Keywords: Dark Matter, Kinetic Energy, Star Velocity, Galactic Rotation, Milky Way

1. INTRODUCTION

The fixed mass-count prescription counts the fundamental time intervals in one second and assigns that dimensionless count to the bound mass count. The quantity 1/t_f itself remains a frequency in s⁻¹. Multiplication by the specified one-second interval gives the dimensionless value used below. This assignment is an MQ model prescription; it is not independently established by the velocity identity.

Our goal is to relate the adopted orbital-speed construction to classical kinetic energy ([6], Appx. AJ); the earlier physical discussion is retained in [8]. We also distinguish the crossing of the effective, modelled and bound velocities from a variational argument. No separate minimization is performed. The empirical application is assessed through the distinct galactic realization law, baryonic source and observational target used in the companion analysis [3].

MQ expands the nomenclature to describe measures using counts of fundamental units. The discreteness and realization rules are assumptions of this framework whose consequences must be tested; the use of count notation alone does not establish physical discreteness. Its construction uses the speed of light, gravitational speed relations and the uncertainty principle [9]. Related MQ applications are discussed in [6, 8, 10–14]. These papers supply the framework’s arguments rather than independent validation of every proposed consequence.

The earlier 1.394 km s⁻¹ quantity was a population standard deviation relative to McGaugh’s modified Newtonian dynamics (MOND) model, not a comparison against independent measured velocities. It is retained here only to identify the historical benchmark. The principal observational discussion below follows the source-separated construction in [3], and its uncertainty qualifications are retained.

The classical kinetic-energy identities and the count prescription provide an internal construction. Testing the physical prediction requires an independently reconstructed baryonic source and a separately determined circular-velocity target. Agreement in that setting would constrain the prediction under its source assumptions; it would not by itself establish a complete explanation of galaxy, cluster, lensing and cosmological evidence.

2. METHODS

This paper uses classical expressions to formulate the count construction and its relation to kinetic energy. We identify that construction as Measurement Quantization. The algebra is checked within the stated definitions and nonrelativistic domain; its physical interpretation remains subject to empirical tests.

The introductory Measurement Quantization paper [6] develops the discreteness assumptions, definitions and derivations on which this presentation depends. Its referenced appendices provide the framework context. An independent observational comparison requires source and target data beyond those internal relationships. The published paper identifies the accompanying Online Resources.

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

Suppl. 1 contains calculations for the introductory paper and its referenced appendices. Suppl. 2 includes the historical speed construction examined here. The companion dark-matter manuscript [3] supplies the newer galactic realization law and reported observational analysis. Its figure and numerical summaries are identified as source results; the underlying force and covariance calculations are not independently reconstructed from the manuscript alone.

3. RESULTS

1. Calculations

We begin by separating the angular and momentum realizations at the lower count bound. Their numerical coefficient is shared under the stated MQ realization convention; their physical dimensions remain distinct. The momentum realization is used to relate the fundamental measures and describe gravitational coupling ([6], Appx. C).

Equation (2) gives the angular realization. Its displayed value is an MQ calculation from the 2022 inverse fine-structure input of the Committee on Data for Science and Technology (CODATA) [7], conditional on the stated realization convention. Parenthetical digits throughout these revised calculations denote one standard uncertainty in the last quoted digits. The reduced Planck quantity has action units J s. Its quotient by twice a length is a momentum expression, as used in Eqs. (4) and (5), rather than an angle ([6], Appxs. P and Q).

Shwartz and Harris [15] give a theoretical construction for polarization-entangled X-ray photons. The parametric down-conversion and wave-mixing studies [16, 17] supply related optical contexts. These works do not independently establish the MQ coefficient as a universal measured angle or equate an angle with linear momentum. Upright ħ and G denote the reduced Planck quantity and gravitational coupling in the limit of large count separation. The electromagnetic demarcation is the fixed interaction count distance used for electromagnetic realization in MQ; the reduced Planck quantity at that demarcation is distinct. In Eq. (3), r is radial separation in m, Q_L is the dimensionless fractional length-count residual, and G is the finite-count coupling. The identity G=2Q_Ln_LG with r=n_Ll_f gives Q_Lrc³/G=l_fc³/(2G). The dimensionless factor in this identity is the Informativity differential, the count-dependent normalization associated with non-Lorentz length contraction under the MQ realization rules. Building on Planck’s historical unit construction [18], the upper-count fundamental-unit identity replaces c³/G by ħ/l_f² ([6], Appx. P).

The two derivations give the same momentum quotient ħ/(2l_f). Equation (6) assigns the angular and momentum realizations separately under the stipulated common numerical coefficient.

The fundamental relation in Eq. (7) follows from p_si=l_fm_f/(2t_f) ([6], Appx. V, Eq. (V.4)). It uses the dimensional momentum realization. The Internal Frame supplies the discrete count configurations, while the System Frame encodes their observable relational structure through the Frames mapping, the kinematic transformation between these descriptions.

Einstein’s historical writings are collected in [19]. We next retain his rest-energy starting expression [20] and resolve the energy E_f of one fundamental mass m_f ([6], Appx. Z). Here m is mass in kg and E is rest energy in J. Equation (8) uses the upper-count fundamental relation and m=n_Mm_f. The explicit fundamental-mass identity avoids treating finite-count and upper-count expressions as interchangeable.

Equation (10) uses v for the escape-speed parameter introduced as v_o in the introduction. This local use is distinct from the general particle speed in Eq. (16). Suppl. 2, tab 31 identifies escape speed, so the factor of two is retained under the root. Substituting R=n_Ll_f and M=n_Mm_f with the upper-count coupling gives the count form ([6], Appx. AF). Here the mass count refers to the source mass M and the positive length count to the radial separation R; in Eqs. (8) and (14), the mass count instead refers to the specified particle mass m. These are scoped count uses, not an identification of the source and particle masses. The numerical specializations below use a separate comparison count.

Equation (12) gives the unscaled comparison speed v_bc=62547.2624022(24) m s^-1 under that count-ratio configuration. Equation (13) multiplies it by the dimensionless coefficient under the adopted metric differential normalization ([6], Appx. T), with coefficient context in ([6], Appx. A). Here the metric differential is the discrete–continuous representation mismatch under the Frames mapping, distinct from the Informativity differential. The speed units come from c. This is the model construction, not an independent measurement of a universal speed bound.

Equation (16) uses v for the general particle speed in m s⁻¹ and gives the classical kinetic-energy relation using the energy and mass counts introduced above. This local use differs from the escape-speed parameter of Eq. (10). The counts remain independent unless the restricted configuration of Eq. (14) is imposed. Positive mass and nonnegative energy are required. The galactic application uses nonrelativistic speeds; the classical expression is not asserted as the exact relativistic kinetic energy.

Equation (17) is the classical kinetic-energy relation recovered by reversing Eq. (16). This algebra checks internal consistency; it does not independently validate the adopted orbital-speed prescription.

The fundamental rest energy sets the energy-count unit. It does not fix the kinetic energy of an arbitrary configuration.

2. Application to Star Velocities

We now examine the adopted fixed bound count and the crossing of the original speed relation, then distinguish the newer galactic realization law used for observational testing. Equation (18) gives the reciprocal fundamental time ([6], Appx. O).

Let n_Mb denote the fixed dimensionless bound mass count ([6], Appx. O). The adopted prescription identifies it with the number of fundamental time intervals in exactly one second, as stated in Eq. (19). This count is distinct from a variable mass count and from the reciprocal comparison count in Eq. (11). Equations (20) and (21) repeat the same comparison configuration as Eqs. (12) and (13). The one-second count definition fixes the normalization; it does not by itself prove that a physical mass above the assigned bound is unobservable.

Equation (21) gives the same normalized model bound speed as Eq. (13). The count coefficient supplies no units and does not replace the dimensional momentum realization used in energy expressions.

For a specified bound speed and radial separation, the mass parameter M_b, in kg, is defined by the circular-speed inversion in Eq. (22) ([6], Appx. AG). This is a mass profile at that radius, distinct from the fixed count prescription. It is not the escape-speed mass inversion in Eq. (10), which would require the additional factor of two in the denominator.

The mass parameter M_e, in kg, describes the adopted mass distribution in the earlier construction ([6], Appx. AH). Equation (23) states its speed relation ([6], Appx. AK). The observational prediction introduced next is a distinct galactic realization law; its baryonic source speed is not silently identified with the earlier modelled-speed variable.

Figure 1 reproduces the companion manuscript’s Milky Way comparison [3]. The circular-velocity target comes from Eilers et al. [5], whose Jeans analysis assumes an axisymmetric potential. The Newtonian source retains McMillan’s baryonic components [4], with the model Navarro–Frenk–White (NFW) halo force excluded before applying Eq. (24). The original McMillan fit includes a halo, so removing its force does not erase source-parameter covariance with the original fit assumptions. Equation (25) then gives the classical kinetic energy of a specified mass at the predicted speed.

The companion analysis [3], Sec. 4.3, reports 38 radial comparisons from 5.27 to 24.82 kpc and root-mean-square (RMS) residuals of 69.03 km s⁻¹ for the Newtonian baryonic curve and 8.90 km s⁻¹ for MQ. These are its reported source results rather than statistics reconstructed from the figure. Its diagonal-error analysis rejects the full sample, while propagation of a specified conditional covariance of the McMillan source normalizations gives a generalized chi-square of 39.57 for 38 degrees of freedom and a right-tail probability of 0.400. The inference is non-rejection under that covariance model. The unpublished Eilers point-to-point covariance and the full joint baryonic-source posterior remain limitations. No observational conclusion follows solely from rewriting kinetic energy.

The following algebra returns to the original speed relation in Eq. (23). Dividing by the positive bound speed squared and imposing equality of the effective and modelled speeds gives the repeated root in Eqs. (26)–(29). It is the point where the three speeds cross. No energy minimization is derived from this equality.

At the crossing in Eq. (29), division by the positive dimensionless coefficient gives Eq. (30). Equation (31) repeats the restricted reciprocal-count form. Neither step supplies an independent observational test or identifies the comparison count with the fixed bound mass count.

For Eq. (32), take m=n_Mbm_f and retain the independent energy count. Rearranging Eq. (33) and using Eq. (9) gives the stated relation. Its validity as a kinetic-energy identity does not determine the numerical value or time normalization of the bound mass count.

Classical kinetic energy can be expressed through independent energy and mass counts. The original speed relation has a repeated crossing at the adopted bound speed, and the one-second prescription fixes the numerical bound mass count within the model. The distinct galactic realization law maps an independently specified baryonic source to a predicted circular speed and then to kinetic energy through Eq. (25). The reported companion comparison is conditional on its source and covariance assumptions. These results establish neither a variational minimum nor a complete removal of the need to explain dark matter evidence beyond rotation curves.

4. DISCUSSION

A dimensionless count coefficient must be tracked separately from a dimensional fundamental measure, even when their SI numerical representations coincide. A one-second count definition specifies normalization without proving a physical limit on measurable mass. The crossing condition is an algebraic property of the original speed relation. The observational realization law requires a baryonic reconstruction, an independently specified acceleration scale and an uncertainty model. Its external evaluation must address source sensitivity and correlated uncertainties, and later tests must examine lensing, cluster dynamics and cosmological predictions [3].

5. APPENDICES

APPENDIX A: TERMS

For the published symbol and term glossary, see Measurement Quantization [6], Appxs. BP and BQ.

• The coefficient and its angular and momentum realizations are introduced before Eq. (1). The revised angular and momentum values appear in Eqs. (2) and (4). The earlier X-ray comparison [15–17] and conserved-quantity discussion ([22], Sec. 4.4) remain distinct source contexts; they do not identify an angle and linear momentum as one dimensional quantity.

• The speed of light is introduced before Eq. (1). The fundamental-unit identity is c=l_f/t_f; the more general count expression is c=n_Ll_f/(n_Tt_f) when the count ratio describes light propagation ([6], Appx. E, Eq. (E.1)). The exact SI value is confirmed in the 2018 and 2022 sources [23, 7].

• The gravitational forms used in Eqs. (3)–(5) are distinguished before Eq. (3). With the fixed 2022 inputs and stated MQ calibration, the upper-count calculation gives

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

  and the electromagnetic-demarcation calculation gives

  $$\mathit{G}=6.6738448317(10)\times {10}^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$$

  [7]. These are MQ calculations with a common input dependency, not two independent CODATA measurements.

• The conditional 2022 MQ fundamental-unit calculations are

  $${\mathit{l}}_{\mathit{f}}=1.61619991148(12)\times {10}^{-35}\mathrm{m},$$

  $${\mathit{m}}_{\mathit{f}}=2.17643254057(17)\times {10}^{-8}\mathrm{kg}$$

  and

  $${\mathit{t}}_{\mathit{f}}=5.39106261131(41)\times {10}^{-44}\mathrm{s}$$

  [7]. The local dimensionless coefficient in Eq. (11) belongs to the stipulated numerical realization and does not carry kg.

• l_p, m_p and t_p are Planck’s units for length, mass and time, in m, kg and s, respectively.

• The general counts, the reciprocal comparison count in Eq. (11), the energy count introduced before Eq. (14) and the fixed bound mass count in Eq. (19) have distinct roles. The frequency in Eq. (18) acquires a dimensionless count only after multiplication by the specified one-second interval.

• The radial count used in Eq. (10) is distinguished from a count of displacement along a trajectory.

• For the radial use in Eq. (10), the separation is R=n_Ll_f.

• n_Lc describes the dimensionless count of l_f representative of a change in position of light measured with respect to the observer’s frame of reference.

• n_θ is the dimensionless System Frame nearest-whole-unit count associated with the electromagnetic demarcation. The mixed brackets denote rounding to the nearest whole unit. The conditional 2022 argument gives n_θ=⌊84.6005456716/2⌉=42.

• The rounding convention in the preceding entry differs from ordinary floor or ceiling alone; the stated argument is away from a half-unit tie.

• The fractional count residual introduced before Eq. (3) enters the finite-count coupling there.

• The dimensionless factor 2Q_Ln_L evaluates the Informativity differential introduced before Eq. (3). Its finite-count role and upper-count limit are distinguished before Eq. (3).

• The upper-count reduced Planck quantity introduced before Eq. (3) has the conditional 2022 calculation

  $$\mathrm{ħ}=1.0545349844417293(56)\times {10}^{-34}\mathrm{J}\mathrm{s}.$$

  At the calibrated electromagnetic demarcation, the conventional reduced Planck input is ℏ=h/(2π) and is exact through the defined h [7]; its decimal begins

  $$1.054571817\times {10}^{-34}\mathrm{J}\mathrm{s}.$$

  Here h is the Planck constant, exactly

  $$6.62607015\times {10}^{-34}\mathrm{J}\mathrm{s},$$

  and π is the circular constant.

• m_e is the mass of an electron, in kg.

• a₀ is the Bohr radius used in the source calibration. The electron mass in the preceding entry and this radius are not treated as independent inputs in the uncertainty calculation. Their exact defining relation cancels them from the adopted single-source MQ calibration.

• α_f^-1, α_p^-1 and α_c^-1 are the fundamental, Planck-like and classical inverse fine-structure forms. The fundamental form

  $${\mathit{\alpha }}_{\mathit{f}}^{-1}=42{\mathit{\theta }}_{\mathit{s}\mathit{i}}$$

  is defined in the Internal Frame; the remaining forms use the System Frame. Their difference involves the Informativity differential. The classically realized 2022 source value used in the present recalculation is

  $${\mathit{\alpha }}_{\mathit{c}}^{-1}=137.035999177(21)$$

  [7].

• γ denotes the electromagnetic transformation parameter retained in the earlier auxiliary calibration context. It is not used in the reviewed equation 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

Suppl. 2 contains the historical comparison with McGaugh’s MOND model. The newer observational construction and Figure 1 are supplied by the companion manuscript [3], with published source and target methods in [4, 5]. The source’s force-integration products and covariance realizations are required to reproduce its reported statistics; the manuscript and figure alone do not provide those numerical products. The present paper distinguishes that reported analysis from calculations reproduced from the fixed 2022 input ledger.

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