Reduced Planck Constant

Physical Significance of Finite Count and Upper Count Realizations

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60607

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We examine the physical significance of finite-count and upper-count reduced Planck quantities in Measurement Quantization (MQ). The 2022 fine-structure, electron-mass and Bohr-radius inputs recommended by the Committee on Data of the International Science Council (CODATA), together with the exact speed of light and the Planck constant of the International System of Units (SI), determine the coefficient, fundamental measures and their propagated uncertainties. Shared-input covariance is preserved through the orbital identity. The finite-count expression recovers the exact SI reduced Planck constant, while the upper-count MQ reduced Planck quantity is 1.0545349844417293(56)×10⁻³⁴ J s. Their ratio describes the count-dependent normalization associated with the Informativity differential. The result specifies how the two realizations are related; it is a consistency calculation using the selected inputs.

Keywords: Reduced Planck Constant, Measurement Quantization, Informativity differential, Fundamental Measures, Fine Structure Constant

1. INTRODUCTION

We examine the reduced Planck quantity through the MQ relations, with Planck’s early radiation work [1] as historical context. The dimensionless coefficient is θ_si, and p_si is its momentum realization in kg m s⁻¹. The adopted SI normalization assigns that realization the coefficient’s numerical value while retaining momentum dimensions. Fundamental length l_f, mass m_f and time t_f have units m, kg and s. The positive realized radial count n_L and right-triangle residual Q_L are dimensionless. Italic ℏ denotes the finite-count action, and upright ħ its upper-count limit, both in J s. The conventional SI action is h/(2π), where h is the exact SI Planck constant and π the dimensionless circular constant. The speed of light c is exact at 299,792,458 m s⁻¹. Equation (1) evaluates the finite-count action at the coherent 2022 blackbody count derived in Eq. (24).

ℏ = \frac{p_{si} l_{f}}{Q_{L} n_{L}} = \frac{h}{2 π} = 1.054571817 … \times 1 0^{- 34} J s .

(1)

The dimensionless mass and time counts n_M and n_T count the fundamental mass and time measures, respectively ([2], Appx. B, E). In the current terminology, the Internal Frame is the discrete count configuration domain. The System Frame is the non-discrete relational encoding of that structure induced by the Frames mapping. The mapping connects discrete count configurations to continuous relational quantities. Reference [2], Appx. K, gives the historical framework construction.

The 2018 CODATA comparison [3] provides the historical context. The present values use the 2022 adjustment [4]. The coefficient follows the fine-structure relation ([2], Appx. AD), and fundamental length follows the orbital relation ([2], Appx. BL). The finite-count action in Eq. (1) incorporates the residual normalization ([2], Appx. D, L, AB). When the same adjusted-constant dependencies are maintained, that normalization cancels and the equation recovers its exact SI action input. The calculation therefore establishes a relation between realizations rather than a discrepancy in the measured or defined Planck constant.

2. METHODS

MQ count representations of gravitational curvature, the speed of light, escape velocity and quantum uncertainty are developed in [2], with applications discussed in [5–10]. Reference [11] provides historical context for quantum uncertainty rather than evidence for the MQ count construction. For the present calculation, the source quantities are the 2022 CODATA fine-structure constant, electron mass and Bohr radius, together with the exact speed of light [4]. Electron mass m_e has units kg, the Bohr radius a₀ has units m, and the classical fine-structure form α_c is dimensionless. The reported source values are

m_{e} = 9.1093837139 (28) \times 10^{- 31} kg

and

a_{0} = 5.29177210544 (82) \times 10^{- 11} m

. We use the recommended inverse form

α_{c}^{- 1} = 137.035999177 (21)

for the coefficient inversion. The exact action input uses

h = 6.62607015 \times 10^{- 34} J s .

Discreteness and countability enter this calculation through the MQ framework developed in Measurement Quantization [2]. Its derivations and reported measurement comparisons supply the framework’s supporting context. The present paper tests the internal consistency of the selected action relations; it does not independently establish the physical discreteness of measure. The referenced appendices and further term definitions are available with [2] and the Online Resources below.

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

Supplementary spreadsheets described with the MQ presentation document calculations for the introductory paper and its applications [2]. The present results are reproduced from the explicit 2022 input chain stated here. Uncertainties follow the same dependency order as central values, with exact definitions held fixed and covariance preserved. The quoted source decimals are rounded representations of adjusted quantities; we retain their defining relationships rather than adding correlated derived inputs as independent measurements. Propagation is conditional on the stated MQ equations and fixed SI realization convention. It quantifies uncertainty inherited from the inputs, not the probability that the physical interpretation is correct.

3. RESULTS

The physical interpretation begins with the normalization connecting count structure to realized length and action. The supporting count construction is presented in Measurement Quantization [2]. Here we retain the distinction between the coefficient, its dimensional realization and the fundamental measures before evaluating the finite-count and upper-count expressions.

1. Nomenclature

The angular realization Θ_si has the coefficient’s numerical value in rad under the angular realization convention and remains distinct from the coefficient and momentum realization ([2], Appx. R, S). Its source comparison uses the calculated phase-matching angle in [12]; the related experiments in [13, 14] concern their respective X-ray configurations. The coefficient used here is calculated in Eq. (16). The 2018 CODATA source [3] and the Morel measurement [15] remain context for source selection, while the numerical chain uses the 2022 adjustment [4]. In [12], Table I, the 3.26239 rad entry is a calculated signal and idler angle for one degenerate-frequency phase-matching configuration. Its rounded agreement with the MQ coefficient under the angular realization convention is a contextual comparison, not an independent measurement of that coefficient. The Morel result [15] is one input to the CODATA assessment; the recommended adjustment [4] accounts for inconsistency among its contributing determinations.

The bound construction ([2], Appx. S) motivates the numerical correspondence used in these realization conventions. Angle and momentum retain their respective dimensions. Chen et al. ([16], Sec. 4.4) discuss angular momentum and center of mass at future null infinity in general relativity. That discussion is separate theoretical context and does not derive the MQ coefficient or its SI momentum realization. The dimensional quantity in the present action equations is the momentum realization defined in the Introduction.

Within MQ, the count-dependent non-Lorentz change in realized separation is described by the Informativity differential ([2], Appx. D), distinct from Lorentz contraction under relative motion [17]. The action expressions use the residual-count product Q_Ln_L; twice that product supplies the differential’s normalization. The residual follows from the Pythagorean relation in Eq. (9). The comparison in [2], Table 2, concerns historical CODATA realizations; the current values and uncertainties are calculated from the stated 2022 inputs. These relations do not supply a rule for adding the radial residual to a Lorentz contraction.

Radial and motion uses of the length count are distinguished by context. The realized radial count determines the separation entering the residual construction ([2], Appx. D). The positive noninteger count is a continuous relational encoding, not a primitive integer configuration. The inversion alone does not establish physical admissibility of an integer configuration. Geometry quantifies the normalization through the Frames mapping.

The residual normalization approaches one half as the positive count increases. At large laboratory separations, the finite-count and upper-count expressions can agree at the reported precision. Electromagnetic applications instead use the count appropriate to the phenomenon ([2], Appx. AB, AE, AN). The fine-structure count is calculated in Eq. (17); the blackbody count is reconstructed from the same coherent input chain in Eq. (24). They have different defining roles even where their values coincide under this calibration.

To display the effect of this normalization, let italic G denote the finite-count gravitational coupling and upright G its upper-count limit, both in m³ kg⁻¹ s⁻² ([2], Appx. M, F). Equation (2) evaluates the former at the count in Eq. (17). Equation (3) takes the upper-count limit explicitly. The distinction is the same one used for the two action realizations. The infinite count in Eq. (3) denotes a mathematical limit at fixed fundamental realizations, rather than an attained physical separation.

G = 2 Q_{L} n_{L} c^{3} \frac{t_{f}}{m_{f}} = 6.6738448317 (10) \times 1 0^{- 11} m^{3} k g^{- 1} s^{- 2} .

(2)

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}} = 6.6740779383 (10) \times 1 0^{- 11} m^{3} k g^{- 1} s^{- 2} .

(3)

Equations (2)–(3) illustrate how a common set of fundamental measures produces different realized couplings through the count normalization. Both values and their uncertainties are MQ calculations from the 2022 inputs; they are not additional CODATA recommendations. Their smaller propagated uncertainties do not constitute a precision test of the gravitational constant against independent measurements.

2. Resolving the Coefficient and Its Momentum Realization

A quantization ratio compares a dimensionless product with its nearest whole-unit value ([2], Appx. AC, T). In Eq. (4), r_q is that ratio and n_θ is the dimensionless System Frame count obtained by rounding half the realized radial count to the nearest whole unit. Mixed brackets denote rounding to the nearest whole-unit value. Neither rounding argument encounters a half-unit tie. The branch used here has n_θ=42, giving r_q=1.000148852639(77). Unity requires an integer product, such as the product in Eq. (8).

r_{q} = \frac{n_{θ} θ_{si}}{⌈ n_{θ} θ_{si} ⌋} .

(4)

The fundamental form α_f and Planck-like form α_p are dimensionless fine-structure quantities ([2], Appx. AA, T). The metric differential describes the mismatch between discrete count structure and its continuous encoding under the Frames mapping. The Planck-like form precedes the application of the Informativity differential; that application gives the classical form introduced in Methods ([2], Appx. AA, D). Equations (5)–(7) express this sequence and use one coefficient and one source edition.

α_{f}^{- 1} = 42 θ_{si} = 137.020392812 (11) .

(5)

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

(6)

α_{c}^{- 1} = 2 Q_{L} n_{L} (84 θ_{si} - ⌈ 42 θ_{si} ⌋) = 137.035999177 (21) .

(7)

The same quantization construction is used in elementary-charge applications ([2], Appx. AM). A count suited to one phenomenon is not substituted automatically for another. The present calculation uses the fine-structure relation to calibrate the coefficient and the blackbody relation to reconstruct the action count.

For this fine-structure construction, the count relation ([2], Appx. AA, AD, AC) fixes the product in Eq. (8). Other MQ applications, including the motion relations in [2], Appx. AU, retain their own physical context. The product 276 is the specified MQ construction used for this calibration, rather than an independently measured input. The solution remains on the nearest-integer branch used in Eqs. (5–7) throughout a ten-standard-uncertainty variation of the inverse fine-structure input.

n_{L} θ_{si} = 276 .

(8)

Equation (9) defines the residual for this realized count ([2], Appx. D). Multiplication by the coefficient gives Eq. (10), and substitution of Eq. (8) gives Eq. (11). These are algebraic steps in the same length-normalization calculation.

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

(9)

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

(10)

Q_{L} = \frac{\sqrt{θ_{si}^{2} + 27 6^{2}} - 276}{θ_{si}} .

(11)

We identify the classical inverse fine-structure quantity with the 2022 source input ([2], Appx. AA; [4]). Equation (12) repeats the relation in Eq. (7) in the form needed for inversion. Rearrangement gives Eq. (13); substitution of the residual gives Eq. (14), and combining its factors gives Eq. (15). The positive solution on the stable nearest-integer branch is Eq. (16).

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

(12)

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

(13)

2 \times 276 α_{c} = \frac{θ_{si}}{(\sqrt{θ_{si}^{2} + 27 6^{2}} - 276)} \frac{θ_{si}}{(84 θ_{si} - ⌈ 42 θ_{si} ⌋)} .

(14)

2 \times 276 α_{c} = \frac{θ_{si}^{2}}{(\sqrt{θ_{si}^{2} + 27 6^{2}} - 276) (84 θ_{si} - ⌈ 42 θ_{si} ⌋)} .

(15)

θ_{si} = 3.26239030504 (25) .

(16)

3. Deriving the Reduced Planck Constant

The coefficient in Eq. (16) determines the fine-structure count through Eq. (17) ([2], Appx. AE). Equation (18) converts that count to a physical separation using the fundamental length calculated below. Parenthetical digits denote one propagated standard uncertainty in the final quoted digits. The central values and uncertainties follow the same 2022 input dependencies, with shared correlations retained and exact inputs held fixed.

n_{L} = \frac{276}{θ_{si}} = 84.6005456716 (65) .

(17)

n_{L} l_{f} = 1.36731394425 (21) \times 1 0^{- 33} m .

(18)

The product in Eq. (19) supplies the count normalization used by the Informativity differential. It remains below one half at finite count and tends to one half at the upper count limit. This difference is responsible for the distinction between the two action expressions.

Q_{L} n_{L} = 0.4999825364172403 (27) .

(19)

The momentum realization gives fundamental mass through the fundamental relation ([2], Appx. V), yielding Eq. (20). The conversion uses the exact speed of light and preserves momentum and mass units.

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

(20)

Equation (21) restates the Planck-like inverse coupling at the precision used for the length calculation. It is derived from the same coefficient as the classical input, while retaining the normalization distinction established in Eqs. (5)–(7) ([2], Appx. AA).

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

(21)

The orbital relation ([2], Appx. BL, BM) then gives fundamental length in Eq. (22). In the coherent 2022 adjustment [4], the exact dependency m_ea₀=h/(2πα_cc) removes electron-mass uncertainty from this product while retaining the fine-structure uncertainty. Fundamental time follows from t_f=l_f/c. These dependencies are used before evaluating the separation in Eq. (18) or the coupling in Eqs. (2)–(3). Electron mass and Bohr radius remain the foundational source quantities; their correlated product is not counted twice.

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

(22)

With the fundamental length and momentum realization, Eq. (23) gives the upper-count action. This is the limit of the finite-count expression as twice the residual-count product tends to unity.

ħ = 2 p_{si} l_{f} = 1.0545349844417293 (56) \times 1 0^{- 34} J s .

(23)

For the blackbody description ([2], Appx. AB), Eq. (24) instead reconstructs the count using the conventional exact SI action and the already calculated momentum and length. Positive momentum and fundamental length are required, and the conventional SI action must exceed twice their product. These conditions hold on the chosen branch. Equation (25) substitutes that count into the finite-count expression.

n_{L} = p_{si} l_{f} \sqrt{\frac{1}{ℏ (ℏ - 2 p_{si} l_{f})}} = 84.6005456716 (65) .

(24)

The resulting finite-count action is

ℏ = \frac{p_{si} l_{f}}{Q_{L} n_{L}} = \frac{h}{2 π} = 1.054571817 … \times 1 0^{- 34} J s .

(25)

In this coherent calibration, the blackbody count in Eq. (24) agrees algebraically with the fine-structure count in Eq. (17), and Eq. (25) recovers the exact SI action. Table 1 compares the physical roles of those counts with the upper-count limit. Their shared numerical value does not make them independent measurements or a universal count for all electromagnetic phenomena.

Table 1. Reduced Planck quantities evaluated with the coherent 2022 input chain. The finite-count rows recover the same exact SI action; the upper-count row gives the MQ limiting action.

Physical role	Coherent 2022 calculation
	Realized count n_L	Reduced Planck quantity (J s)
Fine-structure calibration	84.6005456716(65)	1.054571817…×10^-34 exact
2022 CODATA SI input	—	1.054571817…×10^-34 exact
Blackbody reconstruction	84.6005456716(65)	1.054571817…×10^-34 exact
Upper-count MQ limit	∞	1.0545349844417293(56)×10^-34

The exact SI action is the input in the 2022 CODATA publication [4]. Quoted digits followed by an ellipsis are truncated display digits of an exact value. Parenthetical digits on derived quantities are propagated standard uncertainties.

4. Physical Interpretation

Equation (26) expresses the finite-count action’s departure from the upper-count action through the count-dependent separation. Because the right-hand side is positive for finite positive count, the finite-count action exceeds the upper-count value on the branch used here. This gives the dimensional meaning of the count inversion in Eq. (24).

ℏ (ℏ - 2 p_{si} l_{f}) = {(\frac{p_{si} l_{f}}{n_{L}})}^{2} .

(26)

Equation (27) expresses the same distinction as a dimensionless action ratio. At the finite count, the ratio ℏ/ħ=1.0000349283854689(54). Equivalently, the upper-count action is 0.00349271655194(54)% below the finite-count value. Both comparisons carry the same propagated input uncertainty; no independent comparison is introduced by forming their ratio.

\frac{ℏ}{p_{si} l_{f}} = 2 \frac{ℏ}{ħ} = \frac{1}{Q_{L} n_{L}} .

(27)

5. DISCUSSION

The physical role of the Informativity differential is the count normalization connecting the finite-count and upper-count quantities. Equation (23) gives the latter as twice the momentum realization multiplied by fundamental length; Eq. (27) supplies their ratio at finite count. The conventional SI transform remains h/(2π), with its MQ interpretation discussed in [2], Appx. Z.

Fine-structure calibration and blackbody reconstruction share inputs, so their agreement is an internal consistency result. An empirical test would require an independently specified observable and measurement configuration, with the calibration and realization convention fixed before comparison. The calculated angle in [12] and the distinct configurations in [13, 14] are not such a test of the present action difference.

The present comparison explains the relationship between the two action realizations within MQ, conditional on its count construction and SI normalization. The numerical uncertainties are propagated solely from the 2022 inputs as described in Methods. This establishes neither a measurement inaccuracy nor a replacement for the exact SI definition of the Planck constant.

6. APPENDICES

APPENDIX A: TERMS

The entries below locate the definitions used in this paper. Historical constructions appear in Measurement Quantization [2]; current frame terminology is used here.

The coefficient, momentum and angular realizations are distinguished in the Introduction and nomenclature discussion. The calculated-angle comparison [12] and the distinct X-ray configurations in [13, 14] provide context. The adopted 2 μrad tolerance is an author-specified comparison criterion, not a published standard uncertainty for the angle in [12] or an input to Eq. (16). The conserved-quantity context is [16], Sec. 4.4.

The speed of light and its exact value are introduced before Eq. (1) [4]. The light-count form is discussed in [2], Appx. E; the earlier source [3] remains historical context.

The finite-count and upper-count gravitational quantities are distinguished before Eqs. (2)–(3). Both use the fundamental measures and source chain described in Methods.

The fundamental measures and their units are introduced before Eq. (1). Their numerical dependencies are explained beside Eqs. (20)–(23).

l_p, m_p and t_p denote the conventional Planck units of length, mass and time, in m, kg and s, respectively. They provide the historical comparison context for the fundamental measures.

The mass and time counts are introduced after Eq. (1), alongside the radial count already defined in the Introduction.

Motion and radial uses of the length count are distinguished by the physical context described in the nomenclature discussion.

The radial application of the length count enters Eqs. (9), (17) and (24); the count relations serve their specified phenomena.

n_Lc denotes the dimensionless light-displacement count measured relative to the observer. It is distinct from the radial count used for the action calculation.

The half-count convention and its nearest whole-unit value are stated before Eq. (4).

The mixed nearest-whole-unit brackets are defined before Eq. (4), separately from ordinary floor and ceiling.

The residual is introduced before Eq. (1) and specified algebraically in Eq. (9).

The residual-count product used by the Informativity differential is introduced in the nomenclature discussion and evaluated in Eq. (19).

The finite-count and upper-count actions are distinguished before Eq. (1). Their values appear in Eqs. (23) and (25), with the physical comparison in Eq. (27).

Electron mass is introduced with its 2022 source value and units in Methods. Its covariance role in the orbital product is explained beside Eq. (22).

The Bohr radius is introduced with its 2022 source value and units in Methods. Its defining relation is retained when evaluating the orbital product.

The classical fine-structure input is introduced in Methods; the fundamental and Planck-like forms are introduced before Eqs. (5)–(7). Their dependence is carried through Eq. (22).

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

Input provenance, numerical calculation records and the cumulative revision ledger accompany this manuscript as electronic supplementary material.

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