Discrete Measurement Relations and Three Spatial Dimensions
ABSTRACT
We examine how Measurement Quantization (MQ) relates fundamental measures to gravitational coupling, normalized time counts and a cosmological mass-count rate. The symbols lf, mf and tf denote fundamental length, mass and time, with units m, kg and s; upright G denotes the upper-count gravitational constant in m³ kg⁻¹ s⁻². Its expression contains a cubic length-to-time ratio and a mass-to-time normalization. The remaining examples contain dimensionless cube and cube-root factors after the momentum realization is included. These identities provide a consistent way to compare the stated MQ relations with a description using three spatial dimensions. Their powers do not by themselves establish that other spatial dimensions are impossible. A physical dimensionality conclusion requires assumptions about the realization of geometry and observational evidence beyond the algebraic factorization.
Keywords: Dimensions, Space, Three, Multidimensional, Spatial, Discrete
1. INTRODUCTION
Models with additional spatial dimensions motivate examining which aspects of a description are constrained by measurement [1]. We use the MQ relations [2–8] to separate dimensionless counts from the physical units attached to their realizations. The Internal Frame is the discrete configuration domain in which relational structure is encoded by integer counts of the fundamental units, without an intrinsic continuous geometry. The System Frame is the continuous relational encoding of that structure. The Frames mapping is the kinematic transformation between them, preserving realization consistency and closure conditions. This distinction provides the context for the gravitational, epoch and mass-count examples below; it does not identify either frame with an independent observer or a second physical universe.
The gravitational relation ([2], Appx. F), the mass-count relation ([2], Appx. BE) and the unity expression ([2], Appx. AX) use the fundamental measures introduced in the abstract ([2], Appx. W; [9]). The dimensionless counts nL, nM and nT refer to those measures, and nLc is the length count for light. The speed of light is c. The dimensionless invariant coefficient is θsi; its momentum realization psi carries kg m s⁻¹ and satisfies the fundamental relation
. Equal numerical coefficients in a specified realization do not equate their physical dimensions. The symbol Macr in Eqs. (2) and (29) denotes a dimensionless mass-count rate. Multiplying it by mf/tf produces a mass rate in kg s⁻¹. These definitions distinguish the normalized expressions in Eqs. (2) and (3) from dimensional powers of the fundamental measures.
For comparisons over a common elapsed interval, β is the dimensionless speed ratio in Eq. (4), and v is speed in m s⁻¹.
Equation (1) gives the upper-count gravitational identity, while Eqs. (2) and (3) contain normalized count factors. The fundamental relation reduces the ratio inside the cube in Eq. (2) to twice the dimensionless coefficient. In Eq. (3), the momentum realization makes the first term dimensionless. The Frames mapping supplies the connection to realized observables. Its historical framework context is discussed in Measurement Quantization ([2], Appx. K). The corresponding positive cube-root factor is displayed next.
Equation (5) is the positive normalized time-count ratio used in the epoch construction below. Its cube-root power concerns a dimensionless ratio. We first examine the separate dimensional identity for gravity.
Equation (1) combines a speed factor with an inverse mass-rate factor. The historical labels length frequency and mass frequency refer here to lf/tf in m s⁻¹ and mf/tf in kg s⁻¹, respectively. Neither has ordinary frequency units of s⁻¹. The two factors together give the gravitational units; their algebraic powers do not count independent spatial measurements.
The Informativity differential is the count-dependent, non-Lorentz length contraction associated with the discrete Internal Frame structure and its System Frame realization through the Frames mapping ([2], Appx. D). We use QL for the dimensionless fractional length-count residual in the radial construction. The count geometry must be distinguished from the physical units assigned to its realization. Establishing the number of spatial dimensions from a contraction effect would additionally require a specified physical realization and an observational comparison.
2. METHODS
We examine classical gravitational coupling, escape velocity and time-count expressions written in MQ notation. The fundamental-measure construction and its proposed quantum-measurement context include Heisenberg’s uncertainty principle [10]; MQ applications are discussed in [2–8]. The checks in this paper concern algebra, physical units and the assumptions linking counts to realized quantities.
This paper adopts the MQ count assumptions and examines their internal consequences. Measurement Quantization [2] presents the motivating derivations and proposed physical support. Those arguments do not make the algebra below an independent experimental demonstration of measurement discreteness. The referenced appendices and further definitions are available with that work, and a preprint is listed in Online Resources.
https://www.researchgate.net/publication/361148724_OnlineResources
The prior MQ presentation describes two supplementary calculation spreadsheets. Suppl. 1 covers the introductory paper and its appendices; Suppl. 2 covers applications of MQ. These resources document the proposed calculations. Their existence and agreement with the stated identities do not supply an independent observational test of spatial dimensionality.
3. RESULTS
1. Calculations
The results apply the assumptions identified in Methods to the three relations. Their dimensional and count factors are interpreted separately in the calculations that follow.
We begin with the upper-count gravitational relation ([2], Appx. F). In Eq. (6), psi supplies the dimensional momentum factor. The count-expanded intermediate equality uses nL=nT=1 and nM=1/2; it is not asserted for arbitrary counts.
Substitution of the fundamental relation and c=lf/tf makes the first and last forms of Eq. (6) agree. The reference counts simplify its intermediate representation, without turning a dimensional fundamental time into the number one. The physical status of the escape-speed route requires its separate assumptions below.
For the escape-velocity expression, ve is escape speed in m s⁻¹, M is source mass in kg and R is radial separation in m. Equation (7) is the Newtonian isolated, spherical-source expression with a fixed gravitational coupling; it neglects relativistic corrections and external forces.
For the reference counts nL=nT=1 and nM=1/2 ([2], Appx. I), substitute the count-resolved speed into Eq. (7), square it and solve for the gravitational constant. This reference choice assigns the light speed to the escape-speed variable. At that boundary, the Newtonian expression is a formal algebraic extrapolation, not a reliable relativistic escape-speed prediction or a measurement of the minimum counts.
Equations (8) and (9) recover the same upper-count identity under those reference assumptions. Escape velocity is an additional route to the dimensional relation, rather than an independent measurement of a spatial-dimension count.
The next example combines a time-count ratio with the MQ relation between encoded epoch scales. The cube-root factor follows from that epoch normalization. It must not be read as an unnormalized cube root of dimensional time divided by length and mass.
Measurement Quantization ([2], Appx. AU) writes a special-relativistic clock-rate factor in counts of a common fundamental time. The conventional inertial-clock relation follows from Einstein’s Lorentz transformation [11, Sec. 4]. Its use here requires the same reference interval for the spatial and light-path counts. It does not independently derive the cosmological epoch-volume assumption introduced below.
The dimensionless counts nTo and nTl denote observed and local elapsed durations in units of tf. For the inertial-clock interpretation of Eqs. (10) and (11), the numerator counts ticks of the moving clock and the denominator counts the corresponding reference-frame interval. The positive root applies for a speed below c, with the endpoint used only as an algebraic limit. Dividing Eq. (10) by the nonzero local count gives Eq. (11).
The epoch construction ([2], Appx. BK) compares volume factors built from the expansionary-epoch scale Aee and the SI-equivalent quantum-epoch scale Aqe, expressed in the same time units. The quantum epoch is the pre-referenceable count regime before a valid System Frame exists; its scale is encoded relative to the later realized System Frame, rather than elapsed physical time within that regime. The expansionary scale has realized elapsed-time meaning and is not the present cosmic age. Equation (12) adopts the MQ volume ratio; cancellation of its common factors gives Eq. (13). The subsequent relations are conditional on this assumption and do not independently determine an absolute epoch scale. In the volume formula, π is the dimensionless circular constant.
Identifying the reciprocal encoded epoch-scale ratio from Eq. (13) with the clock-count ratio in Eq. (11) is an additional MQ modeling assumption. Under that identification, Eq. (15) is the epoch-derived count ratio. Equating the nonnegative expressions in Eqs. (11) and (15) gives Eq. (16), and squaring gives the normalized unity relation in Eq. (17).
In Eq. (17), the fundamental relation makes the first ratio
. Both summands are therefore dimensionless. A real count ratio requires the coefficient to be at least one half. The cube-root dependence follows from the epoch-volume construction and its normalization; it does not establish that a dimensional length has been independently measured to have a unique cubic relation to mass and time.
For the mass-count example, we return to the same dimensional fundamental relation ([2], Appx. V). Its momentum realization must be retained before cosmological counts or physical units are introduced.
The quantity DU is the cosmic diameter in the MQ count construction, and AU is its elapsed cosmic duration ([2], Appx. AY, AZ). Equation (19) states the dimensionally complete model relation, with c converting duration to length. The relation is not an independent determination of the observed particle-horizon diameter; no numerical cosmic age is required for the mass cancellation in Eq. (20).
In Eq. (20), the explicit momentum normalization converts the cosmic count ratio into a mass, and substitution of Eq. (19) removes the cosmic age. The calculated fundamental mass is 2.17643254057(17) × 10⁻⁸ kg. Its uncertainty propagates the 2022 inverse fine-structure recommendation of the Committee on Data for Science and Technology (CODATA) [12] through the selected MQ branch and the convention assigning the same numerical coefficient to the International System of Units (SI) momentum realization. It includes no independently evaluated uncertainty for that realization convention or for the MQ model. This value is not the CODATA Planck-mass recommendation or an independent cosmological measurement. The domain calculation below additionally assumes its stated fractions and normalization.
The following calculation uses the MQ domain relations ([2], Appx. BB–BE). Planck’s cosmological parameters are inferred within specified cosmological models from temperature, polarization and lensing data [13]. The alternative-model discussion in [14] and the inflationary account in [15] provide context. The algebra here computes conditional MQ fractions; it neither fits those data nor demonstrates observational agreement with their spectra.
Within the adopted MQ domain model ([2], Appx. BB), the dimensionless fractions are Ωvis for the visible domain, presently referenceable, Ωobs for the observable domain, including the presently and ultimately referenceable portions, and Ωdk for the complementary dark domain, permanently non-referenceable in that construction. The unobserved domain fraction Ωuobs is the difference between the observable and visible fractions. These are model domain definitions, rather than observational classifications established by the calculations below.
The MQ applications [5, 7] propose comparisons between their unobserved and dark fractions and the conventional cold-dark-matter and cosmological-constant fractions, denoted Ωc and ΩΛ. Conventional density fractions in the Planck analysis [13] describe energy components within a fitted cosmological model. Their physical meanings are not supplied by the MQ visibility definitions. A valid identification requires a common normalization and predictions for the observables used in that analysis.
The fundamental-domain mass allocation is Mf=nTumfθsi ([2], Appx. BC). Here nTu=AU/tf is the dimensionless cosmic duration count. The allocation Mf carries kg and is distinct from the fundamental reference mass mf. The symbol Ωf denotes the dimensionless fundamental-domain invariant used to normalize the observable and dark fractions, not an additional additive matter or energy component. Equation (21) relates these quantities using the stated MQ domain relation.
The total-domain fraction is Ωtot=Ωdk+Ωobs. The relation Ωobs=2θsiΩvis ([2], Appx. BB) uses a dimensionless coefficient, rather than a Hubble rate in units per megaparsec. Substituting these relations gives Eq. (22). Equations (23)–(25) are rearrangements of the same domain identity; their denominators must remain nonzero. Equation (26) additionally uses the total-domain normalization to one.
The total mass is Mtot in kg. Apply the factor in Eq. (26) to the mass scale already defined and substitute the observable fraction
([2], Appx. BB). This gives Eq. (27) ([2], Appx. BE). Simplifying its bracket gives Eq. (28), rather than a new independent cosmological constraint.
The dimensionless total-mass count nMu is defined by Mtot=nMumf. Divide Eq. (28) by the nonzero cosmic time count and the fundamental mass to obtain Eq. (29). The result is the dimensionless count rate introduced in the Introduction. Multiplication by mf/tf gives a physical mass rate in kg s⁻¹. These count and dimensional rates must remain distinct.
Equation (30) is the same count rate written using the fundamental relation. The ratio inside its cube is dimensionless because the momentum realization is retained. The unnormalized cube of lfmf/tf would instead carry momentum-cubed units. The accepted expression establishes the normalized MQ count relation; it does not by itself predict how an alternative higher-dimensional physical model would accrete mass.
4. DISCUSSION
The three examples contain cubic powers for different reasons. Gravitational coupling has dimensional speed and mass-rate factors, the epoch relation has a normalized volume ratio, and the mass-count rate has a dimensionless coefficient. Equation (31) retains the gravitational factors to make the units explicit.
Cancelling the common time powers in Eq. (31) recovers Eq. (1), with the same gravitational units. This is a rearrangement of the original identity, rather than a fourth independent constraint.
The identities are consistent with the adopted three-dimensional MQ construction under its reference-count, epoch and domain assumptions. The Newtonian escape-speed relation and the spherical-volume construction already use three-dimensional geometry. Consequently, their cubic factors cannot serve as an independent proof of that geometry or exclude alternative dimensional models. Such exclusion would require predictions that distinguish the alternatives and a measurement comparison not used to fix the same input coefficient.
The Informativity differential supplies a separate count-dependent geometric consideration. Relating its realized length contraction to spatial dimensionality requires specifying the geometry and the Frames mapping and comparing the resulting effect with observation. The present dimensional and count identities supply that construction with consistency conditions; they do not replace the additional physical test.
5. APPENDICES
APPENDIX A: TERMS
The historical terminology context is given in Measurement Quantization [2]. The notes below preserve the representation distinctions introduced earlier and identify the roles of the numerical annotations.
The angular realization Θsi carries rad. Shwartz and Harris calculate signal and idler angles of 3.26239 rad for one frequency-degenerate Bell-state configuration in diamond [16, Table I]. That value is a phase-matching solution, not a measured MQ invariant with a ±2 μrad uncertainty. The related experiments study X-ray down-conversion coincidence rates [17] and X-ray/optical sum-frequency generation [18]; neither establishes that uncertainty for the MQ angular realization. The coefficient and momentum realization introduced above have calculated numerical coefficients 3.26239030504(25) in their respective realizations, using the 2022 CODATA inverse fine-structure input [12]. Parentheses give the propagated standard input uncertainty in the final digits, conditional on the selected MQ branch and realization convention. These values are correlated calculations from the same input, and their dimensions remain distinct.
The vacuum speed of light is exactly 299,792,458 m s⁻¹ [19, 12]. In the MQ reference construction, it can be written as c=nLlf/(nTtf). Selecting nL=nT=1 fixes the reference ratio; the numerical value of c alone does not prove those counts are unique or physically minimal.
Upright G denotes the upper-count realization used in the gravitational identities; italic G denotes a finite-count realization, including evaluation at the electromagnetic demarcation. This demarcation is the fixed interaction count distance at which electromagnetic realization is evaluated under the Frames mapping. The earlier numerical calibrations are omitted here because their length and time calibration and uncertainty chain are not established by the present calculation. The symbolic identities remain conditional MQ relations and do not establish agreement with the conventional CODATA gravitational constant.
The fundamental measures and their units are introduced in the abstract. Their MQ reference construction and relation to conventional Planck units are discussed in [2].
lp, mp and tp are Planck’s reference units for length, mass and time, with SI units m, kg and s, respectively.
The dimensionless measure counts introduced in the Introduction retain the same meanings throughout the paper.
For positional changes, nL is evaluated relative to the observer’s reference frame.
For radial separation, nL specifies the length count from the center of mass.
For the light-path uses, the count already denoted by nLc supplies the corresponding length reference.
At the electromagnetic demarcation, the dimensionless System Frame count nθ is resolved from the Internal Frame length count nL using
. The mixed brackets denote rounding to the nearest whole-unit value, distinct from ordinary floor and ceiling. The recalculated argument is 42.3002728358, away from a half-unit boundary.
The radial residual introduced in the Introduction retains its dimensionless fractional-count meaning.
The product QLnL enters the count-dependent normalization associated with the Informativity differential. It must be distinguished from a standalone dimensional length contraction.
The conventional reduced Planck constant has the decimal approximation 1.054571817 × 10⁻³⁴ J s [19, 12]; its SI value is exact through the defined Planck constant. MQ distinguishes a finite-count reduced Planck realization from its upper-count limit, whose symbol is upright. No calibrated upper-count numerical action value or uncertainty is asserted here.
me is the electron mass, expressed in kg.
a0 is the Bohr radius in m, a length scale for the hydrogen ground state [12].
The dimensionless quantities αf-1, αp-1 and αc-1 are the MQ fundamental, Planck-like and classical inverse fine-structure forms. The fundamental form αf-1=42θsi is defined in the Internal Frame; the other two are realized in the System Frame. The Planck-like form is the intermediate continuous representation that includes the discrete–continuous representational mismatch. The classical form additionally includes the finite-count normalization associated with the Informativity differential described earlier. They are different MQ representation forms, rather than three independent measurements of the conventional fine-structure constant.
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
This paper uses published sources and model calculations and reports no new experimental measurements. The defining equations and input sources are given in the text.
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