Discrete Approach to the Contraction and Dilation of Measure Relative to an Inertial Frame

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We present a count representation of contraction and dilation associated with motion in Measurement Quantization (MQ), with an explicit Special Relativity (SR) interpretation. Lorentz-compatible kinematics supplies the interior square-root factor; the MQ construction expresses it through fundamental measures and bounded counts. A fixed-window clock protocol preserves the multiplying time factor, while the mass parameter represents total energy divided by the square of light speed. The reciprocal mass step includes the local length and mass normalization rather than presuming every object has the fundamental length–mass product. The count calculation uses a common elapsed interval and a real, nonzero-factor domain. The separate Informativity differential remains a non-Lorentz finite-count effect. Numerical context uses the 2022 Committee on Data for Science and Technology (CODATA) adjustment under the stated MQ realization convention, with conditional source uncertainties.

Keywords: Contraction, Dilation, Measure, Motion, Special Relativity, Pythagorean Theorem

1. INTRODUCTION

We examine contraction and dilation through relations between an observer and an observable. An inertial frame is a reference frame in which a free body moves with constant velocity. Einstein’s special-relativistic treatment provides the comparison [1]. The MQ construction relates measurement bounds to the fundamental count frequency associated with expanding-frame geometry ([2], Appx. O, W, AX). In MQ, the fundamental reference measures are derived geometric properties of expanding frames. Their frequency sets the rate bound on changes counted in those measures, providing the limiting-count basis for the relativistic relations developed below. The motion calculation additionally uses a normalized Pythagorean count geometry. Its assignments of time and mass parameters require a measurement prescription, separate from the algebra that follows.

The Internal Frame is the discrete count-configuration domain. The System Frame is its observable continuous relational encoding, induced by the Frames mapping, which relates the discrete configurations to their continuous realization ([2], Appx. K, W). These domains distinguish count configurations from realized observables; they are not an additional universal inertial rest frame. Observer and observable comparisons require their own measurement prescription.

The fundamental reference measures l_f, m_f and t_f have units m, kg and s. Their dimensionless length, mass and time counts are n_L, n_M and n_T ([2], Appx. K, W). The reciprocal of fundamental time specifies the fundamental count frequency in s⁻¹ ([2], Appx. O). This rate must be distinguished from the dimensionless elapsed-time count used here. For the motion comparison, the specialization n_Lm denotes change in realized position of the observed mass, and n_Lc denotes the corresponding light count over the same elapsed interval. Let v and c be the motion speed and the speed of light in vacuum, in m s⁻¹. Then v=n_Lml_f/(n_Tt_f) and c=n_Lcl_f/(n_Tt_f). The reference convention c=l_f/t_f requires n_Lc=n_T.

The Informativity differential ([2], Appx. D) denotes the non-Lorentz contraction in realized separation associated with normalization of conserved count under the Frames mapping. Its finite-count role in the MQ comparison with conventional Planck units [3] is discussed in [2], Appx. X, and Supplement 1. It is distinct from the motion factors developed here.

We call the combination of count notation and realization through these frame domains the MQ approach. Measurement Quantization [2] and its related applications [4,5,6,7,8,9] develop that approach using light speed, escape-speed relations and quantum uncertainty [10] as input context. The present motion calculation tests the consequences of its stated count geometry. These input relations alone do not supply an independent derivation of the clock and mass identifications used below.

Equations (1)–(3) collect the count forms. The subscript l denotes a local measure within the observable’s framework, while o denotes an observed measure within the observer’s framework. These measurement roles do not rename the Internal Frame and System Frame. Thus n_Lo and n_Ll are length counts, n_To and n_Tl are clock-advance counts under the fixed-window protocol specified with Eq. (22), and n_Mo and n_Ml are assigned mass counts. Their corresponding measures are l_o, l_l, t_o, t_l, m_o and m_l, in m, s and kg as appropriate. The reference counts are positive and the common domain is 0≤n_Lm<n_Lc. Integer input configurations belong to the Internal Frame; their continuous System Frame realization does not require independently integer transformed outputs. A separate minimum-output condition, when imposed, gives the conditional finite-reference bounds discussed with Eq. (17).

Lengths, times and masses are converted with their respective reference measures. Multiplying Eq. (1) by l_f gives the length form developed in Eqs. (18) and (21). The corresponding assigned time and mass forms use t_f and m_f. This conversion preserves units while the common reference factors cancel from the ratios. MQ anchors the rest-mass reference to an observer and observable that are not experiencing acceleration. Absence of acceleration does not itself imply zero relative velocity in SR; two inertial bodies can move relative to one another [1]. A comoving comparison, with zero relative velocity, identifies the local rest reference used for that SR comparison. MQ constructs the observed/local relation through maximum and minimum counts of the fundamental measures; the observed mass parameter remains distinct from invariant rest mass. In SR, the longitudinal length comparison uses simultaneous endpoint positions in the inertial reference frame and a local rest length [1]. Matching its scalar factor therefore requires this measurement prescription and does not derive the full Lorentz transformation.

2. METHODS

We evaluate the trial product normalization in Eqs. (4)–(8), compare its two endpoints with the divisor construction in Eqs. (9)–(11), and adopt the unit-normalized geometry in Eq. (12). Equations (13)–(17) give its count and speed forms. The assigned dimensional relations follow in Eqs. (18)–(23), while Eq. (24) records the separate residual construction. Algebraic consequences are evaluated within these model conditions.

Measurement Quantization [2] develops the count interpretation and its calibration comparisons. Those comparisons do not independently test the motion relations derived here. Appendix A specifies the current numerical context, using the 2022 CODATA inputs [11] and a fixed realization convention. The original supplementary-material location follows.

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

The published source [2] describes two Excel calculation supplements. Supplement 1 covers that paper, and Supplement 2 includes its supporting MQ applications. They are referenced as calculation resources; their existence does not establish an experimental test of Eqs. (1)–(24).

3. RESULTS

1. Calculations

The length counts introduced in Sec. 1 use the common reference through l_o=n_Lol_f and l_l=n_Lll_f. In Eqs. (4)–(5), a and b are dimensionless count-coordinate legs, and c is their hypotenuse count coordinate, locally distinct from light speed. The endpoint labels a_max, b_min and c_bound identify selected trial limits for those coordinates. Their positive dimensionless reference counts are n_max and n_min, and n_bound is the dimensionless bound returned by the respective construction. Equation (6) tests a product scaling of the two legs.

For the product construction, setting a=0 and b=n_min gives Eq. (7), while setting a=n_max and b=0 gives Eq. (8). The positive square roots return the squares of the respective scaling counts. Unequal scaling counts therefore give different endpoint bounds, so this product construction does not furnish the common unit bound sought here.

We next divide each leg by its associated nonzero reference count in Eq. (9). This is an alternative normalization, not an algebraic rearrangement of Eq. (6).

At the first endpoint, the first normalized leg vanishes and the second is one. At the second endpoint, those roles are reversed. Equations (10) and (11) consequently give the same unit bound. These checks establish endpoint agreement without determining the allowed interior configurations.

The endpoint tests establish the limiting values, while the interior relation requires a geometric or physical premise. We use n_bound=1 in Eq. (12) for the normalized right-triangle construction ([2], Appx. AU). In an ordinary Euclidean right triangle, dividing both perpendicular legs by its hypotenuse makes their squared sum one by the Pythagorean theorem. Separate division by arbitrary unequal scale counts does not preserve that theorem automatically. A physical route to the required normalization is Lorentz-compatible kinematics. Homogeneity, isotropy, inertial-frame reciprocity and composition of transformations constrain the velocity dependence; invariant vacuum light speed selects the Lorentz case [12]. Rearranging the invariant spacetime interval gives the right-triangle sum of squared motion and proper-time ratios [1,13]. The former is the speed ratio identified in Eq. (17); the latter has the same square-root factor as Eq. (22). Observer-simultaneous rod measurements yield that factor in Eq. (21) as well. Consequently the count relation has a physically supported interior throughout its strict subluminal domain when these SR measurement assignments are used. Frequency and endpoint bounds fix the count domain, but do not replace the Lorentz-compatibility premise.

Equation (13) substitutes n_max=n_Lc and n_min=n_Ll into Eq. (12). The first is the light count over the common elapsed interval; the second is the local length reference. These are the two limiting references of the motion construction. The local length reference is not thereby the universal lower count of one, and the light count is the upper motion reference for the specified interval. MQ’s general finite count bounds do not identify an unrestricted elapsed interval with this particular light-count reference.

Equation (14) replaces the coordinate legs with a=n_Lm and b=n_Lo. Isolating the squared length ratio gives Eq. (15). The nonnegative root in Eq. (16) then fixes the observed length count relative to its positive reference. At zero motion count the two length counts coincide; as the motion-to-light ratio approaches one, the scalar length factor tends to zero.

Equation (17) uses the speed definitions in Sec. 1. Its common elapsed interval cancels, making the count and speed forms equivalent. The light-speed count is fixed for the selected interval; the integer motion count may take any value from one to that count. Rest is the separate zero-motion limit. The fundamental frequency is a count rate in s⁻¹, so its numerical one-second bound must be distinguished from a dimensionless count accumulated over a specified interval. For an interval containing an integer number of fundamental ticks, the corresponding light-speed count is fixed by that tick count. The equal-count endpoint is a formal length/time limit; it is excluded from the reciprocal massive-object expression. Input motion counts belong to the Internal Frame, whereas the System Frame realization need not return independently integer length, time and mass outputs. If the additional requirement that both local and observed length counts be at least one is imposed on a finite local reference, Eq. (14) gives (n_Lm²)/(n_Lc²)≤1-(1)/(n_Ll²) and Eq. (23) gives m_o≤m_ln_Ll. These are conditional finite-reference bounds, not extra restrictions automatically imposed on every selected motion count or a universal mass ceiling.

Multiplying Eq. (16) by l_f gives Eq. (18). Dividing both length measures by c preserves the factor in a time-valued relation. The clock protocol specified below supplies its physical interpretation. The fundamental relation is l_fm_f=2p_sit_f ([2], Appx. V), where p_si is the momentum realization in kg m s⁻¹. For an arbitrary local reference, the local length–mass product also includes its length and mass counts. Multiplying the fundamental product by both local counts gives the common numerator in the revised Eq. (19). Its physical justification is the Lorentz transformation of the same unchanged object. The observer-simultaneous longitudinal length contracts, while total energy divided by the square of light speed increases by the reciprocal factor [1,13]. Their product therefore equals the local length times invariant mass. Here the local mass parameter is invariant mass and the observed parameter is that energy-equivalent quantity. Dividing their common product by the respective mass parameters gives Eq. (19); cancelling its nonzero numerator gives Eq. (20) and then Eq. (23). This normalization applies to arbitrary positive local counts. The earlier fundamental numerator alone is the special case in which the local length-count and mass-count product is one.

Equations (21)–(23) collect the dimensional relations. For Eq. (21), local length is measured in the object’s rest frame, while observed longitudinal length uses endpoint positions simultaneous in the observer’s inertial frame [1]. For Eq. (22), choose a fixed observer coordinate-time window and a clock moving uniformly through it. The local time parameter denotes the at-rest calibrated benchmark advance for a window of that duration; the observed time parameter denotes the moving clock’s accumulated proper-time advance reported in the observer’s frame. The benchmark uses the clock’s local calibration but is not its actual proper duration during motion. The two boundary events lie on the moving clock’s path and are assigned observer times by synchronized observer clocks. Einstein’s moving-clock result then gives exactly the multiplying factor in Eq. (22) [1, Sec. 4]. This is a clock-advance comparison over a common observer window. It preserves the local-reference and observed-report roles without treating Eq. (22) as the conversion from that moving clock’s actual local duration to the observer’s duration. The latter event-duration comparison has the reciprocal orientation. Signal travel-time and Doppler effects must be removed when extracting clock advance from received signals. Equation (23) uses the energy-equivalent mass parameter already specified with Eq. (19), not a change in invariant mass. All relations use the strict massive-object domain; the equal-count pole remains a formal boundary rather than a realized infinite mass.

4. DISCUSSION

The count and speed forms in Eq. (17) describe the same ratio under the stated reference convention. The countability constructions in [2], Appx. I, J, supply the MQ model context. The Informativity Institute’s physical-support and prediction inventories describe tests of the broader framework, including prospective finite-count comparisons [14,15]. They identify research directions rather than supplying a direct experimental test of the particular normalization and clock or mass assignments used here. Independently, muon-storage-ring lifetime measurements have tested the SR time-dilation factor against rest-lifetime measurements [16]. Optical spectroscopy of stored lithium ions also tests the velocity dependence of relativistic time dilation [17]. These experiments supply clock-based benchmarks. They do not by themselves identify the MQ clock and mass parameters or distinguish a count re-expression of the same velocity factor from SR.

The revised derivation supplies three distinct bridges. Lorentz-compatible interval geometry fixes the interior square-root dependence. A rest-calibrated benchmark and an observed clock advance over the same observer window fix the interpretation of Eq. (22). The invariant product of observer-simultaneous longitudinal length and total energy divided by the square of light speed supplies the reciprocal normalization in Eq. (19). These are consequences of established SR kinematics with the stated measurements [1,13,12]. They explain why the dimensional and count forms agree, rather than inferring every premise from agreement at the end. The current MQ presentation likewise treats the motion result as a count representation of relativistic dependence [18]. The clock experiments cited above support the shared velocity dependence but do not independently prove that the fundamental frequency or expanding-frame geometry entails Lorentz symmetry. No independent derivation of that implication has been established here. The Informativity differential remains a distinct non-Lorentz length contraction with its separate finite-count dependency in Eq. (24) [14,15].

The Informativity differential has the separate finite-count role introduced in Sec. 1 ([2], Appx. D). For its right-triangle residual construction, let n_L denote the positive radial count and Q_L the dimensionless mapping-induced residual count contribution. The defining relation is Q_L=√(1+n_L²)-n_L. Squaring Q_L+n_L=√(1+n_L²) and cancelling the radial square gives Eq. (24). This is a relation within the residual construction, rather than a replacement for the motion-speed factor.

For positive radial count the selected residual is positive. Rationalization gives Q_L=1/(√(1+n_L²)+n_L), an exact form that avoids subtraction of nearly equal quantities at large count. Equation (24) gives Q_Ln_L=(1-Q_L²)/2, so the product remains below one-half at every finite positive count. As the radial count increases, the residual tends to zero and the product tends to one-half. Its finite-count normalization must therefore be distinguished from its limiting value. Neither relation supplies a rule for adding the residual directly to a Lorentz contraction.

5. APPENDICES

APPENDIX A NUMERICAL AND NOTATION CONTEXT

The entries below supply numerical context and refer back to the notation already introduced. Measurement Quantization [2] gives a fuller discussion of its terms. The present calibration uses the 2022 adjustment [11], with exact International System of Units (SI) light speed and Planck’s constant and an uncertain inverse fine-structure input. Its quoted standard uncertainties are conditional on the fixed MQ realization convention; they exclude uncertainty in the physical validity of that convention. The base-term dependencies in [2], Appx. BM, provide the starting point for propagating uncertainty and covariance through the fundamental measures and the realized constants. Historical numerical inputs in that appendix are updated here to the stated 2022 calibration, and its notation follows current MQKB501 definitions. A complete experimental uncertainty budget must also address the measurement model and relevant influence quantities [19]. The present source-input propagation therefore cannot be interpreted as a numerical bound on model discrepancy or as a probability that the MQ interpretation is correct.

• θ_si is the dimensionless invariant coefficient. Θ_si is its angular realization, with units rad, whereas the momentum realization defined before Eq. (19) has units kg m s⁻¹. The specified calibration gives θ_si=3.26239030504(25) and p_si=3.26239030504(25) kg m s^-1 [11]. The same numerical coefficient is used under the realization convention without equating physical dimensions. Parenthetical digits denote one standard uncertainty in the final quoted digits. Shwartz and Harris [20] calculate polarization-entangled X-ray production and beam-angle solutions; Shwartz et al. [21] study down-conversion coincidence rates with loss; Glover et al. [22] demonstrate X-ray and optical sum-frequency generation. Chen et al. [23] discuss conserved quantities at future null infinity in general relativity (GR). These works do not report an independent measurement of the MQ coefficient or its momentum realization.
• The light-speed convention uses the exact SI value c=299792458 m s^-1 [24,11]. This value is unchanged between the cited 2018 and 2022 recommendations. The current calculation uses the latter adjustment; the common-interval convention is the one already established in Sec. 1.
• G, upright, denotes the upper-count gravitational coupling, calculated as G=6.6740779383(10)×10^-11 m³ kg^-1 s^-2. The electromagnetic demarcation is the fixed electromagnetic interaction count distance used in this MQ realization; Appendix A specifies its radial argument below. G, italic, denotes finite-count coupling; its electromagnetic-demarcation value is G=6.6738448317(10)×10^-11 m³ kg^-1 s^-2. The latter retains the residual normalization, while the former uses its upper-count limit. Both are conditional MQ calculations using 2022 inputs [11]. They are not CODATA recommendations or independent source measurements.
• Under the same calibration, the fundamental references used in the count conversion have calculated values l_f=1.61619991148(12)×10^-35 m, m_f=2.17643254057(17)×10^-8 kg and t_f=5.39106261131(41)×10^-44 s [11]. These outputs share the inverse fine-structure input and are not independent measurements.
• l_p, m_p and t_p are the conventional Planck reference length, mass and time, in m, kg and s. Their role differs from the MQ reference calibration above.
• The count notation in Sec. 1 distinguishes the integer configuration domain from its continuous realization. The scalar relations remain subject to the domain and admissibility conditions stated there.
• The motion specialization introduced in Sec. 1 remains distinct from the radial count used in Eq. (24).
• Radial separation uses the count notation introduced before Eq. (24).
• The light count retains the common-interval convention used in Eq. (17). It is distinct from both the motion count and the radial count in Eq. (24).
• n_θ is dimensionless and is the System Frame count of the invariant coefficient resolved from the Internal Frame demarcation after adjustment for the metric differential, the mismatch between discrete count structure and its continuous encoding, rather than a physical deformation in itself. In the fixed electromagnetic calibration, the radial argument is n_L=276/θ_si=84.6005456716(65). This noninteger calibration parameter is distinct from an assertion of an allowed integer motion configuration. The mixed brackets denote rounding to the nearest whole unit, giving n_θ=⌊(84.6005456716)/(2)⌉=42. Propagated source uncertainty stays well away from a half-unit tie [11].
• The mixed-bracket convention introduced in the preceding entry differs from ordinary floor or ceiling alone.
• The residual notation used in Eq. (24) retains the dimensionless meaning introduced in the Discussion.
• The residual-count product in Eq. (24) contributes to the count-dependent normalization associated with the Informativity differential; its limiting value must not be substituted as an exact finite-count value.
• ℏ, italic, denotes the finite-count reduced Planck quantity in the electromagnetic calibration, with units J s. The conventional SI input is exactly h/(2π), where h is Planck’s constant in J s and π is the dimensionless circular constant. The exact input h=6.62607015×10^-34 J s is shared with the current calibration [11]. ħ, upright, denotes its upper-count MQ quantity, calculated as ħ=1.0545349844417293(56)×10^-34 J s. The exact source input and the conditional upper-count output must not be conflated.
• m_e is the electron mass, in kg, and enters the electromagnetic reference calibration. a₀ is the Bohr radius, in m, and is the associated atomic length scale. Their product can be eliminated through the exact Bohr relation, so their adjusted uncertainties are not treated as independent source uncertainties here [11].
• The elimination described in the preceding entry leaves one uncertain inverse fine-structure source input for the reported conditional calibration. This common dependence must be retained when comparing its outputs. For two derived outputs, the product of their derivatives with respect to that shared input, multiplied by its variance, gives their first-order covariance. A calculation with several uncertain inputs retains their full covariance matrix. Algebraic cancellation of shared inputs precedes propagation, preventing double counting. These conditional numerical uncertainties do not supply an uncertainty for untested physical assumptions.
• α_f, α_p and α_c are dimensionless fine-structure forms that distinguish the fundamental Internal Frame form from the Planck-like and classical System Frame realizations. The inverse classical form differs from the inverse Planck-like form through the finite-count normalization. In this calibration the measured source input is α_c^-1=137.035999177(21) from the 2022 adjustment [11]. The MQ relations α_f^-1=42θ_si, α_p^-1=84θ_si-⌊42θ_si⌉ and α_c^-1=2Q_Ln_Lα_p^-1 complete the coefficient inversion using the fixed radial argument and Eq. (24). Reusing the source input in that inversion supplies no independent experimental test.
• The historical auxiliary electromagnetic parameter is not used in Eqs. (1)–(24).

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 numerical calibration in Appendix A is reproducible from the stated 2022 inputs and MQ relations. Supporting calculation resources are identified in Sec. 2 through the original Online Resources location. These resources are distinct from independent measurements of the motion factors.

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