The research result and its present status

Section 5.3 of the current galactic manuscript develops the lensing consistency calculation; Appendix E limits its evidentiary status. The public lensing explanation carries the same central qualification. The construction links a radial acceleration law to a spherical light-deflection integral under specified weak-field assumptions. It then gives an ESO 325-G004 aperture benchmark. Neither step substitutes for solving the metric of an extended lens and predicting its resolved images. [1] [4]

In MQ terminology, the Internal Frame describes discrete count configurations. The System Frame is the realized relational description in which physical measures arise through the Frames mapping. The manuscript proposes a shared MQ variational principle for metric and matter. Those structural proposals explain where a relativistic bridge is intended to come from; they do not alone establish the particular weak-field potential relation needed for the numerical lensing benchmark. [1] [5]

Why rotation curves cannot settle lensing

A galaxy rotation curve tests the radial gravitational response experienced by slowly moving matter. In a stationary weak-field metric this is associated with the gradient of the temporal gravitational potential. Photon deflection depends on the combination of temporal and spatial potentials along the light path. A successful fit to one therefore leaves information about the other undetermined. Gravitational slip describes a possible difference between these potentials in the chosen convention. [1]

The source’s spherical calculation assumes the potential relation that gives the standard general-relativistic deflection coefficient. Its negligible-stress or negligible-slip restriction is part of the calculation, not an observational conclusion from fitting rotation curves. For the extended MQ solution, the relation must follow from field equations, source coupling and appropriate boundary conditions. Figure 1 separates that derivation from the measurements used to test it. [1]

What the spherical deflection calculation means

The source begins with the conventional exterior weak-field point-mass result. The physical bending angle is four times the gravitational constant times source mass, divided by light speed squared and impact parameter. Impact parameter is the perpendicular offset of the unperturbed ray from the lens center. This exterior expression cannot be applied indiscriminately to a ray passing through an extended stellar distribution by inserting a projected aperture mass. [1]

For the static spherical MQ consistency class, the source instead integrates the realized radial acceleration along the unperturbed line of sight. The transverse component is the radial acceleration multiplied by impact parameter divided by radius. The weak-field photon coefficient contributes twice the integral of that transverse acceleration divided by light speed squared. This description is the content of source Eq. (171), subject to the assumed relation between metric potentials. Eq. (172) inserts the exact galactic realization law; it is not an independently fitted lensing response function. [1]

The physical deflection at the lens must also be distinguished from the reduced angular deflection in the observed lens equation. The latter includes the lens-to-source divided by observer-to-source angular-diameter distance ratio. Thus a bending angle cannot be equated directly to an observed ring radius. Lens and source redshifts, a specified distance prescription, alignment and the angular mass distribution enter the image calculation. [1]

In the isolated finite-mass outer limit, the source obtains a physical deflection equal to two pi times the asymptotic circular speed squared divided by light speed squared. This follows from the same low-acceleration response as the outer rotation curve. It is an asymptotic spherical consistency relation. It does not put the inner Einstein aperture of a compact stellar lens into that regime, and it does not establish the potential relation for every source geometry. [1]

ESO 325-G004 and the measured inputs

ESO 325-G004 is a nearby elliptical lens with a compact, stellar-dominated Einstein aperture. Smith and Lucey provide the photometric and population inputs in Table 1. The lensing mass is inferred through a conventional lens model; it must be reserved for comparison rather than inserted as an independently known baryonic MQ source. [1] [2]

The luminosity and mass-to-light ratio refer to the same F814W band and aperture. Their product supplies the stellar mass for the preliminary source construction. The normalization assumes a Milky Way-like initial mass function, the distribution of stellar birth masses. Calling this input independent of lensing does not remove its stellar-population assumptions or their systematic uncertainty. [1] [2]

Reproduced central bookkeeping and reported predictions

Multiplying 4.07 × 10¹⁰ by 3.01 gives a central aperture stellar mass of 1.22507 × 10¹¹ solar masses. Multiplying that value by the manuscript’s reported 1.0475 enhancement gives 1.283260825 × 10¹¹ solar masses under the point-mass deflection normalization. This is a deflection-equivalent comparison mass, not additional stellar material or a directly measured dark component. Table 2 separates this arithmetic from the source’s numerical predictions. [1]

The equivalent central mass falls below the conventional lensing comparison by about 0.2167 × 10¹¹ solar masses. Figure 2 shows the bookkeeping without attaching a probability to it. The reported ring radius is about 6.3 percent below the observed angular radius. Neither fractional difference by itself supplies a goodness of fit or a rejection significance. The resolved image likelihood and correlated nuisance parameters are absent from this spherical summary. [1]

The factor at one impact parameter cannot simply be held fixed while solving for a different ring radius. Deflection changes with impact parameter, and an extended source changes the relevant profile. The manuscript reports solving the spherical lens equation for its radius. This Brief does not claim to reproduce that numerical solve, its distance implementation or the line-of-sight integral. A reproducible test should release the source profile, constants, integration controls and lens-equation implementation. [1]

The uncertainty difference and the quoted shortfall

The MQ text uses a mass-to-light uncertainty of ±0.25. The observational paper’s default result gives ±0.27 with index systematics included. The central normalization agrees, but the error budget does not. This distinction is material when a sigma estimate is quoted; the discrepancy must be resolved against the precise population fit selected for the benchmark. [1] [2]

As a diagnostic only, ordinary first-order propagation of independent luminosity and normalization errors gives a stellar-mass uncertainty of 0.10456 × 10¹¹ solar masses with ±0.25, or 0.11250 × 10¹¹ with ±0.27. These are reproduced input-product errors, not full lensing-model errors. They omit covariance, population-model changes and the mass dependence of the MQ enhancement. They therefore cannot certify or replace the source’s quoted shortfall. [1]

The correct propagation must pass each permitted source realization through the same gravitational and lensing calculation. It must include distance and environment assumptions, photometric calibration, stellar-population systematics and the likelihood used for the observed images. Treating the enhancement as a fixed multiplier for all mass perturbations would be an additional approximation, not a consequence of its central reported value. No revised sigma claim is made here.

What a complete extended-lens test must deliver

The source prescription should be fixed from the observed stellar light and independently selected population normalization before the arc constraints are used. Any relevant gas or other baryonic contribution and measured environmental effect should be documented. Feeding the lensing-inferred total mass back into that source would blur the distinction between prediction and reconstruction. A new halo profile or lens-specific acceleration scale would change the claimed common-calibration test. [1]

Convergence describes the isotropic focusing part of the lens mapping. Shear describes its directional distortion, and magnification depends on the local mapping as a whole. Their spatial patterns and the ring morphology test more than a spherical aperture total. Table 3 identifies the missing outputs rather than assuming that a rotation-curve law determines every one of them. [1]

Strong lensing describes multiple imaging and substantial image distortion. It does not require a relativistically strong local gravitational field. A galaxy can therefore produce a strong lens while still being modeled through weak-field metric potentials. The issue here is whether MQ derives the correct potentials and their projection, not whether the phrase strong lensing invalidates a weak-field calculation. [1] [3]

What the independent gravity study establishes

Collett and colleagues combined Hubble Space Telescope lens images with spatially resolved MUSE stellar kinematics. Their 2018 Science study reported spatial curvature per unit mass relative to the general-relativistic prediction of 0.97 ± 0.09 at 68 percent confidence, consistent with the general-relativistic value of one. That result constrains gravity in its own modeling framework. It is not an MQ likelihood or evidence that the MQ spherical benchmark has passed the same test. [3]

For MQ, the useful comparison is methodological as well as numerical. A single source-and-metric model should explain the stellar dynamics and the lensed images together. Transferring a constraint from another gravity model requires showing that the parameter definitions, source assumptions and predicted observables match. Agreement with a broad constraint cannot replace the extended calculation described in Table 3. [1] [3]

Chronology and reporting guidance

The archive records September 8, 2026 as the completed galactic-manuscript milestone. The current public text includes later scope qualifications. This Brief was prepared November 2, 2026; it does not establish that every qualification or this wording appeared on the archive date. The independent observational papers were published in 2013 and 2018. Journal publication of the current MQ manuscript is not established. [1] [2] [3]

For a story, identify the conditional spherical calculation as the achieved step and the common metric plus resolved-lens comparison as the outstanding test. The central mass arithmetic has been checked here. The enhancement, radius and significance remain source-reported at their stated level. Request the potential derivation, frozen source profile, calculation files and reconciled uncertainty budget before describing a successful lensing validation. The observations provide a demanding independent target; they do not yet supply that conclusion.