Galactic Rotation and Dark Matter in Measurement Quantization

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We summarize the Measurement Quantization (MQ) galactic realization law and the principal rotation-curve comparisons reported in the companion preprint. Within the stated MQ construction, an externally specified Newtonian baryonic source determines the predicted circular velocity through a fixed acceleration scale. The System Frame expansion rate supplies this scale; the complementary Internal Frame rate retains its separate role. With the preprint’s epoch normalization held fixed, the 2022 CODATA calibration and measured cosmic microwave background temperature give an acceleration of 1.20037(33) × 10⁻¹⁰ m s⁻². This is a conditional input-propagated result; the absolute temporal normalization and model discrepancy are not included in its uncertainty. The local saturation model recovers the Newtonian response at high acceleration and the asymptotic baryonic Tully-Fisher fourth-power relation for a finite baryonic source. Reported galactic rotation-curve comparisons test the fixed normalization against externally derived kinematic curves. Their interpretation remains conditional on source reconstructions, kinematic assumptions and uncertainty models; aggregate agreement does not establish precision agreement at every radius.

Keywords: Galactic Rotation, Star Velocities, Dark Matter, Gravitation, Discrete

1. INTRODUCTION

Galactic rotation motivates a comparison between the acceleration inferred from stellar motion and the Newtonian acceleration of the externally reconstructed baryonic source. Measurement Quantization (MQ) develops this comparison from discrete counts and their continuous realization [1, 2]. The Internal Frame is the domain of discrete count configurations. The System Frame represents observable continuous relational separations induced by the Frames mapping, which relates those count configurations to their realized geometry. The underlying construction uses the speed of light, escape velocity and Heisenberg’s uncertainty principle [3]. The present paper gives a concise overview of the galactic argument in [2], retaining the calibration and mathematical limits needed to interpret its observational tests.

We use dimensionless counts n_L, n_M and n_T of the fundamental length, mass and time measures l_f, m_f and t_f. In the source construction, these counts denote radial separation, source mass and elapsed time, respectively. Their realizations in the International System of Units (SI) have units m, kg and s. The speed of light c = n_L l_f/(n_T t_f) has units m s⁻¹ for the light-travel count relation. Count configurations and dimensional quantities remain distinct. In the point-source escape-speed construction, radial n_L > 2n_M corresponds to escape speed below c; equality is the formal escape-speed boundary. This count criterion alone is not a derivation of a relativistic event horizon.

Within MQ, the inverse fundamental time supplies a count-resolution scale [4]. Planck’s historical work [5] provides the conventional fundamental-unit context, rather than independent evidence for the MQ count interpretation. The galactic prediction is expressed as a realization of an externally specified baryonic acceleration. Earlier MQ Milky Way comparisons [6] used McGaugh’s modelled velocities [7, 8]; the current comparison uses separate source and kinematic constructions [2].

In the galactic law, gravitational availability denotes the Newtonian baryonic source relation before realization through the Frames mapping. Here R is galactocentric radius in m, v_New(R) is the Newtonian baryonic circular velocity in m s⁻¹, v_real(R) is the realized circular velocity in m s⁻¹, g_bar(R) is the baryonic radial acceleration in m s⁻², and a_MQ is the characteristic acceleration in m s⁻². The positive circular-speed branch is

v_{real} (R) = \frac{v_{New} (R)}{\sqrt{1 - \exp [- \sqrt{\frac{g_{bar} (R)}{a_{MQ}}}]}} .

(1)

Equation (2) retains the historical escape-speed expression for a point mass or the exterior of a spherical source, with potential zero at infinity. Here v_m denotes the historical velocity proxy in m s⁻¹, M is the source mass in kg, and G is the gravitational coupling in m³ kg⁻¹ s⁻². The function f(m) denotes the speed-valued mass dependence in m s⁻¹ at fixed R, with mass argument m = M in kg. Enclosed mass alone does not determine escape speed within an extended source. This proxy is distinct from the baryonic circular velocity v_New(R).

v_{m} = \sqrt{\frac{2 G M}{R}} = f (m) .

(2)

The current comparison applies Eq. (1) to a specified baryonic circular source and tests it against an external kinematic target [2, 9, 10]. Agreement between two historical model curves [7, 8] is not an independent test.

2. METHODS

The calculation separates three tasks. Fundamental measures are calibrated from the 2022 fine-structure input, the cosmological relation fixes the acceleration scale before galaxy comparisons, and a published baryonic source supplies the radius-dependent Newtonian circular velocity. Earlier MQ work develops the count and realization framework [1, 11, 12, 6, 13, 14]. Its supporting definitions and derivations are collected in Measurement Quantization [4]; the current galactic derivation and empirical analysis are given in [2].

The finite-count inversion uses the dimensionless classical inverse fine-structure input

{α_{c}}^{- 1}

= 137.035999177(21) from the 2022 CODATA adjustment, with the defined SI speed of light and Planck constant [15]. The electron-mass and Bohr-radius product in Eq. (6) is eliminated through the exact conventional Bohr identity before uncertainty propagation. These shared dependencies prevent treating the fundamental outputs as independent measurements. Their numerical agreement with their calibration inputs is not an independent observational test.

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

For a galaxy comparison, the adopted source reconstruction is fixed before examining the MQ velocity residuals. Its photometric and dynamical assumptions remain relevant, as do uncertainties in distance, inclination, stellar normalization, gas and kinematics. No galaxy-specific MQ parameter is fitted in the reported tests [2]; this does not mean that the published source and kinematic reconstructions are parameter-free. The existing online materials provide background calculations.

3. RESULTS

The fundamental-time frequency is calculated from the same calibration as the other fundamental measures. It is a dimensional frequency, with

\frac{1}{t_{f}} = 1.85492188108 (14) \times 1 0^{43} s^{- 1} .

(3)

The calibration uses the dimensionless coefficient θ_si and its momentum realization p_si, in kg m s⁻¹. Their numerical coefficients coincide under the adopted MQ SI realization, while their dimensions differ [2]. The finite-count inversion gives θ_si = 3.26239030504(25). The Planck-like fine-structure quantity α_p is dimensionless, m_e is electron mass in kg and a₀ is the Bohr radius in m. The mixed brackets below denote rounding to the nearest whole unit. Parenthetical digits give one standard uncertainty in the final quoted digits, propagated from the common inverse fine-structure input with exact SI defining constants. Shared derived quantities are correlated. These conditional uncertainties exclude unquantified MQ model discrepancy; they are neither confidence intervals nor measures of the probability that MQ is correct. Uncertainties are rounded to two significant digits and central values to the same decimal place. The mass, coupling, length and time dependencies are

m_{f} = \frac{2 p_{si}}{c} = 2.17643254057 (17) \times 1 0^{- 8} kg .

(4)

{α_{p}}^{- 1} = 84 θ_{si} - ⌊ 42 θ_{s i} ⌉ = 137.040785623 (21) .

(5)

l_{f} = \frac{m_{e} a_{0} α_{p}}{m_{f}} = 1.61619991148 (12) \times 1 0^{- 35} m .

(6)

t_{f} = \frac{l_{f}}{c} = 5.39106261131 (41) \times 1 0^{- 44} s .

(7)

The finite-count gravitational coupling supplies the source-level connection to the count construction [4]. Here r is radial separation in m, Q_L is the dimensionless residual count contribution at that separation, and G(n_L) is the finite-count coupling in m³ kg⁻¹ s⁻², with r = n_L l_f in this context. The galactocentric R in Eq. (1) and source separation r are distinct radial contexts. The upright upper-count G in Eq. (2) is the limit of the finite-count coupling. Dimensional consistency requires the momentum realization p_si, giving

\frac{Q_{L} c^{3}}{r p_{s i}} = \frac{G (n_{L})}{r^{2}} .

(8)

As the radial count grows, Eq. (8) approaches the upper-count coupling within the stated MQ realization. This supplies the model gravitational normalization. The circular response then follows from Eq. (1), once the baryonic source and acceleration scale are fixed [2].

The count form of the escape-speed starting relation is retained to show its connection with Eq. (2). It is a foundation relation rather than the circular-velocity prediction.

At the upper-count limit, the point-source escape-speed construction reduces to a ratio of mass and radial length counts. In Eq. (9), v_e denotes escape velocity in m s⁻¹. Its factor of two belongs to escape speed; it is absent from the corresponding point-source circular-speed relation.

v_{e} = \sqrt{2 \frac{G M}{R}} = \sqrt{2 \frac{l_{f}^{3}}{t_{f}^{3}} \frac{t_{f}}{m_{f}} \frac{n_{M} m_{f}}{n_{L} l_{f}}} = \sqrt{2 \frac{l_{f}^{2}}{t_{f}^{2}} \frac{n_{M}}{n_{L}}} = c \sqrt{2 \frac{n_{M}}{n_{L}}} .

(9)

The galactic acceleration scale is fixed before applying Eq. (1). The coefficient appearing in Eq. (10) is dimensionless. H_sys is the System Frame expansion rate, converted to s⁻¹ when used with c in SI units. H_int remains the complementary Internal Frame rate; it is not the rate entering this galactic acceleration scale [2].

a_{MQ} = \frac{c H_{sys}}{2 θ_{si} - 1} = 1.20037 (33) \times 1 0^{- 10} m s^{- 2} .

(10)

The frame-resolved rate uses the System Frame universe age A_U in s and the dimensionless circle constant π. Reproducing the MQ age relation [2, Sec. 5.5] with its epoch normalization held fixed, the measured cosmic microwave background temperature T_CMB = 2.72548(57) K [16] and shared 2022 calibration gives A_U = 13.7943(38) Gyr. The absolute time normalization underlying that reconstruction remains an unverified premise. The quoted result is conditional on that premise; the measured temperature alone does not establish it. One Gyr is 10⁹ Julian years and one Julian year is 31,557,600 s. The applicable rate is reported in conventional astronomical units below; Mpc denotes megaparsec. Its SI value is obtained by converting km to m and Mpc to m before insertion into Eq. (10).

H_{sys} = \frac{π}{θ_{si} A_{U}} = 68.259 (19) km s^{- 1} Mp c^{- 1} .

(11)

The complementary rate is H_int = 73.508(20) km s⁻¹ Mpc⁻¹ under the same fixed age normalization [2, Sec. 3.12]. The uncertain inputs are the inverse fine-structure calibration and T_CMB, treated as uncorrelated across their separate source experiments. The age is derived rather than an additional independent input, so its shared dependence is propagated through the rates and acceleration. The CMB temperature dominates the conditional input uncertainty. The two rates share that dependence and cannot supply independent combined evidence. The time-normalization premise and unquantified model discrepancy are excluded from the quoted uncertainty, rather than assigned experimentally established zero uncertainty. All values remain conditional calculations, rather than independent measurements of cosmic age or expansion.

For positive baryonic acceleration and positive a_MQ, χ is the dimensionless realization depth and P is the dimensionless gravitational realization fraction in the System Frame. The local construction gives

χ = \sqrt{\frac{g_{bar} (R)}{a_{MQ}}} .

(12)

P (R) = 1 - \exp [- \sqrt{\frac{g_{bar} (R)}{a_{MQ}}}] .

(13)

The baryonic circular acceleration is

g_{bar} (R) = {v_{New}}^{2} (R) / R .

The realized radial acceleration g_real(R), in m s⁻², follows by dividing this source by the realized fraction.

g_{real} (R) = \frac{g_{bar} (R)}{1 - \exp [- \sqrt{\frac{g_{bar} (R)}{a_{MQ}}}]} .

(14)

For positive realization depth, the denominator in Eqs. (1) and (14) is positive and below one. Equation (14) therefore has acceleration units and Eq. (1) has velocity units. This domain check connects the count-based construction with the observable circular response.

3.1. Local Realization

Within the MQ construction [2], Eq. (12) is the dimensionless ratio of the source circular-speed magnitude to the reference speed supplied by a_MQ at the same radius. Its square-root form follows from the adopted positive quadratic System Frame norm and circular kinematics. Subdivision consistency means that the unrealized fractions multiply when realization depth is subdivided. With continuous composition and the unit initial response specified by the scalar remaining-availability assumption, this gives the unit-rate saturation law below. The differential equation and boundary condition determine a unique fraction within this model; their mathematical consistency does not by itself establish the proposed realization mechanism observationally.

\frac{d P}{d χ} = 1 - P .

(15)

P (0) = 0 .

(16)

The boundary condition excludes a realized fraction at zero depth. Separation of variables gives the unique regular saturation solution for χ ≥ 0,

P (χ) = 1 - \exp [- χ] .

(17)

Substituting Eq. (12) into Eq. (17) gives Eq. (13). The independently adopted source sets

g_{bar} (R) = {v_{New}}^{2} (R) / R;

division by the realized fraction gives Eq. (14). Circular motion then gives Eq. (1) on the positive-speed branch. The observed rotation curve supplies the test of this prediction, rather than an input to its normalization [2].

2. Asymptotic Galactic Velocity

For a finite baryonic source in the exterior circular-motion limit, let M_bar denote the total baryonic mass in kg. As R increases, the enclosed baryonic mass approaches M_bar and g_bar(R) approaches the Newtonian inverse-square exterior field. The exact limiting statements are

\lim_{R \to \infty} R g_{real} (R) = \sqrt{G M_{bar} a_{MQ}} .

(18)

\lim_{R \to \infty} {v_{real}}^{4} (R) = G M_{bar} a_{MQ} .

(19)

Equation (19) is the asymptotic baryonic Tully-Fisher fourth-power relation. It gives a radius-independent limiting speed for a specified finite baryonic mass, rather than a universal speed shared by all galaxies. At high baryonic acceleration, the Newtonian response is recovered exactly as a limit,

\lim_{\frac{g_{bar}}{a_{MQ}} \to \infty} \frac{g_{real}}{g_{bar}} = 1 .

(20)

In the weak-acceleration limit, the exponential realization fraction has unit first derivative at χ = 0. Therefore the realized field obeys the exact limit

\lim_{\frac{g_{bar}}{a_{MQ}} \to 0} \frac{g_{real}}{\sqrt{g_{bar} a_{MQ}}} = 1 .

(21)

Combining this limit with exterior Newtonian acceleration and circular motion gives Eqs. (18) and (19). The assumptions of finite total baryonic mass and circular motion are essential; this argument does not extend those limits to an arbitrary nonspherical mass distribution at finite radius.

The square-root law requires a nonnegative total inward baryonic circular acceleration and a positive acceleration scale. Individual source components can exert outward radial forces, as in some gas distributions [17], and must be combined with their physical sign. A negative total inward source is outside the stated circular-motion branch.

At a zero-source boundary, the realized acceleration is interpreted through its continuous weak-source limit. An independently specified baryonic source remains necessary throughout the calculation.

3. Observational Comparisons

The exponential functional form of Eq. (14) was previously reported as the empirical radial acceleration relation (RAR) [18]. The MQ claim concerns its derivation within the framework and its normalization fixed from non-galactic inputs under the stated epoch premise [2]. An empirical fit to the same functional shape does not independently determine a_MQ, and its fitted scale is not treated here as a measurement of the MQ quantity.

Figure 1 reproduces the Milky Way comparison in [2, Fig. 3]. The source is constructed without an RAR or modified Newtonian dynamics (MOND) prescription. It uses McMillan’s bulge, stellar disks and gas components [10], with the halo force excluded. McMillan fitted baryonic and halo parameters jointly using Galactic dynamical constraints; the retained baryonic parameters therefore carry that modelling context. Eilers et al. [9] inferred the comparison curve using axisymmetric Jeans modelling of stellar tracers. The constructions are separate and no parameter is refitted to the Eilers residuals in [2], but this does not make the source gravity-independent or prove statistical independence. For 38 radius bins from 5.27 to 24.82 kpc, [2] reports a root-mean-square (RMS) discrepancy of 8.90 km s⁻¹ and mean absolute error (MAE) of 7.43 km s⁻¹, compared with a Newtonian baryonic RMS of 69.03 km s⁻¹.

Milky Way circular-velocity comparison reproduced from the companion preprint — **Figure 1.** Milky Way circular-velocity comparison reproduced from [2, Fig. 3]. Blue points use Eilers et al.’s inferred curve [9]; error bars retain the total uncertainties adopted in [2]. Orange is the McMillan baryonic Newtonian contribution [10] and green the MQ prediction. The plot retains the preprint’s nominal normalization; the updated conditional acceleration is given in Eq. (10).

The diagonal-error treatment in [2] formally rejects the complete Milky Way sample. The source-covariance calculation varies the thin-disk, thick-disk and bulge density normalizations using a three-parameter conditional covariance derived from McMillan’s published posterior standard deviations and eight-parameter correlation matrix [10]. Disk scale lengths and other source parameters remain fixed. The reported 200,000-draw propagation gives source standard deviations of 2.09–3.17 km s⁻¹ across the 38 radii. Adding that correlated source contribution to the adopted observational diagonal gives a generalized chi-square statistic of 39.57 for 38 degrees of freedom and a right-tail probability of 0.400 [2]. This probability is not the probability that MQ is correct. Eilers et al. [9] identify tracer-model and Galactic-parameter systematics that can affect multiple radii coherently. The full observational point-to-point covariance and complete source posterior have not been jointly propagated. The reported non-rejection remains conditional on this specified covariance model.

For Spitzer Photometry and Accurate Rotation Curves (SPARC), [2] applies the law to all 175 catalog galaxies and 3,366 radial measurements using the published baryonic reconstructions [17]. This complete-catalog diagnostic is broader than the primary survey’s selected dynamical science sample, which excludes strongly disturbed and nearly face-on systems [17]. The preprint reports pooled RMS and MAE values of 22.82 and 16.20 km s⁻¹, and a ratio of mean MQ to mean observed velocity of 1.00019. The latter describes aggregate normalization, not pointwise precision. For the 99 galaxies classified as high quality by the catalog, the galaxy-level geometric ratio is 1.0042 and the marginal velocity-dependent normalization residual is not statistically significant. The full catalog retains a significant velocity-dependent residual under simultaneous quality and property controls, with strong velocity–luminosity collinearity [2]. Quality dependence alone does not establish that lower-quality residuals are observational errors.

The separate LITTLE THINGS dwarf-galaxy survey comparison in [2] uses 15 prospectively screened galaxies outside SPARC and 288 radial comparisons extracted from published figures [19, 20]. O’Brien et al. [20] fitted stellar masses in their alternative-gravity curves. The preprint therefore reconstructs the gas contribution from those curves and replaces the fitted stellar contribution using Oh et al.’s spectral-energy-distribution stellar masses [19], rather than adopting the plotted fitted baryonic curve unchanged. It reports a galaxy-weighted geometric MQ-to-observed velocity ratio of 0.9907 and a 95% galaxy-block-bootstrap interval of 0.868–1.130. The bootstrap resamples galaxies together with their radial measurements. This is a population-level normalization comparison. Digitization, source reconstruction and individual curve scatter preclude a precision pointwise likelihood.

These statistics and Figure 1 are reported results of [2], calculated there with the frozen nominal acceleration 1.20040492178 × 10⁻¹⁰ m s⁻². The present shared-input calculation differs by 0.11 conditional standard uncertainties and is reported in Eq. (10) with the same epoch premise held fixed. At an unchanged baryonic source, Eq. (1) bounds the decrease in every positive predicted velocity from this acceleration revision below 0.000750%. This also bounds its fractional effect on positive weighted-mean and geometric velocity normalizations. The small common calibration uncertainty is correlated across radii; its conditional first-order contribution to velocity standard uncertainty is at most 0.00698%. These bounds do not quantify the time-normalization premise or missing source and observational covariance. The source figure, catalog residuals and tail probabilities have not been rerun; their source and covariance qualifications remain.

4. DISCUSSION

The realization law predicts a circular response from an externally specified baryonic source and an upstream acceleration scale. Here independently specified means fixed without calibration to the MQ velocity residuals. It does not imply statistical independence of the source and kinematic reconstructions. The high-acceleration limit, weak-source limit and asymptotic fourth-power relation describe complementary regimes of this same law. The companion preprint’s rotation-curve comparisons test its fixed normalization conditional on the adopted sources and epoch premise [2].

The asymptotic limits require a finite exterior baryonic source and circular motion, as stated in Section 3. At finite radius, the actual force distribution must be used. A total mass alone does not determine the circular field of an arbitrary nonspherical source.

The reported comparisons test the galactic construction under specified source and uncertainty assumptions, with material population residuals. Conventional halo models and other gravitational prescriptions also describe related rotation-curve data [10, 20]; this overview does not perform comparative model selection. Galactic rotation alone does not establish agreement with lensing, cluster dynamics or cosmological dark-matter constraints. The age-dependent acceleration in [2, Sec. 5.5] supplies a prospective redshift test once the expansion history and independent baryonic reconstructions are fixed; that test has not yet been performed in this form.

5. APPENDICES

APPENDIX A: TERMS

The glossary uses the current MQ terminology and realization distinctions in [2, Appendix A]. Reference [4] provides the published foundational treatment.

The dimensionless calibration coefficient, momentum realization and angular realization remain distinct. The angular quantity Θ_si has units rad. Shwartz and Harris [21, Table I] calculate signal and idler angles of 3.26239 rad for one degenerate-frequency Bell-state geometry in diamond using the (111) lattice direction. This is a calculated phase-matching angle, not a measurement of the MQ coefficient or its uncertainty. The X-ray studies [22, 23] and the conserved-quantity discussion [24] supply their stated physical contexts; none is used here as independent calibration evidence.

The SI values c = 299,792,458 m s⁻¹ and h = 6.62607015 × 10⁻³⁴ J s are exact definitions [15]. The count realization of c was introduced in Section 1.

The upright upper-count gravitational value used here is the MQ calculation G = 6.6740779383(10) × 10⁻¹¹ m³ kg⁻¹ s⁻² from the 2022 calibration. The finite electromagnetic-demarcation calculation is 6.6738448317(10) × 10⁻¹¹ m³ kg⁻¹ s⁻². It retains the finite-count interpretation of G(n_L), whereas the upright value is its upper-count limit. These are MQ calculations, not direct CODATA recommendations.

The fundamental measures introduced in Section 1 are distinguished from the conventional Planck measures. The MQ calculation includes the count-realization normalization rather than merely renaming the conventional units.

The conventional Planck length l_p, mass m_p and time t_p have units m, kg and s, respectively.

The dimensionless motion count n_Lm records the change in realized position of an observed mass, in fundamental-length units.

The radial count n_L introduced in Section 1 describes center-of-mass separation when used in Eq. (8) or (9), rather than the motion increment n_Lm.

The dimensionless light-motion count n_Lc records a change in realized light position, in fundamental-length units.

The dimensionless System Frame auxiliary count n_θ belongs to the demarcation normalization. Its nearest-whole-unit prescription uses the mixed floor-ceiling brackets, so the historical argument 84.60055/2 selects 42. The brackets preserve the original argument without an added half-unit term.

The product Q_L n_L enters the radial count normalization. In the current galactic construction, the Informativity differential is the non-Lorentz departure 1-2Q_Ln_L of the finite-count realization factor from unity [2, Sec. 3.6]. It vanishes in the upper-count limit and is distinct from curvature.

The conventional reduced Planck constant is h/(2π), using the exact h given above [15]. The MQ upper-count action calculation is

1.0545349844417293 (56) \times 10^{- 34} J s

under the shared calibration. Its conditional standard uncertainty is 5.6 × 10⁻⁴⁹ J s, propagated from the common inverse fine-structure input. These are distinct quantities and are not interchanged; the MQ result is not an independent determination of the SI Planck constant.

The electron mass in Eq. (6) shares the conventional Bohr-product dependency used in the calibration.

The Bohr-radius quantity a₀ enters the length relation in Eq. (6).

The dimensionless fundamental inverse fine-structure quantity

{α_{f}}^{- 1}

belongs to the Internal Frame. The previously introduced Planck-like quantity

{α_{p}}^{- 1}

and classical calibration quantity

{α_{c}}^{- 1}

belong to System Frame realizations. The distinction between the latter two includes the finite-count normalization. Reusing the classical calibration input is not an independent comparison with CODATA [15].

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 reported galaxy analyses and their detailed source and uncertainty construction are documented in [2]. The primary kinematic and baryonic sources are [9, 10, 17, 19, 20]. Figure 1 is reproduced from [2] and retains its nominal calculation. The historical McGaugh source [7, 8] is available at https://astroweb.case.edu/ssm/models/MW2018model.dat. The present overview reproduces the shared calibration and cosmological normalization conditional on the stated epoch premise and bounds the velocity effect of its small acceleration revision; it does not supply a new pointwise catalog analysis or a complete covariance likelihood.

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