Discrete Approach to the Contraction and Dilation of Measure with Respect to a Gravitational Mass

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We present a count-based model of relations between an observer and an observable in Measurement Quantization (MQ), with General Relativity (GR) providing the comparison context. The model adopts a normalized count geometry based on the Pythagorean theorem. Its equations give a common contraction factor for observed length and time and a reciprocal dilation factor for observed mass, relative to the local measures of the observable. Eliminating the shared speed parameter from the inertial and gravitational forms gives the count-only geometric equivalence used in MQ. The reference measures cancel from that relation. The model is applied on its real nonzero-factor domain with a minimum observable-count condition. Numerical calibration uses inputs from the 2022 adjustment of the Committee on Data of the International Science Council (CODATA) and distinguishes the dimensionless coefficient from its momentum realization. The results describe the adopted model; comparison with GR additionally requires the corresponding observer and measurement prescription.

Keywords: Discrete, Relativity, Gravity, Mass, Discrete Gravity, Gravitational Mass

1. INTRODUCTION

We describe contraction and dilation through relations between an observer and an observable. Einstein’s formulation of GR [1] supplies the comparison context, while the specific kinematic and static gravitational comparisons are addressed below [2,3]. In current MQ terminology, the Internal Frame is the discrete count-configuration domain. The System Frame is its observable continuous relational encoding. The Frames mapping is the kinematic transformation relating the integer configurations of the Internal Frame to the continuous relational structure of the System Frame. The equations below constitute the adopted model of these relations.

The fundamental measures $l_{f}$ , $m_{f}$ and $t_{f}$ have units m, kg and s, respectively. The dimensionless counts $n_{L}$ , $n_{M}$ and $n_{T}$ denote radial length, source mass and elapsed-time counts in the applications below. The corresponding dimensional quantities use these measures as their reference units. Conventional Planck units [4] provide a comparison, while the MQ construction includes the Informativity differential ([5], Appx. D). This term denotes the non-Lorentz contraction in realized separations induced by conserved discrete count under the Frames mapping. It is distinct from the motion and gravitational factors developed here.

The MQ calibration uses the speed of light, an escape-speed relation and quantum uncertainty. Heisenberg’s paper [6] concerns limits on simultaneous determination of conjugate quantities; it does not by itself establish discrete length or elapsed-time counts. The MQ interpretation and calibration are developed in Measurement Quantization ([5], Appx. I, W), with related applications in [7,8,9,10,11,12]. The present calculation uses the 2022 CODATA inputs specified in Appendix A. Agreement produced through those shared inputs is a calibration consistency check, rather than an independent experiment on these transformation factors.

Let $c$ be the speed of light in vacuum, in m s⁻¹. We use $n_{L m}$ for the dimensionless count associated with the change in realized position of the observed mass and $n_{L c}$ for the corresponding light count over the same elapsed interval. The speed parameter $v$ , in m s⁻¹, is then v=n_Lml_f/(n_Tt_f), while c=n_Lcl_f/(n_Tt_f). With c=l_f/t_f, the common-interval convention requires $n_{L c} = n_{T}$ . The common elapsed-time factor cancels from the speed ratio. This distinguishes motion and radial separation before the comparison with special relativity (SR) and the gravitational substitution.

Equations (1)–(3) state the gravitational forms developed below. The subscript $o$ denotes the measure observed after the contraction or dilation is accounted for, and $l$ denotes the measure in the local frame of the observable. Thus $l_{o}$ and $l_{l}$ are their length parameters in m, $t_{o}$ and $t_{l}$ their time parameters in s, and $m_{o}$ and $m_{l}$ their mass parameters in kg. These are the observed and local quantities related by the model. For a real nonzero factor and a finite reciprocal mass expression, the equations require $n_{L} > 2 n_{M}$ . The model is applied only to configurations satisfying this domain and the minimum observable-count condition discussed below.

l_{o} = l_{l} {(1 - 2 \frac{n_{M}}{n_{L}})}^{1 / 2} .

(1)

t_{o} = t_{l} {(1 - 2 \frac{n_{M}}{n_{L}})}^{1 / 2} .

(2)

m_{o} = m_{l} / {(1 - 2 \frac{n_{M}}{n_{L}})}^{1 / 2} .

(3)

The reference units and their realized geometrical relations must be distinguished. The observable relational structure belongs to the System Frame. A reference unit is not itself a spacetime metric, and this distinction does not exclude curvature in the realized geometry. The scalar factors below leave the continuous metric undetermined.

2. METHODS

We follow the count algebra in Eqs. (4)–(38). The normalized geometry, the observed/local mass and time relations, and the common speed parameter specify the adopted model. Appendix A records the numerical calibration and the 2022 input edition. The calculation derives the transformations within these stated model conditions.

Measurement Quantization [5] proposes discrete count configurations and compares its calculations with existing data. In this paper, that interpretation is the model context, not an independently established experimental premise for Eqs. (1)–(38). The cited comparisons do not directly measure the length, clock and assigned-mass transformations developed here. The published source contains the external appendices cited below, while Online Resources gives the original supplementary-material location.

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

The published source [5] describes two calculation spreadsheets. Supplement 1 covers the introductory paper and appendices, and Supplement 2 covers applications. The supplements are updated as the associated papers change. They are referenced here as maintained supplementary materials, rather than as a fixed dataset attached to this paper. The numerical results retained here are reproducible from the specified 2022 inputs and calibration relations in Appendix A. Earlier spreadsheet outputs are not independent source inputs for this calculation.

3. RESULTS

1. Calculations

In the length construction, the dimensionless count $n_{L o}$ corresponds to $l_{o}$ , and $n_{L l}$ is the local reference count corresponding to $l_{l}$ . Their conversion uses the common length reference, so $l_{o} = n_{L o} l_{f}$ and $l_{l} = n_{L l} l_{f}$ . The local reference count remains distinct from the radial count defined above. In Eqs. (4)–(5), $a$ and $b$ are dimensionless count-coordinate legs, and $c$ is the hypotenuse count coordinate, locally distinct from light speed. The labels $a_{\max}$ and $b_{\min}$ identify the selected endpoint legs, and $c_{bound}$ their hypotenuse. The nonzero positive counts $n_{\max}$ and $n_{\min}$ scale the trial construction, while $n_{bound}$ denotes its resulting bound. These coordinate and scaling roles lead to Eqs. (4)–(6).

a^{2} + b^{2} = c^{2},

(4)

a_{\max}^{2} + b_{\min}^{2} = c_{bound}^{2},

(5)

{(a n_{\max})}^{2} + {(b n_{\min})}^{2} = n_{bound}^{2} .

(6)

The product construction in Eq. (6) is a trial normalization. Setting $a = 0$ and $b = n_{\min}$ gives Eq. (7); setting $a = n_{\max}$ and $b = 0$ gives Eq. (8). The resulting endpoint values depend on the selected scaling counts, so the product does not give a common unit bound for unequal endpoints.

n_{bound} = {({(0 n_{\max})}^{2} + {(n_{\min} n_{\min})}^{2})}^{1/2} = n_{\min}^{2} .

(7)

n_{bound} = {({(n_{\max} n_{\max})}^{2} + {(0 n_{\min})}^{2})}^{1/2} = n_{\max}^{2} .

(8)

We next consider the divisor construction in Eq. (9). Each leg is normalized by its associated nonzero reference count.

{(\frac{a}{n_{\max}})}^{2} + {(\frac{b}{n_{\min}})}^{2} = n_{bound}^{2},

(9)

Equations (10) and (11) evaluate the two endpoints of the divisor construction. Both yield a unit bound exactly. These endpoint checks illustrate the normalization adopted for the model; they are not used to infer all interior configurations from the endpoints alone.

n_{bound} = {({(\frac{0}{n_{\max}})}^{2} + {(\frac{n_{\min}}{n_{\min}})}^{2})}^{1/2} = 1 .

(10)

n_{bound} = {({(\frac{n_{\max}}{n_{\max}})}^{2} + {(\frac{0}{n_{\min}})}^{2})}^{1/2} = 1 .

(11)

We adopt the normalized relation $n_{bound} = 1$ in Eq. (12) as the geometric model for the observer–observable comparison. It specifies the allowed normalized relation used in the following algebra.

{(\frac{a}{n_{\max}})}^{2} + {(\frac{b}{n_{\min}})}^{2} = 1 .

(12)

Equation (13) replaces the upper normalization count by the light count and the other normalization count by the local length reference. The discussion of count bounds in [5], Appx. O, supplies the broader context. Here the local reference need not equal a universal minimum count; it is the nonzero reference used in the normalized construction.

{(\frac{a}{n_{L c}})}^{2} + {(\frac{b}{n_{L l}})}^{2} = 1 .

(13)

Equation (14) identifies the first leg with the motion count and the second with the observed length count. Rearrangement gives Eq. (15), and its nonnegative root gives Eq. (16). The reference count stays positive, and the motion-to-light ratio must lie between zero and one for a real length factor.

{(\frac{n_{L m}}{n_{L c}})}^{2} + {(\frac{n_{L o}}{n_{L l}})}^{2} = 1,

(14)

\frac{n_{L o}^{2}}{n_{L l}^{2}} = 1 - \frac{n_{L m}^{2}}{n_{L c}^{2}},

(15)

n_{L o} = n_{L l} {(1 - \frac{n_{L m}^{2}}{n_{L c}^{2}})}^{1 / 2} .

(16)

Equation (17) expresses the count ratio through speeds over their common elapsed interval. Multiplication by the fundamental length gives Eq. (18). In SR, the corresponding longitudinal length comparison is made at equal times in an inertial frame [2]. Matching its scalar factor does not supply that simultaneity prescription or a derivation of the adopted count geometry.

\frac{n_{L m}^{2}}{n_{L c}^{2}} = {(\frac{n_{L m} l_{f}}{n_{T} t_{f}})}^{2} {(\frac{n_{T} t_{f}}{n_{L c} l_{f}})}^{2} = \frac{v^{2}}{c^{2}} .

(17)

The fundamental relation is $l_{f} m_{f} = 2 p_{s i} t_{f}$ ([5], Appx. V), where $p_{s i}$ is the momentum realization in kg m s⁻¹. It relates the fundamental reference quantities. Equation (19) applies the model’s reciprocal-mass representation to the observed and local mass parameters. Cancelling its common nonzero momentum-time numerator yields Eq. (20), and solving for the observed mass yields the reciprocal factor collected in Eq. (23).

l_{o} = l_{l} {(1 - \frac{n_{L m}^{2}}{n_{L c}^{2}})}^{1 / 2},

(18)

\frac{2 p_{s i} t_{f}}{m_{o}} = \frac{2 p_{s i} t_{f}}{m_{l}} {(1 - \frac{n_{L m}^{2}}{n_{L c}^{2}})}^{1 / 2},

(19)

m_{l} = m_{o} {(1 - \frac{n_{L m}^{2}}{n_{L c}^{2}})}^{1 / 2} .

(20)

Equations (21)–(23) collect the conditional motion forms. Equation (22) assigns the length factor to the two time parameters; Eq. (23) rearranges the assigned masses in Eq. (20). In SR, a moving clock’s elapsed proper time is compared with the elapsed coordinate time of the inertial reference frame [2]. The observed and local labels alone do not establish which of those intervals is represented here. Nor does the reciprocal mass factor establish a transformation of invariant rest mass.

l_{o} = l_{l} {(1 - \frac{n_{L m}^{2}}{n_{L c}^{2}})}^{1 / 2} .

(21)

t_{o} = t_{l} {(1 - \frac{n_{L m}^{2}}{n_{L c}^{2}})}^{1 / 2} .

(22)

m_{o} = \frac{m_{l}}{{(1 - \frac{n_{L m}^{2}}{n_{L c}^{2}})}^{1 / 2}} .

(23)

For the gravitational substitution, upright $G$ is the upper-count gravitational coupling, in m³ kg⁻¹ s⁻², with G=t_fc³/m_f. In this part of the comparison, $v$ is the escape-speed parameter. Equation (24) is the historical fundamental-unit starting expression. The general counted source mass and radial separation enter Eq. (25); substituting the upper-count coupling and $l_{f} = c t_{f}$ gives Eq. (26). Dividing by $c^{2}$ gives Eq. (27). Equation (24) at unit mass and radial counts lies outside the subluminal domain used for the nonzero factors, so it is not an admissible example of those factors.

v = {(\frac{2 G m_{f}}{l_{f}})}^{1 / 2},

(24)

v^{2} = \frac{2 G n_{M} m_{f}}{n_{L} l_{f}} = n_{M} m_{f} \frac{2}{n_{L} l_{f}} \frac{t_{f} c^{3}}{m_{f}},

(25)

v^{2} = \frac{2 n_{M} t_{f} c^{3}}{n_{L} l_{f}} = \frac{2 n_{M} c^{3}}{n_{L} c} = \frac{2 n_{M} c^{2}}{n_{L}},

(26)

\frac{v^{2}}{c^{2}} = 2 \frac{n_{M}}{n_{L}} .

(27)

Equation (17) expresses the model’s inertial count ratio through the speed ratio, while Eq. (27) expresses its gravitational count ratio through the same model parameter. Eliminating that parameter gives Eq. (28), the count-geometric equivalence between the inertial and gravitational descriptions ([5], Appx. AV, AW). The relation contains only dimensionless counts; the fundamental reference measures cancel. It establishes the equivalence used within the adopted MQ geometry. A corresponding physical comparison must still specify the observer and the measures being compared.

\frac{n_{L m}^{2}}{n_{L c}^{2}} = 2 \frac{n_{M}}{n_{L}} .

(28)

Substituting this conditional relation into Eqs. (21)–(23) gives Eqs. (29)–(31). They preserve the length and time factor and the reciprocal assigned-mass factor. Their domain is the strict inequality in Eq. (32), which excludes a zero denominator.

l_{o} = l_{l} {(1 - 2 \frac{n_{M}}{n_{L}})}^{1 / 2} .

(29)

t_{o} = t_{l} {(1 - 2 \frac{n_{M}}{n_{L}})}^{1 / 2} .

(30)

m_{o} = m_{l} / {(1 - 2 \frac{n_{M}}{n_{L}})}^{1 / 2} .

(31)

A positive nonzero square-root factor requires the escape-speed parameter to remain below $c$ . Equation (27) then gives the strict count condition in Eq. (32), consistent with the escape-bound discussion ([5], Appx. H). Equality at the bound is excluded from this domain rather than removed by count discreteness.

1 > 2 \frac{n_{M}}{n_{L}} .

(32)

Equations (33)–(34) retain the frequency-bound construction, with $v_{p L}$ and $v_{p M}$ as its length- and mass-related frequency labels. The intended common reference rate is (1)/(t_f), the reciprocal of the fundamental MQ time ([5], Appx. O). The author’s interpretation uses this rate to seek upper observable counts of length and mass per fundamental-time increment. To evaluate the displayed subtractions as dimensionless count operations, a specified reference interval must convert each frequency into a count and the time term into a normalized duration. The frequency identification alone does not supply that interval or establish a finite maximum count, so the historical expression is not used as a validated bound or a numerical input here.

v_{p L} > 2 (\frac{v_{p M}}{2} - 1) .

(33)

1 > \frac{2}{v_{p L}} (\frac{v_{p M}}{2} - 1) = \frac{2}{t_{f}^{- 1}} (\frac{t_{f}^{- 1}}{2} - 1) = (1 - 2 t_{f}) = 0.9999999999 (99) . . . .

(34)

The decimal string in Eq. (34) is the historical estimate from that construction, not a current 2022 calibration result. It does not establish a maximum count or a restriction to even mass counts. For example, $n_{M} = 1$ and $n_{L} = 3$ satisfy Eq. (32), so the strict inequality alone permits an odd positive mass count.

Physically admissible Internal Frame configurations use whole counts no smaller than one. The local and observed quantities are related through the Frames mapping. A realized quantity below a fundamental reference measure cannot be admitted as a positive physical count merely by extending a continuous expression. This model restriction is distinct from a parity condition on source mass counts.

For the dimensional form, let $M$ be the source mass in kg and $R$ the radial separation from its center in m. With $M = n_{M} m_{f}$ and $R = n_{L} l_{f}$ , the ratio in Eqs. (29)–(31) becomes n_M/n_L=(M/R)(l_f/m_f). Equations (35)–(37) apply this exact substitution and retain the same positive-root domain.

l_{o} = l_{l} {(1 - 2 \frac{M}{R} \frac{l_{f}}{m_{f}})}^{1 / 2} .

(35)

t_{o} = t_{l} {(1 - 2 \frac{M}{R} \frac{l_{f}}{m_{f}})}^{1 / 2} .

(36)

m_{o} = m_{l} / {(1 - 2 \frac{M}{R} \frac{l_{f}}{m_{f}})}^{1 / 2} .

(37)

The upper-count coupling gives l_f/m_f=G/c², so Eqs. (35)–(37) are dimensional restatements of the conditional count relations. A GR comparison additionally needs a spacetime metric, an observer and a measurement prescription. In the static, spherically symmetric vacuum exterior described by Schwarzschild [3], the temporal and radial metric coefficients are reciprocal, and the angular geometry has a separate coefficient. A common scalar factor therefore does not establish an isotropic contraction of measured lengths. The present time and length labels have not been identified with the proper and coordinate intervals of that geometry.

4. DISCUSSION

The equations describe the adopted count geometry relating observed measures to the local measures of an observable. The resulting transformations and the inertial–gravitational count relation follow within that model. Their physical comparison uses the stated admissibility and measurement conditions.

In the right-triangle residual construction, $Q_{L}$ is the dimensionless mapping-induced residual count contribution, with Q_L=√(1+n_L²)-n_L for positive radial count ([5], Appx. D, Eq. (D.5)). Its count-dependent normalization enters the Informativity differential. Current MQ definitions distinguish that differential from emergent curvature in the System Frame. Neither the residual nor the scalar transformation factors determine the continuous metric required for the GR comparison.

The domain condition in Eq. (32) is essential to the reciprocal mass expression. The count-bound and discreteness constructions are discussed in [5], Appx. I, J. The minimum-count condition is applied to the physically counted observable as well as to the source configuration. The boundary $n_{L} = 2 n_{M}$ gives a zero length and time factor and is therefore excluded from the model’s admissible observed counts. With fixed local reference counts, the minimum observable count prevents arbitrarily small resolved length or time. This does not establish a universal separation from the boundary across configurations whose reference counts are unbounded.

Equation (38) restates Eq. (28), with the motion count on the left and the radial and mass counts on the right. It records the inertial–gravitational equivalence of the adopted MQ count geometry ([5], Appx. AW). The absence of dimensional reference measures makes this a relation between count configurations.

\frac{n_{L m}^{2}}{n_{L c}^{2}} = 2 \frac{n_{M}}{n_{L}} .

(38)

Count ratios are dimensionless, while the factors multiplying them retain the units and physical roles of their parameters. The normalization and substitutions explain how the same algebra is expressed in count and dimensional forms. They do not erase the distinction between a discrete configuration and its observable realization.

Lengths are converted with $l_{f}$ , masses with $m_{f}$ and times with $t_{f}$ . Equations (19) and (22) supply the additional assigned mass and time identifications. An experimental test would require an independently specified observable and a physical realization of those identifications; the conditional calibration values alone do not provide that test.

5. APPENDICES

APPENDIX A NOTATION AND NUMERICAL CONTEXT

The following entries record numerical context and refer back to definitions already introduced. A fuller set of MQ term and symbol definitions is given in Measurement Quantization [5].

The dimensionless invariant coefficient $θ_{s i}$ is calculated from the 2022 CODATA inverse fine-structure input as $θ_{s i} = 3.26239030504 (25)$ . Its angular realization $Θ_{s i}$ has units rad. For the momentum realization introduced before Eq. (19), the stipulated numerical calibration in the International System of Units (SI) gives $p_{s i} = 3.26239030504 (25) kg m s^{- 1}$ . Shwartz and Harris [13] analyze the generation of polarization-entangled X-ray photons; Shwartz et al. [14] study down-conversion coincidence rates with loss; Glover et al. [15] observe X-ray and optical sum-frequency generation. Chen et al. [16] discuss conserved quantities at null infinity in GR. These sources address distinct quantities and do not independently measure the MQ coefficient or establish its momentum realization. Parenthetical digits give one standard uncertainty in the last quoted digits, propagated from the 2022 input conditional on the fixed realization calibration [17]. They do not include uncertainty in the physical validity of that calibration.

The light-speed convention introduced in Sec. 1 uses the exact SI value $c = 299792458 m s^{- 1}$ . This source value is unchanged between the cited 2018 and 2022 recommendations [18,17]. Over a common elapsed interval, the light and time counts agree; this does not restrict the general comparison to a single fundamental tick.

The upper-count coupling used in Eqs. (24)–(27) is calculated as G=6.6740779383(10)×10^-11 m³ kg^-1 s^-2. Italic $G$ denotes finite-count coupling. At the electromagnetic demarcation, the fixed interaction count used in the MQ calibration, the count normalization gives G=6.6738448317(10)×10^-11 m³ kg^-1 s^-2. Both values are conditional MQ calculations using 2022 source inputs [17], rather than CODATA recommendations or independent measurements.

Under the same calibration, the reference measures used throughout the derivation are calculated as 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 [17]. Their uncertainties share the inverse fine-structure input; they are not independent source measurements.

The conventional Planck units $l_{p}$ , $m_{p}$ and $t_{p}$ denote reference length, mass and time, in m, kg and s, respectively. Their comparison role in Sec. 1 is distinct from the MQ calibration above.

The count notation introduced in Sec. 1 is dimensionless. Integer configurations belong to the Internal Frame; their observable relational realization belongs to the System Frame. Applying a scalar count-times-reference substitution does not by itself establish every physical mapping assumption.

The radial count convention in Eqs. (25)–(38) remains distinct from the motion count in Eqs. (14)–(23). The local reference length count is retained separately in Eqs. (13)–(16).

Equation (17) evaluates the motion and light counts over their shared nonzero elapsed-time count. The light-count specialization agrees with the exact SI light speed and avoids treating the radial separation as the motion count.

The dimensionless $n_{θ}$ is the System Frame count of the invariant coefficient used at the electromagnetic demarcation. The mixed brackets below denote rounding to the nearest whole unit. With the recalculated demarcation count $84.6005456716 (65)$ , n_θ=⌊(84.6005456716)/(2)⌉=42.

This mixed-bracket convention differs from ordinary floor or ceiling alone. The recalculated argument and its propagated source uncertainty remain away from a half-unit tie.

The residual introduced in the Discussion enters the electromagnetic normalization through its product with the demarcation count. Its conditional recalculated value is $Q_{L} n_{L} = 0.4999825364172403 (27)$ [17]. The same radial-count notation is evaluated here at the fixed calibration parameter rather than at an arbitrary gravitational separation. This noninteger calibration parameter is not an assertion of an allowed integer gravitational configuration.

The Informativity differential retains the non-Lorentz realization role stated in Sec. 1. It must not be identified with the motion-speed factor merely because both affect a length description.

Italic $ℏ$ denotes the finite-count reduced Planck quantity in the electromagnetic calibration, in J s. It is calibrated to the conventional exact SI reduced Planck value h/(2π), where $h$ is Planck’s constant in J s and $π$ is the dimensionless circular constant. Upright $ħ$ denotes the upper-count MQ quantity, calculated as ħ=1.0545349844417293(56)×10^-34 J s. The conventional input and the conditional upper-count output are distinct [17].

The electron mass $m_{e}$ , in kg, enters the electromagnetic length calibration together with the Bohr radius introduced in the next entry. The current calculation does not treat their product as two independent measured inputs.

The Bohr radius $a_{0}$ , in m, is the conventional atomic length scale used in that calibration. Here $α_{c}$ is the dimensionless classically realized System Frame fine-structure parameter, identified with the conventional 2022 source input. The exact source relation m_ea₀=h/(2πα_cc) eliminates their shared dependence in the numerical calculation [17].

The dimensionless quantities $α_{f}$ and $α_{p}$ are the fundamental Internal Frame and intermediate Planck-like fine-structure forms, respectively. The latter is the intermediate realization before the finite-count normalization associated with the Informativity differential. The classically realized source parameter in the preceding entry has inverse α_c^-1=137.035999177(21) for the 2022 adjustment [17]. These MQ realizations remain distinct; they do not add an independent measurement to the gravitational comparison.

The conventional electromagnetic transformation parameter $γ$ belongs to the historical auxiliary-input approach. In its conventional SI representation it has units m⁶ C⁻² s⁻⁴, with C the coulomb. It is not used in Eqs. (1)–(38) or as an independent numerical input here.

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

Supporting calculations are referenced through Supplement 1 and Supplement 2 at the original supplementary-material location given in Sec. 2. These materials are updated as the associated papers change. The numerical calibration reported here uses the 2022 source inputs and the relations specified in Appendix A.

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