Three Fundamental Constants

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60607

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We present the fundamental measures l_f, m_f and t_f, in m, kg and s, and their momentum relation. Their dimensionless counts are n_L, n_M and n_T, normalized to these references. At the joint uncertainty, light-speed and escape-velocity bound, the count equations give n_L=n_T=1 and n_M=1/2. The Internal Frame is the discrete count domain, and the System Frame is its continuous relational encoding under the Frames mapping. We distinguish the invariant dimensionless coefficient θ_si from its momentum realization p_si, in kg m s⁻¹, and angular realization Θ_si, in rad. The fine-structure inversion uses 2022 CODATA inputs. The realizations share a numerical coefficient in these units, with a unit conversion that leaves it unchanged. A selected calculated signal and idler propagation angle provides a separate numerical comparison. An independent angular determination could test the reverse fine-structure relation; the physical connection remains unexplained.

Keywords: Constants, Fundamental Measure, Planck Units, Fundamental Measure, Measure, Discrete

1. INTRODUCTION

Within Measurement Quantization (MQ), the fundamental measures introduced in the abstract are irreducible references derived through the count relations ([1], Appx. V, W), defined in the Internal Frame count domain ([1], Appx. I, J). They are not directly observable reference objects. Observable quantities are encoded relationally in the System Frame, which has no independent fundamental measure. The Frames mapping transforms discrete count configurations into this continuous relational encoding [2]. Physical geometry and observables belong to that realization. Agreement among dependent expressions is an algebraic check; independent tests concern the resulting observable predictions.

The realization maps R_p and R_L assign momentum and angular dimensions to the coefficient introduced in the abstract. The momentum expression is related to the Planck momentum [3]. The angular realization is compared with selected calculated propagation angles in Shwartz and Harris [4] ([1], Appx. A, Q-S). Photon-pair coincidences [5] and X-ray/optical sum-frequency generation [6] concern distinct experimental settings. The conserved-quantity discussion of Chen et al. [7] concerns quantities at null infinity. These sources do not derive the conversion used below or establish a mechanism for the numerical correspondence ([1], Appx. AX, S).

We investigate how the physical constants considered here can be expressed through the three fundamental measures and what fundamental means in this framework. Irreducibility concerns the reference measures themselves; expressing a physical constant through them does not prevent it from being constant. The following derivations distinguish this structural proposal from the numerical calibration and the separate X-ray comparison.

Planck units [3] and Stoney units [8] provide historical constructions of natural units. The MQ proposal instead relates its reference measures to the stated count structure and calibration. Choosing natural units, or representing measured constants with them, does not by itself establish a smallest physically realizable length, time or mass.

The cited Planck-scale studies address different limits. Wendel et al. [9] obtain model-dependent bounds on a possible fundamental clock period. Hossenfelder [10] examines operational limits to spatial resolution through gravitational thought experiments. Neither provides an experimental determination of the MQ reference measures or establishes that quantities below the Planck scale have no physical significance. The uncertainty principle [11] constrains statistical spreads of observables; identifying those spreads with discrete measure counts requires the additional assumptions used below.

We examine whether the proposed count relations give a consistent representation of the constants considered here. Their numerical calibration, their interpretation as fundamental reference measures and their comparison with an external angular calculation are separate questions.

2. METHODS

MQ combines classical gravitational relations, light propagation and escape velocity with a count representation of the uncertainty relation. Its applications in the publication record [1, 12, 13, 14, 15, 16] provide the framework for this calculation. The existence of those applications is distinct from independent experimental validation of their predictions.

The introductory paper, Measurement Quantization [1], develops the count argument deductively from the uncertainty relation at its bound, light propagation and the escape-velocity boundary. Appendix I resolves the joint count values. Within MQ, Appendix J applies the minimum measurable difference to admissible comparisons throughout the physical domain. Fractional normalized counts are not additional primitive physical configurations; their relational use is governed by the Frames mapping. The derivation and its admissibility conditions provide the framework used here. A publication copy and supporting resources are linked in Online Resources.

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

The companion spreadsheets organize the numbered MQ calculations. Suppl. 1 addresses the introductory publication. In Suppl. 2, the Title worksheet maps a preprint title to an ID, which selects the numbered worksheet. For this paper the ID is 8. The supplied index uses the earlier title Three Constants of Nature, and worksheet 8 is headed Three New Constants of Nature. That worksheet labels original equations (1)-(43), includes parameter-dependent entries, and redirects some calculations to Suppl. 1. It is a companion to the earlier preprint, rather than a replacement for the revised equations and uncertainty treatment here. The resources are listed in Online Resources. The present numerical values use the identified 2022 CODATA inputs [17] and dependencies specified in Results.

3. RESULTS

Within MQ, we begin with the residual count Q_L of l_f per radial length count n_L and fundamental time increment t_f ([1], Appx. C). Here r=n_Ll_f is the radial separation in m and c=l_f/t_f is the exact speed of light, 299,792,458 m s⁻¹ [17]. The factor c/p_si converts the acceleration measure in Eq. (1) to a field per unit source mass. Equation (3) is the MQ gravitational relation. Italic G denotes its finite-count coupling in m³ kg⁻¹ s⁻², while upright G denotes its upper-count limit.

\frac{Q_{L} l_{f}}{n_{L} {t_{f}}^{2}}

(1)

\frac{Q_{L} l_{f}}{n_{L} {t_{f}}^{2}} (\frac{c}{p_{si}}) = \frac{Q_{L} c^{2}}{n_{L} t_{f} p_{si}} = \frac{Q_{L} l_{f} c^{2}}{n_{L} l_{f} t_{f} p_{si}} = \frac{Q_{L} c^{3}}{r p_{si}}

(2)

\frac{Q_{L} c^{3}}{r p_{si}} = \frac{G}{r^{2}}

(3)

l_{f} = \frac{G p_{si}}{Q_{L} n_{L} c^{3}} = \frac{2 G p_{si}}{c^{3}}

(4)

Equations (5)-(7) use the MQ upper-count gravitational quantity and momentum realization calculated from the 2022 CODATA fine-structure calibration [17], as developed in Eqs. (37)-(48). The derived-measure construction follows [1], Appx. W, BM, while the coefficient inversion follows Appx. AD. Parenthetical digits denote one standard uncertainty in the final reported digits.

l_{f} = \frac{2 G p_{si}}{c^{3}} = 1.61619991148 (12) \times 1 0^{- 35} m

(5)

t_{f} = \frac{l_{f}}{c} = \frac{2 G p_{si}}{c^{4}} = 5.39106261131 (41) \times 1 0^{- 44} s

(6)

m_{f} = \frac{2 p_{si}}{c} = \frac{t_{f} c^{3}}{G} = 2.17643254057 (17) \times 1 0^{- 8} kg

(7)

Equations (5)-(7) follow the derived-measure construction ([1], Appx. W, BM), with the input and covariance treatment developed below [17]. The fundamental measures are derived within MQ from that calibration. Their reported uncertainties describe propagation of the specified source uncertainty through the accepted equations, rather than independent measurements or an assessment of all possible model uncertainty.

The earlier example used the 2014 CODATA gravitational input [18]. The present calculation derives the MQ upper-count gravitational quantity from the 2022 calibration. The last equality in Eq. (4) follows from 2Q_Ln_L→1 at the upper-count limit ([1], Appx. D); it does not identify finite-count coupling with its limit.

The joint-bound deduction combines the MQ count form of the uncertainty relation ([1], Appx. B), the light-speed relation and the escape-velocity boundary ([1], Appx. H, I). At that boundary n_L=2n_M; light propagation gives equal length and time counts, and equality in the count uncertainty relation gives Eq. (8). The count representation identifies the uncertainty-product bound with the stated geometric measures. Both earlier length-count roles coincide at this boundary; n_L is used there without identifying distinct counts away from it. The statistical uncertainty principle [11] and its MQ count interpretation are distinguished in this deduction.

2 n_{L} n_{M} n_{L} = n_{T}

(8)

2 n_{L} n_{M} = 1

(9)

2 (2 n_{M}) n_{M} = 1

(10)

n_{M}^{2} = \frac{1}{4}

(11)

n_{M} = \frac{1}{2}

(12)

Using the saturated-bound condition n_M=1/2 in Eq. (8) gives Eq. (13). The light-speed condition then gives Eq. (14), and the positive-count solution is Eq. (15) ([1], Eq. (B.7)).

2 n_{L} \frac{1}{2} n_{L} = n_{T}

(13)

n_{L} = \frac{n_{T}}{n_{L}} = 1

(14)

n_{L} = n_{T} = 1

(15)

The joint-bound algebra derives the length and time counts as one and the normalized mass count as one-half ([1], Appx. I). Appendix J excludes fractional primitive length configurations by comparison with the minimum measurable difference. Within MQ, the required differences and integer translations are admissible throughout the physical domain. This closure condition supplies the step from a minimum difference to an integer configuration domain. A minimum uncertainty and a minimum difference between count configurations are distinct notions [19]. Equations (8)-(15) establish the joint-bound values; fractional normalized parameters do not thereby become independently realizable primitive configurations.

We use fundamental for the irreducible reference measures already introduced. Within MQ, a smallest reference cannot be directly measured against a smaller reference. The reference measures are resolved mathematically through the count geometry of the expanding frame and the relations in Appendices V and W [1]. Their numerical evaluation in this paper uses the calibration stated below. The coefficient describes a relation among realizations and is not an additional fundamental measure. Observable consequences of these derived references provide the physical tests.

The coefficient has distinct momentum and angular realizations ([1], Appx. Q, S). Their numerical values agree in the stated units by the MQ realization convention. The angular value is compared separately with selected X-ray angles [4]. We first establish the count geometry and momentum expression, then examine the angular construction and the scope of that comparison.

1. Non dimensional Coefficient and Count Geometry

The unity expression ([1], Appx. AX) represents the stated dimensionless count geometry within MQ. It uses the coefficient rather than its momentum realization. Its algebraic unity is not a separate observational test of the geometry.

{(\sqrt[3]{\frac{1}{2 θ_{si}}})}^{2} + {(\frac{n_{L}}{n_{Lc}})}^{2} = 1

(16)

In Eq. (16), the expansion count n_L and light-motion count n_Lc are dimensionless counts of l_f with distinct reference roles. The first describes expansion at the first fundamental time increment; the second describes light propagation in vacuum. The cube-root term and their ratio enter the stated Pythagorean construction. These roles are local to Eq. (16) and are not interchangeable with the radial separation count away from the bound.

2. Momentum Measure Using Discrete Gravity

Equation (17) expresses the momentum realization through finite-count coupling and then its upper-count limit ([1], Appx. C, D, Q). The limit uses upright G and the same calibrated fundamental length.

p_{si} = \frac{Q_{L} r c^{3}}{G} = (\frac{c^{3}}{G}) Q_{L} n_{L} l_{f} = \frac{l_{f} c^{3}}{2 G} = 3.26239030504 (25) kg m s^{- 1}

(17)

3. Angular Measure Using Entangled X Rays

The signal, idler and pump use dimensionless coefficients n_sx, n_sy, n_ix, n_iy, n_px and n_py; the subscripts identify the photon and coordinate. The angular parameter θ is in rad. The vector $\vec{G}$ , with components G_x and G_y, is the normalized reciprocal-lattice vector, distinct from gravitational coupling. For the reverse reduction of the coplanar phase-matching geometry in [4], all wavevector magnitudes and the reciprocal-lattice vector are divided by one positive reference magnitude, in inverse metres. Each photon has the same normalized magnitude coefficient in both coordinate projections; the trigonometric factors supply the signs. This defines the local coefficients consistently with the source vector balance, without identifying them with primitive integer count labels ([1], Appx. A). The circular constant is π.

θ = 2 π - \arccos (\frac{G_{x}}{n_{sx} + n_{ix} - n_{px}})

(18)

θ = π - \arcsin (\frac{G_{y}}{n_{sy} + n_{iy} + n_{py}})

(19)

Equations (18)-(24) are read in reverse order. Equation (24) is the normalized vector balance, specialized to equal signal and idler directions and a reflected pump direction. Expanding the reflected pump gives Eqs. (22) and (23), which are identical here. Expanding the component sums gives Eq. (21), and collecting terms gives Eq. (20). Equations (18) and (19) are conditional inversions of those components.

(\cos (θ) (n_{sx} + n_{ix} - n_{px}), \sin (θ) (n_{sy} + n_{iy} + n_{py})) = \vec{G}

(20)

(n_{sx} \cos (θ) + n_{ix} \cos (θ) - n_{px} \cos (θ), n_{sy} \sin (θ) + n_{iy} \sin (θ) + n_{py} \sin (θ)) = \vec{G}

(21)

(n_{sx} \cos (θ), n_{sy} \sin (θ)) + (n_{ix} \cos (θ), n_{iy} \sin (θ)) - (n_{px} \cos (θ), - n_{py} \sin (θ)) = \vec{G}

(22)

(n_{sx} \cos (θ), n_{sy} \sin (θ)) + (n_{ix} \cos (θ), n_{iy} \sin (θ)) - (n_{px} \cos (θ), - n_{py} \sin (θ)) = \vec{G}

(23)

(n_{sx} \cos (θ), n_{sy} \sin (θ)) + (n_{ix} \cos (θ), n_{iy} \sin (θ)) = (n_{px} \cos (2 π - θ), n_{py} \sin (2 π - θ)) + \vec{G}

(24)

The angles θ_s, θ_i and θ_p specify signal, idler and pump propagation relative to the atomic planes normal to the reciprocal-lattice vector [4]. Here θ_s=θ_i=θ and θ_p=2π-θ, with the selected branch π<θ<3π/2. The proposed MQ comparison identifies the numerical value of this common angle in rad with θ_si, through the angular realization already defined. H and V specify polarization in and normal to the scattering plane, respectively. The selected source configuration uses diamond (111), a 25 keV pump and degenerate 12.5 keV daughters, with the pump polarized normal to that plane. The nonzero, equal-magnitude polarization-pair amplitudes have the relative sign of the symmetric Bell state in Table 3 [4]. Equal propagation angles alone do not establish that state. The inversions in Eqs. (18) and (19) require nonzero denominators and real ratios from −1 to 1. A vanishing denominator leaves the unreduced component balance valid but prevents that inverse formula from determining the angle.

4. Angular Interpretation Using the Reduced Planck Constant

C_{p θ} = \frac{p_{si}}{Θ_{si}} = 1 kg m s^{- 1} ra d^{- 1}

(25)

Figure 1 from the original preprint — Fig. 1. Arc length L_si of a circle of radius l_f subtending Θ_si. Adapted from [12].

Equation (25) defines C_pθ, in kg m s⁻¹ rad⁻¹. The shared numerical coefficient makes its numerical value one in these units. It converts units without altering the coefficient or introducing an adjustable parameter. SI quantity calculus distinguishes a quantity from its numerical value in a chosen unit [20]. Thus the momentum and angular realizations remain different quantities even when their numerical values agree; a unit change also changes the numerical value of the conversion factor. For the circle in Eq. (26), r is evaluated at l_f, and L_si is its arc length in m. Radians follow the SI dimensionless-unit convention [20]. Figure 1 depicts this arc relation, adapting [12]. Upright ħ denotes the MQ upper-count reduced Planck quantity in J s. The conventional italic quantity is ℏ=h/(2π), where h is the exact SI Planck constant [17].

L_{si} = r Θ_{si} = l_{f} Θ_{si}

(26)

Θ_{si} = \frac{p_{si}}{C_{p θ}} = 3.26239030504 (25) rad

(27)

G = c^{3} \frac{t_{f}}{m_{f}} = \frac{2 t_{f}}{l_{f} m_{f}} \frac{l_{f}}{2} c^{3} = \frac{l_{f}}{2 p_{si}} c^{3}

(28)

p_{si} = \frac{l_{f}}{2} (\frac{c^{3}}{G}) = \frac{ħ}{2 l_{f}} = 3.26239030504 (25) kg m s^{- 1}

(29)

Θ_{si} = θ_{si} rad = 3.26239030504 (25) rad

(30)

5. Relation of Angular and Momentum Realizations at the Measurement Bound

The momentum in Eqs. (29) and (35) is one-half of the MQ fundamental momentum m_fc. The corresponding angular realization retains the same numerical coefficient through Eq. (25). This half-momentum relation concerns the fundamental measures; it does not identify that momentum with the momentum of a daughter X-ray photon.

Θ_{si} = \frac{ħ}{2 l_{f} C_{p θ}} = 3.26239030504 (25) rad

(31)

Equation (32) restates the gravitational expression in Eq. (3). Solving for momentum gives Eq. (33); its last form uses the upper-count relation already specified after Eq. (4) ([1], Appx. C, D).

\frac{Q_{L} c^{3}}{r p_{si}} = \frac{G}{r^{2}}

(32)

p_{si} = \frac{Q_{L} r c^{3}}{G} = Q_{L} n_{L} l_{f} \frac{c^{3}}{G} = \frac{l_{f}}{2} \frac{c^{3}}{G}

(33)

The upper-count gravitational identity is G=c³t_f/m_f ([1], Appx. F). The fundamental relation uses l_fm_f=2p_sit_f ([1], Appx. V), and the upper-count reduced Planck quantity obeys ħ=2p_sil_f. This MQ action quantity is distinct from the conventional exact SI quantity defined before Eq. (26); its calibration does not revise the SI Planck constant or constitute an independent measurement of it.

p_{si} = \frac{l_{f}}{2} (\frac{c^{3}}{1}) (\frac{m_{f}}{c^{3} t_{f}}) = \frac{2 l_{f}}{4} (\frac{m_{f}}{t_{f}}) (\frac{l_{f}}{l_{f}})

(34)

p_{si} = \frac{2 l_{f}}{2 l_{f}} (\frac{l_{f} m_{f}}{2 t_{f}}) = \frac{ħ}{2 l_{f}}

(35)

Substituting the upper-count gravitational identity into Eq. (33) gives Eq. (34). Using c=l_f/t_f and the fundamental relation reduces it to Eq. (35). The resulting reduced Planck quotient has momentum units. Equation (36) applies the unit conversion already defined in Eq. (25).

p_{si} = C_{p θ} Θ_{si}

(36)

The electromagnetic and momentum derivations in Appendices P-S [1] reduce to the common reduced Planck quotient ħ/(2l_f). Equations (29) and (35) demonstrate their equality within MQ; it is not merely an agreement between rounded decimal values. In the dimensionally explicit notation used here, that quotient gives p_si, while Eq. (31) gives its angular realization through the unit conversion in Eq. (25). The calculations inherit precision and standard uncertainty from the reported calibration inputs [17]. Their shared numerical coefficient is established by the derivations. Its appearance in quantities with different dimensions is the correspondence under investigation. Why the selected crystal propagation angle in Shwartz and Harris [4] shares this value remains an open physical question; no connecting experiment has yet been performed.

6. Precise Solution to θ_si

The fundamental inverse fine-structure form $α_{f}^{- 1} = 42 θ_{si}$ belongs to the Internal Frame description ([1], Appx. AA). The Planck-like and classical forms, $α_{p}^{- 1}$ and $α_{c}^{- 1}$ , are dimensionless System Frame quantities. The metric differential is the representation mismatch between integer count structure and its continuous encoding; here it supplies the nearest-whole-unit remainder ([1], Appx. T). The Informativity differential describes the count-dependent, non-Lorentz contraction of realized spatial separation; here its normalization factor is 2Q_Ln_L. The metric differential is not a physical deformation in itself, and the Informativity differential is distinct from curvature. The mixed brackets ⌊x⌉ mean rounding the dimensionless argument x to the nearest whole unit. Equations (37)-(43) invert these relations to resolve the coefficient ([1], Appx. AD).

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

(37)

Q_{L} θ_{si} = \sqrt{{θ_{si}}^{2} + {(θ_{si} n_{L})}^{2}} - n_{L} θ_{si}

(38)

Q_{L} = \frac{\sqrt{{θ_{si}}^{2} + {(n_{L} θ_{si})}^{2}} - n_{L} θ_{si}}{θ_{si}}

(39)

Equation (37) supplies the residual used in the fine-structure expression, Eq. (40) ([1], Appx. AA). Equations (38) and (39) multiply and divide by the positive coefficient without changing that residual. Inverting Eq. (40) and regrouping its factors gives Eqs. (41) and (42).

\frac{1}{α_{c}} = 2 Q_{L} n_{L} (84 θ_{si} - ⌈ 42 θ_{si} ⌋)

(40)

2 α_{c} = \frac{1}{n_{L}} \frac{θ_{si}}{θ_{si}} \frac{1}{Q_{L} (84 θ_{si} - ⌈ 42 θ_{si} ⌋)}

(41)

2 α_{c} = \frac{1}{n_{L} θ_{si}} \frac{θ_{si}}{Q_{L} (84 θ_{si} - ⌈ 42 θ_{si} ⌋)}

(42)

For this calibration, the electromagnetic demarcation is the fixed interaction scale specified by n_Lθ_si=276 ([1], Eq. (AC.5)). Its normalized count is distinct from a freely selectable fractional primitive configuration. The 2022 CODATA input is $α_{c}^{- 1} = 137.035999177 (21)$ [17]. Substituting Eq. (39) into Eq. (42) gives the three factors in Eq. (43), including both numerator factors of θ_si. The positive solution on the stated nearest-whole-unit branch gives Eq. (44). The earlier source edition was 2018 [21].

2 α_{c} = \frac{1}{n_{L} θ_{si}} \frac{θ_{si}}{\sqrt{{θ_{si}}^{2} + {(n_{L} θ_{si})}^{2}} - n_{L} θ_{si}} \frac{θ_{si}}{84 θ_{si} - ⌈ 42 θ_{si} ⌋}

(43)

θ_{si} = 3.26239030504 (25)

(34)

Equation (44) gives the dimensionless coefficient and its propagated standard uncertainty. Equations (45-48) give dependent fundamental measures and the Planck-like inverse fine-structure value from the same calibration. The realizations retain the coefficient in the stated units. Conversely, Eq. (40) can resolve α_c from an independently determined θ_si, with the same n_Lθ_si=276 relation and specified nearest-whole-unit branch ([1], Appx. AD). That reverse calculation becomes an independent test only when the angular identification and input uncertainty are established without reusing the fine-structure calibration.

The inversion uses the fine-structure input to determine the coefficient. An orbital expression ([1], Appx. BL) involving electron mass, Bohr radius and the exact speed of light can represent the same adjusted input, but it does not provide an independent prediction when those constants share the CODATA adjustment.

The SI values of elementary charge and the Planck constant are exact [17], while electromagnetic constants such as vacuum permittivity depend on the adjusted fine-structure value. Independently measured observables outside the calibration can test MQ predictions and can constrain inferred input values when the relations are solved in reverse. Independence depends on the underlying measurements and shared calibration dependencies, not only on excluding particular constant names. Laboratory gravitational measurements offer one comparison after the finite-count or upper-count domain is matched. The 2018 edition [21] remains historical source context.

A separate cosmological relation uses the universe radius R_U in m and age A_U in s ([1], Eq. (BI.4)). It requires an independently justified cosmological input and is not used in the calibration below. In Eq. (47), m_e is electron mass in kg and a₀ is the Bohr radius in m. The exact adjusted-constant identity m_ea₀=ℏ/(α_cc) [17] eliminates their shared electron-mass dependence before uncertainty propagation.

m_{f} = \frac{2 p_{si}}{c} = 2.17643254057 (17) \times 1 0^{- 8} kg

(45)

{α_{p}}^{- 1} = 84 θ_{si} - ⌈ 42 θ_{si} ⌋ = 137.040785623 (21)

(46)

l_{f} = \frac{m_{e} a_{0} α_{p}}{m_{f}} = 1.61619991148 (12) \times 1 0^{- 35} m

(47)

t_{f} = \frac{l_{f}}{c} = 5.39106261131 (41) \times 1 0^{- 44} s

(48)

7. Comparison with X-Ray Calculations

Tables 1 and 2 give equal numerical angular realizations from the gravitational and reduced Planck momentum derivations ([1], Appx. P-S). Both constructions reduce to the common quotient demonstrated in Eqs. (29) and (35), with the angular unit conversion in Eq. (25). Their equality follows from the derivations and the same calibrated inputs. Table 3 gives selected Shwartz and Harris solutions [4] for diamond (111), a 25 keV pump and degenerate 12.5 keV daughters.

Table 1. MQ angular realizations from the gravitational momentum expression and Eq. (25), updating the comparison adapted from [15]. ^a

Angle	θ_p	Pump (radians)	θ_s	Signal (radians)	θ_i	Idler (radians)
Offset branch	$\frac{l_{f} c^{3}}{2 G C_{p θ}} - π rad$	0.12079765145(25)	$π rad - \frac{l_{f} c^{3}}{2 G C_{p θ}}$	-0.12079765145(25)	$π rad - \frac{l_{f} c^{3}}{2 G C_{p θ}}$	-0.12079765145(25)
Main branch	$2 π rad - \frac{l_{f} c^{3}}{2 G C_{p θ}}$	3.02079500214(25)	$\frac{l_{f} c^{3}}{2 G C_{p θ}}$	3.26239030504(25)	$\frac{l_{f} c^{3}}{2 G C_{p θ}}$	3.26239030504(25)

^a Calculated from the shared 2022 calibration using Eq. (25). Standard uncertainties propagate the input within MQ and exclude unquantified model uncertainty. The external angles are calculated source values [4].

Table 2. MQ angular realizations from the reduced Planck momentum expression and Eq. (25), updating the comparison adapted from [15]. ^a

Angle	θ_p	Pump (radians)	θ_s	Signal (radians)	θ_i	Idler (radians)
Offset branch	$\frac{ħ}{2 l_{f} C_{p θ}} - π rad$	0.12079765145(25)	$π rad - \frac{ħ}{2 l_{f} C_{p θ}}$	-0.12079765145(25)	$π rad - \frac{ħ}{2 l_{f} C_{p θ}}$	-0.12079765145(25)
Main branch	$2 π rad - \frac{ħ}{2 l_{f} C_{p θ}}$	3.02079500214(25)	$\frac{ħ}{2 l_{f} C_{p θ}}$	3.26239030504(25)	$\frac{ħ}{2 l_{f} C_{p θ}}$	3.26239030504(25)

^a Uses the same coefficient and conversion as Table 1. Agreement of these two representations is an algebraic consistency check. The external angular calculation is [4]; the experiments [5, 6] have the distinct scope stated in the Introduction.

Table 3. Selected calculated propagation angles in rad for the stated Bell configuration at frequency degeneracy, adapted from Table I of Shwartz and Harris [4]. The polarization labels are defined in Section 3.3.

Bell State	θ_p	θ_s	θ_i
(\|H_s, V_i> +\|V_s, H_i>)/√2	0.1208	-0.1208	-0.1208
	3.02079	3.26239	3.26239

The main-branch daughter value in Tables 1 and 2 rounds to the 3.26239 rad calculated in Table 3 [4]. The source pump value differs in its final reported digit when the MQ result is rounded to the same decimal place. The MQ values derive from reported physical-constant inputs with precision and standard uncertainty [17], as propagated above. The shared result of the two MQ constructions is established mathematically; its comparison with the selected crystal solution is a further physical correspondence. The source table supplies a theoretical angular value without a measurement standard uncertainty and therefore does not test the additional MQ digits. An experiment connecting this selected angle to half the fundamental momentum has not been performed. An independently calibrated angular determination with verified Bell-state conditions could test that connection and the reverse fine-structure calculation. The fundamental references are tested through mapped observables, and their momentum realization is distinct from the daughter photon momentum. Related experiments [5, 6] retain the separate scope stated in the Introduction.

4. DISCUSSION

Within MQ, dimensional physical constants are expressed through the fundamental measures, while dimensionless constants describe relations among them. The reviewed equations retain this distinction. Changing a realization or its units changes the representation of a numerical coefficient; it does not erase the dimensions of momentum, angle or action. The conversion in Eq. (25) makes those units explicit.

Within MQ, the three derived reference measures represent the constants considered here. The joint count-bound deduction and calibration address different parts of that construction. Appendix J supplies the admissible-difference argument over the physical domain, while the fine-structure input supplies the numerical coefficient. Fractional normalized values encode relations and do not establish fractional primitive physical configurations. The Frames mapping connects these configurations to observable quantities. The reviewed electromagnetic and momentum derivations establish their common reduced Planck quotient and its numerical coefficient. Their calibrated values inherit the reported input precision and uncertainty. Agreement among calculations using shared adjusted constants establishes consistency within that construction; independent observations can test its physical consequences and constrain inferred inputs through inverse relations. The occurrence of the common numerical value in angular and momentum realizations is the dimensional correspondence identified here. Its physical interpretation and connection to the selected crystal-angle solution remain open research questions, with experimental tests reserved for future work.

5. APPENDICES

APPENDIX A: SYMBOL DEFINITIONS

The entries below identify where the terms and symbols are introduced in this paper. Measurement Quantization [1] supplies the extended derivations and historical terminology; the frame and differential meanings used here follow the distinctions already stated in the narrative.

The coefficient and its realizations are introduced in the abstract and related in Eqs. (25)-(36). The source daughter angle 3.26239 rad [4] is a calculated entry without a stated standard uncertainty. The separate experimental settings [5, 6] are identified in the Introduction. The discussion of conserved quantities at null infinity ([7], Sec. 4.4) does not derive the cross-context relation considered here.

The exact speed of light and its fundamental-measure relation are introduced before Eq. (1). The earlier 2018 source [21] used the same exact speed value.

The finite-count and upper-count gravitational meanings are introduced before Eq. (1). Under the 2022 calibration, the MQ upper-count calculation is $G = 6.6740779383 (10) \times 10^{- 11} m^{3} {kg}^{- 1} s^{- 2}$ , while the matched electromagnetic-demarcation calculation is $G = 6.6738448317 (10) \times 10^{- 11} m^{3} {kg}^{- 1} s^{- 2}$ .

The fundamental measures and their units are introduced in the abstract; Eqs. (5)-(7) and (45)-(48) give their recalculated values.

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

The counts and their normalization are introduced in the abstract. The half-count mass value in Eqs. (8)-(15) is relative to m_f at the joint boundary. It is a normalized parameter, not a fractional primitive physical configuration. The length-count roles coincide there and remain distinct in their other stated contexts.

The observer-relative and radial length-count roles are fixed by the local context. Equation (16) uses the distinct expansion role described immediately after that equation.

The radial length count is introduced before Eq. (1) using n_L. The local expansion and light-count roles are distinguished after Eq. (16).

The light-motion count is introduced after Eq. (16).

The dimensionless System Frame count n_θ incorporates the electromagnetic demarcation and the metric differential already described. It is distinct from the radial count n_L. The present calibration gives ⌊84.6005456716(65)/2⌉=42.

The mixed brackets have the nearest-whole-unit meaning introduced before Eq. (37).

The residual count is introduced before Eq. (1) and resolved in Eq. (37).

The normalization Q_Ln_L uses the residual and radial count introduced before Eq. (1); its role in the Informativity differential is discussed before Eq. (37).

The conventional and MQ upper-count reduced Planck quantities are distinguished before Eq. (26). The conventional value is exactly $1.054571817 \dots \times 10^{- 34} J s$ [17]. The MQ upper-count calculation is $ħ = 1.0545349844417293 (56) \times 10^{- 34} J s$ under the stated calibration.

Electron mass and its role in the adjusted-constant product are introduced before Eq. (45).

The Bohr radius and its role in the adjusted-constant product are introduced before Eq. (45).

The three inverse fine-structure forms and their frame assignments are defined before Eq. (37). The classical form includes the finite-count normalization.

The historical symbol γ denotes the earlier electromagnetic transformation in [1], Appx. AS. It is retained only as source notation and does not occur in the present derivations.

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

Calculation resources associated with [1] are linked in Online Resources. The numerical values reported here use the 2022 CODATA edition [17] and the calibration dependencies specified in Results.

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