Fundamental Measures and Their Relation to the Planck Units

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60607

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We derive fundamental length, mass and time within Measurement Quantization (MQ) and compare them with the conventional Planck units. The calculation retains the finite count factor in gravitational coupling and reduced Planck constant, and uses the 2022 fine-structure constant recommended by the Committee on Data of the International Science Council (CODATA), with exact constants of the International System of Units (SI) for calibration. A single calibrated set of fundamental measures is used throughout. At the upper count limit the Planck root expressions recover these measures; at finite count the mass expression retains an additional normalization factor. Comparisons with the recommended gravitational constant are reported separately from calibration identities. The derived uncertainties describe propagation of the stated inputs and do not establish the physical discreteness of measure or independently validate the assumed realization maps.

Keywords: Fundamental Measures, Planck Units, Discrete Measure, Measure, Bounds

1. INTRODUCTION

The conventional Planck length l_P, mass m_P and time t_P combine the speed of light in vacuum c, gravitational constant G and reduced Planck constant ℏ=h/(2π), where h is the Planck constant [1]. These combinations provide natural dimensional scales. The gravitational constant has units m³ kg^-1 s^-2, reduced Planck constant has units kg m² s^-1, and the corresponding Planck scales have units m, kg and s [2].

Planck and Stoney units [1,3] use different choices of physical constants to construct natural scales. A unit construction alone does not establish that its length or time is an experimentally resolved minimum. The present comparison concerns the algebraic relation between these conventional scales and the fundamental measures defined within MQ.

The purpose is to derive the MQ fundamental measures, identify the assumptions required for their calibration, and compare the resulting finite count and upper count expressions with the conventional Planck forms. Numerical consistency with an input used in calibration is distinguished from comparison with an independently adjusted quantity.

The MQ derivation combines gravitational coupling, the speed of light, escape velocity and the uncertainty principle [4,5]. The fundamental measures l_f, m_f and t_f have units m, kg and s. The dimensionless coefficient θ_si has angular realization Θ_si=θ_si rad and momentum realization p_si in kg m s^-1. Their dimensional conversion factor is C_pθ=p_si/Θ_si in kg m s^-1 rad^-1, so p_si=C_pθΘ_si. The shared numerical coefficient used here fixes this conversion normalization; it is a stipulated realization convention rather than an independent SI momentum measurement. Equations (1)–(3) summarize the calibration developed in Eqs. (18)–(31), using the MQ upper count gravitational output.

l_{f} = \frac{2 G p_{si}}{c^{3}} = 1.61619991148 (12) \times 10^{- 35} m .

(1)

t_{f} = \frac{2 G p_{si}}{c^{4}} = 5.39106261131 (41) \times 10^{- 44} s .

(2)

m_{f} = \frac{2 p_{si}}{c} = 2.17643254057 (17) \times 10^{- 8} kg .

(3)

Using the MQ terminology adopted here, the Internal Frame is the discrete count configuration domain. Continuous geometry and physically meaningful observables belong to the System Frame through the Frames mapping (for the earlier framework discussion, see [5], Appx. K), the kinematic transformation from discrete count configurations to continuous relational structure. The latter introduces no independent fundamental measure. At the upper count limit, the momentum realization is half the fundamental momentum, as expressed by Eq. (9). Its angular comparison is treated separately below ([5], Appx. A, Q–S).

The relevant angular comparison is the theoretical branch of Shwartz and Harris [6], Table I. It uses diamond with the (111) reciprocal lattice vector, a 25 keV pump, and equal signal and idler energies. The selected pump angle is 3.02079 rad; the calculated signal and idler propagation angles are both 3.26239 rad relative to the specified atomic planes. Other branches occur in the same table. Thus the comparison is configuration dependent and does not independently validate the additional digits of the MQ coefficient. The related studies concern X-ray/optical wave mixing [7] and down-conversion coincidence rates [8], rather than a universal MQ momentum measurement. Reference [9] discusses conserved quantities at null infinity in general relativity; its supertranslation results do not determine the realization conversion introduced above.

The fundamental expressions differ from the conventional Planck forms, with their comparison given below ([5], Appx. X). The Informativity differential is the non-Lorentz length contraction associated with discrete count structure under the Frames mapping ([5], Appx. D). Here n_L is a positive dimensionless count of fundamental lengths, and

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

is its dimensionless residual. The residual is used in the count-dependent expressions below.

2. METHODS

Measurement Quantization applies discrete count relations to gravitational coupling and kinematic identities. The present derivation follows the MQ relations cited in [5] and retains their count assumptions. Other applications are described in [10,11,12,13,14,15]; their existence does not supply independent validation of the particular calibration evaluated here.

Discrete count structure is a model premise of this calculation. The introductory MQ treatment [5] develops proposed derivations and physical interpretations of discreteness. These supply the internal basis for the construction. The two-mode torsion-balance comparison below provides experimental evidence consistent with the proposed gravitational count-regime distinction, with an apparatus-specific interpretation and stated limitations. The cited appendices and supplementary-material descriptions are available in that publication and its Online Resources.

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

Reference [5] describes two supplementary spreadsheets, one for its own expressions and another for applications. Their contents have not been independently audited here. The present numerical results are reproducible from the explicit relations, branch selection and 2022 inputs stated below.

3. RESULTS

Let r denote radial separation in m, with the length count n_L introduced earlier. Upright G denotes the upper count gravitational coupling, while italic G denotes its finite count evaluation. Both have units m³ kg^-1 s^-2. The corresponding distinction for reduced Planck constant is upright ħ at the upper count limit and italic ℏ at the specified finite count; both have units kg m² s^-1. The conventional SI reduced Planck input retains the standard notation introduced in the Introduction and is identified explicitly when used for calibration.

The notation distinctions above apply throughout the MQ calculation. Electromagnetic interactions are evaluated at a fixed interaction count distance called the electromagnetic demarcation ([5], Appx. AA, AE). The retained condition for that distance is specified in Section 3.2. This MQ interaction scale is distinct from the formal upper count limit; it is not a universal assignment of gravitational experiments to a count regime.

1. Resolving the Fundamental Measures

Beginning with the expression for discrete gravity ([5], Appx. C), then

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

(4)

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

(5)

Equation (6) retains the finite-count coupling. Equation (7) is the upper-count expression, with c=l_f/t_f. These regimes remain distinct.

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

(6)

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

(7)

The cubic factor in Eq. (7) follows algebraically from c=l_f/t_f. It combines with t_f/m_f to give the dimensions of gravitational coupling. This factorization does not establish a one-to-one relation between independent mass and time counts.

We equate the two upper-count expressions for G in Eqs. (5), (7), obtaining the momentum identity in Eqs. (8), (9) ([5], Appx. F).

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

(8)

l_{f} m_{f} = 2 p_{si} t_{f} .

(9)

Equation (9) is the fundamental relation ([5], Appx. V, Eq. (V.4)) in the momentum realization introduced earlier. It follows from the two upper count gravitational expressions and the speed relation, so it is an internal algebraic result.

Continuing the same reduction, Eqs. (10)–(14) substitute the upper count gravitational expression from Eq. (7) and cancel the common count factors.

\frac{Q_{L} c^{3}}{r p_{si}} = \frac{Q_{L} c^{3}}{n_{L} l_{f} p_{si}} = \frac{Q_{L}}{n_{L} l_{f} p_{si}} \frac{c^{3} t_{f}}{m_{f}} \frac{m_{f}}{t_{f}} .

(10)

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

(11)

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

(12)

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

(13)

l_{f} = G \frac{1}{c^{2}} \frac{2 p_{si} t_{f}}{l_{f}} = \frac{2 G p_{si}}{c^{3}} .

(14)

The numerical evaluations of Eqs. (15)–(17) use the common calibration derived below. The upper count gravitational output is

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

It shares the fine-structure input with p_si and must not be treated as an independent input in uncertainty propagation. The substitution uses unrounded values internally; displayed central values are rounded only for presentation.

l_{f} = \frac{2 G p_{si}}{c^{3}} = \frac{2 (6.6740779383 \times 10^{- 11}) (3.26239030504)}{{(299792458)}^{3}} = 1.61619991148 (12) \times 10^{- 35} m .

(15)

t_{f} = \frac{2 G p_{si}}{c^{4}} = \frac{2 (6.6740779383 \times 10^{- 11}) (3.26239030504)}{{(299792458)}^{4}} = 5.39106261131 (41) \times 10^{- 44} s .

(16)

m_{f} = \frac{2 p_{si}}{c} = \frac{2 (3.26239030504)}{299792458} = 2.17643254057 (17) \times 10^{- 8} kg .

(17)

The fine-structure calculation below uses the 2022 CODATA adjustment [2]. Earlier CODATA reports [16,17,18] remain relevant to the historical comparison in Table 1. Reference [19] supplies historical context for fine-structure phenomenology, and [20] discusses precision determinations. The recommended gravitational constant is reserved for the separate comparison in Table 2.

2. Fundamental Measures as a Measure of the Fine Structure Constant

The retained electromagnetic demarcation condition is n_Lθ_si=276 ([5], Appx. AA, Eq. (AA.5)). It selects the count for this calibration, rather than being inferred from the CODATA gravitational recommendation. This selection is an assumption of the present evaluation. It is not imposed on the free count variable in the upper count limit. The following relations express the residual before inserting the interaction condition.

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

(18)

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

(19)

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

(20)

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

(21)

The dimensionless classical fine-structure quantity α_c enters the form in [5], Appx. AA. Equations (22)–(25) give its inversion. The mixed brackets ⌊x⌉ mean rounding x to the nearest whole-unit value. For the recalculated coefficient, the argument 42θ_si rounds to 137 and remains away from a half-unit boundary across the propagated uncertainty.

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

(22)

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

(23)

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

(24)

We identify

α_{c}^{- 1}

with the 2022 CODATA input 137.035999177(21) [2] and retain the interaction condition. The positive solution of Eq. (25) is chosen on the branch where the nearest count is 137, giving Eq. (26). This is a conditional inversion, not a unique solution over all possible nearest-count branches. Equation (25) follows by substituting Eq. (20) into Eq. (24); both factors of θ_si in its numerator are required by that substitution.

2 α_{c} = \frac{1}{276} \frac{θ_{si}}{84 θ_{si} - ⌈ 42 θ_{si} ⌋} (\frac{θ_{si}}{\sqrt{θ_{si}^{2} + 27 6^{2}} - 276}) .

(25)

θ_{si} = 3.26239030504 (25) .

(26)

The dimensionless MQ fine structure form α_p is specified by Eq. (28). To obtain a dimensional length, Eq. (29) uses the electron mass m_e and Bohr radius a₀. Their product is evaluated through the exact CODATA relation

m_{e} a_{0} = ℏ α_{c}^{- 1} / c

[2], rather than by multiplying two rounded and correlated recommendations as independent inputs. The conventional reduced Planck constant in this calibration is exact through the SI definition of h. Thus the stated uncertainties propagate the single adjusted fine-structure input; they exclude uncertainty in the MQ realization assumptions.

m_{f} = \frac{2 p_{si}}{c} = 2.17643254057 (17) \times 10^{- 8} kg .

(27)

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

(28)

l_{f} = \frac{m_{e} a_{0} α_{p}}{m_{f}} = 1.61619991148 (12) \times 10^{- 35} m .

(29)

t_{f} = \frac{l_{f}}{c} = 5.39106261131 (41) \times 10^{- 44} s .

(30)

G = c^{2} \frac{l_{f}}{m_{f}} = 6.6740779383 (10) \times 10^{- 11} m^{3} k g^{- 1} s^{- 2} .

(31)

3. Correlating the Fundamental Measures with Planck’s Expressions

The fundamental measures (left) can now be compared with Planck’s unit expressions (right).

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

(32)

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

(33)

To compare the fundamental measures with the conventional Planck expressions, we now take the upper-count limit, where 2Q_Ln_L tends to unity. The fundamental measures and the momentum realization remain fixed during this formal limit. Upright G and ħ denote the resulting upper-count quantities.

The square roots below are the conventional Planck length, time and mass combinations, evaluated using these MQ upper-count quantities. The existing speed relation c=l_f/t_f and the fundamental relation l_fm_f=2p_sit_f establish their correspondence with the fundamental measures.

\frac{2 G p_{si}}{c^{3}} = \sqrt{\frac{ħ G}{c^{3}}} .

(34)

\frac{2 G p_{si}}{c^{4}} = \sqrt{\frac{ħ G}{c^{5}}} .

(35)

\frac{2 p_{si}}{c} = \sqrt{\frac{ħ c}{G}} .

(36)

Squaring any of Eqs. (34)–(36) and rearranging gives the same relation,

4 G p_{si}^{2} = ħ c^{3} .

(37)

Thus the three comparisons are algebraically consistent. In the upper-count limit, the Planck length, time and mass combinations recover the corresponding fundamental measures. The finite-count normalization in Eqs. (32) and (33) must be retained when evaluating quantities away from this limit.

Restating the common relation for the subsequent reduction gives

4 G p_{si}^{2} = ħ c^{3} .

(38)

For the following reduction we use the MQ upper-count identity ħ=2p_sil_f ([5], Appx. L, Eq. (L.7)).

4 G p_{si}^{2} = 2 θ_{si} l_{f} c^{3},

(39)

2 G p_{si} = l_{f} c^{3} .

(40)

Substituting the upper-count gravitational expression G=c³t_f/m_f then gives

2 (c^{3} \frac{t_{f}}{m_{f}}) θ_{si} = l_{f} c^{3} .

(41)

l_{f} m_{f} = 2 θ_{si} t_{f} .

(42)

The reductions recover the fundamental relation established earlier and demonstrate algebraic consistency among the upper-count expressions. They supply no additional independent calibration and do not determine gravitational field dynamics from the scalar coupling alone.

3.4 CALIBRATION AND CURRENT COMPARISON

At the calibrated interaction count, n_L=84.6005456716(65) and 2Q_Ln_L=0.9999650728344806(54). Finite reduced Planck constant recovers the conventional exact input by calibration, while upper count reduced Planck constant is an output. CODATA determines its recommended gravitational constant separately from the other adjusted constants [2]; no MQ count assignment is inferred from that procedure.

Table 1 preserves the historical comparison with the 2010, 2014 and 2018 CODATA editions [16,17,18]. Its historical MQ rows are archival evaluations of the conventional Planck root expressions using the constants quoted in each row, and are not the current calibration. Their row labels are descriptive identifiers. Table 2 gives the current results and distinguishes the finite and upper count evaluations.

Table 1. Historical comparison preserved from the earlier analysis. The CODATA rows are historical recommendations; the historical MQ rows are archival evaluations and are not current validation results.

Year	ℏ (10⁻³⁴ J s)	l_P (10⁻³⁵ m)	t_P (10⁻⁴⁴ s)	m_P (10⁻⁸ kg)	G (10⁻¹¹ m³ kg⁻¹ s⁻²)
2010 ^a	1.054571726(47)	1.616199(97)	5.39106(32)	2.17651(13)	6.67384(80)
2014 ^a	1.054571800(13)	1.616229(38)	5.39116(13)	2.176470(51)	6.67408(31)
2018 ^a	1.054571817	1.616255(18)	5.391247(60)	2.176434(24)	6.67430(15)
MQ 1 b	1.0545349844(45)	1.6161716871(50)	5.3909684650(90)	2.1764705491(26)	6.6738448362(53)
MQ 2 b	1.0545349844(45)	1.6161999120(35)	5.3910626131(72)	2.1764325398(24)	6.6740779428(56)
MQ 3 b	1.0545718176(46)	1.6162281374(12)	5.3911567628(97)	2.1764705491(26)	6.6740779428(56)
MQ 4 b	1.0545718176(46)	1.6161999120(35)	5.3910626131(72)	2.1765085590(91)	6.6738448362(53)

^a CODATA publications for 2010, 2014 and 2018 [16,17,18].

^b Archival MQ values retained for provenance; their earlier input covariance is not reconstructed here. Table 2 supplies the current evaluation and uncertainty treatment.

Table 2. Current calibration and conventional comparison using the 2022 CODATA edition. Length, time and mass entries are Planck-root evaluations with the constants in the same row. At the upper count limit they are the fundamental measures; the finite count mass root differs from fundamental mass.

Evaluation	Reduced Planck constant 10⁻³⁴ J s	Length 10⁻³⁵ m	Time 10⁻⁴⁴ s	Mass 10⁻⁸ kg	Gravitational coupling 10⁻¹¹ m³ kg⁻¹ s⁻²
CODATA 2022	1.054571817… (exact)	1.616255(18)	5.391247(60)	2.176434(24)	6.67430(15)
MQ upper count	1.0545349844417293(56)	1.61619991148(12)	5.39106261131(41)	2.17643254057(17)	6.6740779383(10)
MQ interaction count	1.054571817… (exact closure)	1.61619991148(12)	5.39106261131(41)	2.17650855984(17)	6.6738448317(10)

Parentheses give standard uncertainty in the final digits. The exact reduced Planck constant is h/(2π); the ellipsis is decimal truncation. MQ uncertainties propagate the single adjusted fine-structure input with shared covariance, conditional on fixed realization assumptions. The interaction count satisfies n_Lθ_si=276. Conventional recommendations are quoted from [2], including its published Planck time rather than a recomputation from rounded gravitational coupling.

The current upper count gravitational output is 1.48 combined standard uncertainties below the 2022 recommendation; the interaction-count output is 3.03 below it. These comparisons use the separately adjusted CODATA gravitational uncertainty [2] and exclude MQ model uncertainty. CODATA reports substantial inconsistency among the gravitational measurements and retains expanded uncertainties. The offsets therefore are numerical comparisons, not significance tests establishing a count regime or rejecting a physical theory. The conventional Planck length uncertainty is governed chiefly by the gravitational recommendation; the smaller MQ propagated uncertainty remains conditional on its calibration assumptions.

The 2026 redetermination using the torsion balance of the International Bureau of Weights and Measures (BIPM) at the National Institute of Standards and Technology (NIST) reports a lower servo-mode gravitational coupling than the free-deflection result for both copper and sapphire test masses [21], Tables 16–18. Free deflection infers gravitational torque from the torsional response, whereas servo operation balances it with an electrostatic torque. For copper, the published values are

6.674021 \times 10^{- 11} m^{3} k g^{- 1} s^{- 2}

in free deflection and

6.673642 \times 10^{- 11} m^{3} k g^{- 1} s^{- 2}

in servo operation. Their servo-normalized difference is 56.8 ± 29.4 parts per million, using the published relative standard uncertainties and correlation 0.42.

The proposed apparatus-specific MQ interpretation assigns the free-deflection result to the upper count gravitational evaluation and the servo result to the electromagnetic interaction-count evaluation. The retained relations give a servo-normalized separation 10⁶(G/G-1)=34.9284 parts per million. This lies 0.74 experimental standard uncertainties from the copper difference. The regime distinction predates the 2026 result [5], Appx. N; the apparatus-specific mode assignment is retrospective. This constitutes quantitative concordance with the proposed count-regime interpretation, while leaving competing instrumental explanations open. The experimental authors report that the cause of the recurring mode offset remains unknown [21], Section 8. The comparison does not establish an independently measured momentum normalization or a universal count assignment for every gravitational apparatus.

4. DISCUSSION

The retained MQ equations relate the fundamental measures to the Planck root expressions. The finite factor changes the mass root, while the length and time roots recover the calibrated fundamental measures. The formal upper count limit removes the mass factor. These are conditional algebraic conclusions, not measurements of a minimum physical length or time.

The realization distinctions preserve the different dimensions of angle and momentum. The optical comparison remains a specified theoretical branch, and recovery of reduced Planck constant is a dependent calibration identity. The NIST/BIPM comparison supplies physical evidence consistent with the proposed count-regime interpretation. Its quantitative agreement is evaluated separately from the CODATA calibration. The experiment does not uniquely identify the cause of the mode difference or independently measure the dimensional angle-to-momentum conversion.

The count condition and selected nearest-count branch remain assumptions of this evaluation. The broader charge realization construction describes an integer basis, but a general uniqueness theorem for that selection remains open. This paper neither proves that theorem nor promotes a scalar coupling identity to common gravitational and electromagnetic field dynamics. Its uncertainty budget covers shared-input propagation; realization and model uncertainties are not quantified.

5. APPENDICES

Appendix A. NOTATION INDEX

This index points back to notation introduced in the text. MQ terminology follows [5]; entries do not add new physical assumptions.

The coefficient, its two realizations and their conversion factor are introduced before Eq. (1). The qualified optical and conserved-quantity contexts are given in the Introduction.

The speed of light is introduced in the Introduction; its exact SI value is 299792458 m s⁻¹ [2].

The upper count gravitational output in Eq. (31) is
$G = 6.6740779383 (10) \times 10^{- 11} m^{3} k g^{- 1} s^{- 2} .$
The interaction-count output follows Eq. (33) and Table 2. Upright upper count notation and italic finite count notation denote distinct evaluations.

The fundamental length, mass and time are introduced before Eq. (1) and evaluated consistently in Eqs. (27), (29), (30).

The conventional Planck scales are introduced in the Introduction and compared with the fundamental measures in Eqs. (34)–(38).

The length count is introduced in the Introduction. If mass and time counts are required in the broader MQ notation [5], n_M and n_T denote dimensionless counts of fundamental masses and times respectively; they are not additional inputs to this calculation.

The length count refers to the specified radial separation in fundamental lengths; the calibrated interaction value is reported in Section 3.4.

The radial separation is introduced at the start of the Results and uses the same length count as Eq. (4).

For broader MQ count notation [5], n_Lc denotes the dimensionless count of fundamental lengths corresponding to a change in the realized position of light relative to the observer. It is distinct from the calibrated interaction count and is not an additional input to this calculation.

The nearest whole count operation uses mixed floor and ceiling brackets in Eqs. (22)–(25); no separate count symbol is required.

The bracket operation and its selected branch are introduced with the fine-structure calculation.

The dimensionless count residual is introduced in the Introduction and written explicitly in Eq. (18).

The finite normalization follows the residual and count, and its free count limit is described in Section 3.3.

The conventional reduced Planck input is exact. The upper count output is
$ħ = 1.0545349844417293 (56) \times 10^{- 34} kg m^{2} s^{- 1};$
the interaction-count recovery of the conventional input is a calibration identity.

The electron mass is first used in the calibration preceding Eq. (27).

The Bohr radius is first used in the same calibration paragraph.

The classical and MQ fine-structure forms are introduced in Section 3.2; their numerical identification is an explicit calibration assumption.

All reported MQ uncertainties are conditional on the specified realization and count assumptions.

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 introductory publication [5] describes the Online Resources and supplementary spreadsheets. For this calculation, the reproducibility inputs are the stated 2022 calibration, count condition and nearest-count branch. No claim is made that every expression in the broader supplementary collection has been independently audited.

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