required by the crystal geometry, and polarization-current amplitudes for entangled X-rays. Related work measures down-conversion coincidence rates [17] and X-ray/optical wave mixing [18]. Equation (12) is retained as a residual count-vector balance. Here n_sx, n_sy, n_ix, n_iy, n_px and n_py are dimensionless signed count coefficients for the signal, idler and pump in two coordinates; θ is the angular parameter in rad. The vector G⃗ is the dimensionless residual required to balance these count coordinates. Its quantitative identification with a reciprocal-lattice vector requires a specified dimensional normalization. The angle comparison below uses the published solutions, not a normalization inferred from Eq. (12). The circular constant π sets the angular offset in the pump terms.

(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} .

(12)

For the selected branch, Table 3 records equal signal and idler propagation angles and the corresponding pump angle from [13]. Frequency degeneracy identifies equal daughter frequencies, not a fundamental-mass count of one half. The symbols θ_p, θ_s and θ_i denote pump, signal and idler propagation angles relative to the atomic planes. In the Bell-state notation, H and V denote polarization in and normal to the scattering plane; subscripts s and i identify the signal and idler photons. The selected solutions use a diamond (111) configuration with a 25 keV pump and 12.5 keV daughter photons [13]. They are two branches among the published solutions, not a unique universal angle.

Table 3. Selected propagation-angle solutions for a maximally entangled Bell state at frequency degeneracy. Adapted from Table I of [13]; angles are in rad.

Bell State	θ_p	θ_s	θ_i
$(\| H_{s}, V_{i} > + \| V_{s}, H_{i} >) / \sqrt{2}$	0.1208	-0.1208	-0.1208
	3.02079	3.26239	3.26239

Equation (13) recovers the momentum realization through the upper-count gravitational and reduced Planck identity. To construct an angular comparison, Table 4 divides this momentum expression by the conversion factor in Eq. (10). The angular offsets use π rad. The selected branch labels organize the comparison; the table does not derive the Bell-state phase-matching solutions from MQ alone.

p_{s i} = \lim_{n_{L} \to \infty} \frac{Q_{L} n_{L} l_{f} c^{3}}{G} = \frac{l_{f} c^{3}}{2 G} = \frac{ħ}{2 l_{f}} .

(13)

Table 4. Angular realizations obtained from the reduced Planck momentum expression and the conversion in Eq. (10). This updates the comparison in [2], Table P.1. a

Branch

θ_p

Pump (radians)

θ_s

Signal (radians)

θ_i

Idler (radians)

Offset branch

\frac{ħ}{2 l_{f} C_{p θ}} - π rad

0.120797651(36)

π rad - \frac{ħ}{2 l_{f} C_{p θ}}

−0.120797651(36)

π rad - \frac{ħ}{2 l_{f} C_{p θ}}

−0.120797651(36)

Main branch

2 π rad - \frac{ħ}{2 l_{f} C_{p θ}}

3.020795002(36)

\frac{ħ}{2 l_{f} C_{p θ}}

3.262390305(36)

\frac{ħ}{2 l_{f} C_{p θ}}

3.262390305(36)

a Calculated from the quoted angular realization through Eq. (13), with the conversion factor defined in Eq. (10). The entries share the same input and are correlated. Parenthetical uncertainties are conditional on the quoted coefficient uncertainty; they are not independent experimental error bars.

Equation (14) obtains the same momentum realization using fundamental length, mass and time. Table 5 applies the same conversion factor to its gravitational form. Thus Tables 4 and 5 express one angular mapping through two algebraically equivalent momentum formulas ([2], Appx. C, D, F). Their agreement is not two independent experimental tests.

p_{s i} = \lim_{n_{L} \to \infty} \frac{Q_{L} n_{L} l_{f} c^{3}}{G} = \frac{l_{f} c^{3}}{2 G} = \frac{l_{f} m_{f}}{2 t_{f}} .

(14)

Table 5. Angular realizations obtained from the gravitational momentum expression and the conversion in Eq. (10). This updates the comparison in [2], Table Q.1. a

Branch

θ_p

Pump (radians)

θ_s

Signal (radians)

θ_i

Idler (radians)

Offset branch

\frac{l_{f} c^{3}}{2 G C_{p θ}} - π rad

0.120797651(36)

π rad - \frac{l_{f} c^{3}}{2 G C_{p θ}}

−0.120797651(36)

π rad - \frac{l_{f} c^{3}}{2 G C_{p θ}}

−0.120797651(36)

Main branch

2 π rad - \frac{l_{f} c^{3}}{2 G C_{p θ}}

3.020795002(36)

\frac{l_{f} c^{3}}{2 G C_{p θ}}

3.262390305(36)

\frac{l_{f} c^{3}}{2 G C_{p θ}}

3.262390305(36)

a Calculated through Eq. (14) using the same defined conversion and quoted coefficient as Table 4. These entries have the same correlations and conditional uncertainty interpretation.

Tables 4 and 5 restate the same angular realization through two algebraically equivalent momentum expressions. Their equality follows from Eq. (10) and the momentum identities. The numerical proximity to the selected solutions in [13] provides a comparison, but no independent validation of a physical angle–momentum relationship or of the quoted MQ uncertainty.

4. DISCUSSION

This comparison supplies an angular realization and a bound-related momentum calculation. It supplies no joint measurement of those quantities in one X-ray interaction. Testing a physical association would require an independently specified conversion and measurements of both propagation angle and photon momentum, with the geometry and uncertainty model held explicit.

The shared numerical coefficient depends on the assigned realization maps and the stated units. A change of units changes the numerical value of a dimensional quantity without changing its physical value. The conditional algebraic consistency established here therefore does not supply a unit-independent experimental law.

Angular momentum and linear momentum are different quantities. The reduced Planck constant has action units, which are also angular-momentum units; its division by a length in Eqs. (6), (8) and (13) yields linear momentum. This does not turn the same quotient into an angle. Questions concerning conserved quantities in general relativity, including their definitions at null infinity [19], belong to a different physical setting.

The remaining question is whether the numerical angular comparison reflects a physical relationship beyond the chosen representation. Equation (15) records the bound-related reduced Planck identity. It is a consequence of the momentum realization and does not determine a measured photon momentum from a propagation angle.

ħ = 2 p_{s i} l_{f} .

(15)

Equation (15) retains action dimensions on both sides. Equations (7) and (10) separately specify arc length and the dimensional conversion. Conditional on the MQ definitions and count assumptions, these expressions are internally consistent. The comparison with selected X-ray calculations does not establish a unique physical origin of the coefficient, a photon momentum assignment, or the quoted precision. An independent physical test remains necessary.

5. APPENDICES

APPENDIX A: NOTATION AND SOURCE CONTEXT

The published symbol and term context is given in [2], Appx. BP and BQ. This appendix indexes notation introduced in the text and defines supplementary notation retained for source context, without duplicating the main first-use explanations.

θ_si, Θ_si and p_si are introduced before Eq. (1). Their distinct representations are used in Eqs. (6), (7) and (10). The X-ray comparison is discussed in Sec. 3.2 and Tables 3-5; it does not assign a beam momentum from the angular value.

c is introduced in Sec. 1. The time and length relation used in Eq. (9) is the fundamental-measure relation described there.

G and G are introduced before Eqs. (2)-(5). Their typography distinguishes finite-count coupling from the upper-count limit. The current upper-count value appears in Eq. (5); Eq. (4) retains the specified historical finite-count evaluation.

l_f, m_f and t_f are introduced in Sec. 1. The momentum derived from them is used in Eqs. (6), (9) and (14).

l_p, m_p and t_p denote the conventional Planck units of length, mass and time, in m, kg and s. Table 2 retains the historical comparison with those units.

n_L, n_M and n_T are introduced in Sec. 1 as the dimensionless counts associated with the fundamental measures.

The count n_L in the displayed radial expressions is evaluated for the separation r introduced before Eq. (6).

The center-of-mass reference for a radial separation fixes the local interpretation of the count; it does not introduce a second length-count symbol.

n_Lc denotes the dimensionless count of fundamental lengths associated with a change in the position of light relative to the observer, as used in the source nomenclature.

n_θ denotes the dimensionless System Frame count associated with the demarcation and adjusted for the metric differential. This differential describes the discrete–continuous representation mismatch under the Frames mapping. The retained count example is

⌊ \frac{84.60055}{2} ⌉ = 42 .

The mixed left-floor/right-ceiling brackets denote nearest-whole-unit rounding; this example is not a half-unit tie.

A pair of floor brackets denotes floor, and a pair of ceiling brackets denotes ceiling. The mixed pair used above has the declared nearest-whole-unit meaning.

Q_L and its positive-count domain are introduced before Eq. (2). The residual-count product occurs in the finite expressions and their upper-count limits.

The normalization associated with the Informativity differential is explained before Eqs. (2) and (3). The limiting step is used explicitly in Eqs. (5), (8), (11), (13) and (14).

ℏ and ħ are introduced before Eq. (2). Equation (11) gives the upper-count identity. Historical numerical values in Table 2 retain their original calibration.

m_e denotes electron mass, in kg, in the source derivations.

a₀ denotes the Bohr radius, in m, in the source derivations.

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

denote the dimensionless fundamental, Planck-like and classical inverse fine-structure forms in the MQ source nomenclature. The fundamental form is defined relative to the Internal Frame, and the other two relative to the System Frame. Their published normalization is supplied in [2], Appx. AA; they are not interchangeable with the momentum realization.

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 numerical values used for the angular comparison are included in Tables 1–5. Reference [2] describes supplementary calculation resources. This paper presents no newly acquired experimental data.

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