Appendix J: Geometric Framework Neutrality and Dual Applications

Revision note (v13.0). The geometry numbers in this appendix were wrong and are corrected. The ratio $R_T/r_T=6.4$ does not match the Earth's radius ratio (1.0034) and gives $J_2=0.376$ or $0.503$ against the measured $1.08263\times10^{-3}$ , about 350 or 465 times too large. The pole-asymmetry null of zero is replaced by the fact that standard geodesy already measures about 45 m. The claim that odd harmonics are forbidden is withdrawn, because the real Earth has measured odd harmonics. The geomagnetic estimate $2\times10^{-14}$ is withdrawn (the formula as written gives $2.5\times10^{-25}$ ). The statement that "the heliocentric model is a consequence of MCE" is withdrawn: the field equations are geometry-neutral. The classical tests are Level 0 results inherited from General Relativity. The solar-system scalar row now records the first-pass thin-shell estimate; the full calculation is open. The HUST-Grace2026s noise floors and the signal-to-noise columns are removed. See the main document for the architecture and Global Geometry Hypothesis Tests for the computations.

Preamble: Why Geometry Matters

General Relativity is written on a pseudo-Riemannian four-manifold, and its solutions depend on the global boundary conditions chosen. A theory whose predictions depend on a global geometric assumption inherits that assumption.

The field equations of MCE are locally valid on any smooth Riemannian or pseudo-Riemannian manifold, so the theory makes no mandatory statement about the global shape of the Earth, the configuration of the solar system or the topology of the universe. Predictions come from the local field equations and from the boundary conditions imposed by the matter distribution. Neutrality of the equations does not make every global geometry compatible with data. Each geometry is a separate hypothesis and has to be tested, as in Global Geometry Hypothesis Tests. Earth-fixed coordinate descriptions are legitimate relabellings of the same physics and are set out in Reference Frame Equivalence.

This appendix serves two purposes:

It states the geometric framework neutrality of the MCE field equations.
It links to the two standalone framework documents:
- Toroidal Field Framework for MCE
- Standard Heliocentric Framework for MCE

The detailed toroidal (TF) and Sun-centred (SH) cases are separate documents, so the core effective theory can be assessed without framework cross-talk. This appendix keeps the formal neutrality statement and the comparison logic.

1. Geometric Framework Neutrality: The Formal Statement

1.1. The field equation in covariant form

In the Level 1 sector of v13.0 the scalar field obeys, for non-relativistic matter of density $\rho$ ,

\nabla^\mu\nabla_\mu\phi = V'(\phi) + \frac{\beta}{M_{\rm Pl}}\,\rho\,e^{\beta\phi/M_{\rm Pl}},

where $\nabla_\mu$ is the covariant derivative compatible with the metric $g_{\mu\nu}$ , $V(\phi)$ is the scalar potential, $\beta$ is the species coupling, and the metric obeys the Einstein equations with $G$ taken from experiment. The static linearised limit is electrostatics in a screening medium (see Section 2.1 of the screened scalar document). The v12 form of this appendix, $\nabla^\mu\nabla_\mu\phi-m_\phi^2\phi=-\kappa\,T$ with a fixed mass $m_\phi$ and a separate vector-field equation, is superseded. The vector field is withdrawn from the core: ordinary electromagnetism already supplies repulsion between like charges, and a baryon-number vector would be a separate fifth force constrained by torsion-balance tests.

The key observation. The equations contain no term that specifies the topology or global geometry of the manifold $\mathcal{M}$ . The metric and its connection appear, and they are determined locally by the matter distribution. The global topology enters only through boundary conditions.

1.2. The principle of local equivalence

Statement (local equivalence). Let $(\mathcal{M}_1, g_1)$ and $(\mathcal{M}_2, g_2)$ be smooth manifolds that are locally isometric in an open neighbourhood $\mathcal{U}$ of a point $p$ (there is a local diffeomorphism $\psi:\mathcal{U}_1\to\mathcal{U}_2$ with $\psi^*g_2=g_1$ ). Then the field equations and their solutions restricted to $\mathcal{U}$ are identical, given the same boundary data on $\partial\mathcal{U}$ .

Consequence. A local experiment cannot distinguish a manifold with toroidal global topology from one with spherical global topology if the local metric and the boundary data are the same. This is a statement about the equations. It does not say that the local metric and boundary data of a torus and a sphere are the same, and in practice they are not: the external gravitational multipoles differ, and they are measured (section 4 of Global Geometry Hypothesis Tests).

1.3. Companion framework documents

Toroidal Field Framework for MCE isolates the TF boundary conditions and the satellite-gravimetry tests.
Standard Heliocentric Framework for MCE isolates the conventional weak-field and solar-system reading.

A null result against TF-specific signatures constrains the TF application, not the local field equations or the SH application.

Global Geometry Hypothesis Tests for MCE tests the flat-disc, toroidal, non-rotating and firmament hypotheses against measured quantities.
Reference Frame Equivalence for MCE sets out when an Earth-fixed description reproduces the same instrument readings.

2. The Toroidal Field (TF) Framework

2.1. Physical motivation

The TF framework posits that the dominant large-scale structure of the Earth's mass distribution, and of the field sourced by it, is toroidal. The motivation offered in v12 was the toroidal component of the geomagnetic field. The Earth's magnetic field does have a toroidal component inside the core and mantle, generated by differential rotation of conducting fluid, in addition to its dipole (poloidal) component. That is a fact about the magnetic field. It does not imply that the mass distribution is toroidal, and no argument linking the two was derived.

A toroidal solenoid of major radius $R_T$ and minor radius $r_T$ carrying a uniform current has interior field $B=\mu_0 nI/(2\pi r)$ , confined within the torus body, with zero exterior field. In the MCE context that analogy does not make the mass distribution toroidal. Outside a mass distribution the dominant long-range acceleration is the Newtonian field of the metric sector, and the departures from spherical symmetry appear as gravitational multipoles. The torus has much larger multipoles than the Earth, which is what excludes the nominal parameters (section 2.3).

2.2. The TF boundary conditions

In the TF framework the Earth is modelled as a toroidal mass distribution with major radius $R_T$ (the radius of the centreline), minor radius $r_T$ (the radius of the tube) and density $\rho(\mathbf{x})$ concentrated within the torus body. The boundary conditions for the scalar are

\phi(\mathbf{x}) \Big|_{\partial \mathcal{T}} = \phi_{\text{surface}},\qquad \hat{n} \cdot \nabla \phi \Big|_{\partial \mathcal{T}} = -\frac{\beta_0}{M_{\rm Pl}}\,\sigma_{\text{eff}},

where $\partial\mathcal{T}$ is the torus surface, $\hat{n}$ is the outward normal and $\sigma_{\text{eff}}$ is an effective surface source density. The v12 text wrote the coupling as $\kappa$ , which is withdrawn; $\beta_0/M_{\rm Pl}$ is the corresponding quantity in v13.0. The interior scalar field would have to be solved numerically. No solver is included in the current document set, and the v12 reference to a code appendix is removed.

2.3. TF model claims and their status

The quadrupole moment $J_2$ is the largest departure of the Earth's gravity field from spherical symmetry. For a uniform solid torus,

J_2 = \frac{I_z - I_x}{M a^2}, \qquad I_z = M\left(R_T^2 + \tfrac{3}{4}r_T^2\right), \quad I_x = M\left(\tfrac{R_T^2}{2} + \tfrac{5}{8}r_T^2\right),

where $a$ is a reference radius.

$R_T/r_T$	Reference radius $a$	$J_2$ (torus)	Ratio to measured $1.08263\times10^{-3}$
6.4	$R_T+r_T$	0.376	348
6.4	$R_T$	0.503	465

The v12 text took $R_T/r_T=6.4$ as "matching Earth's mean radius to semi-minor axis ratio". The Earth's equatorial-to-polar radius ratio is 1.0034 (6378.137 km to 6356.752 km, about 3963.2 miles to 3949.9 miles). A torus with $R_T/r_T=6.4$ is a thin ring. The nominal TF parameter set is therefore excluded by the measured $J_2$ , under the stated assumptions (uniform density, a single connected body). The TF framework stays listed as a falsifiable branch. A revised parameter set must reproduce the measured $J_2$ before its other signatures are worth computing.

The v12 claims in the table below are therefore recorded with their status.

Observable	Standard expectation	v12 TF claim	v13.0 status
Surface gravity variation	$g$ varies by about 0.5% from equator to pole (rotation and flattening)	A toroidal anisotropy, stronger near the inner equator and weaker near the outer equator	Open: not computed for any parameter set that reproduces $J_2$
Chandler wobble period	About 433 days	Shifted by $\delta T$ depending on $R_T/r_T$	Open: not computed
Gravitational anisotropy	Isotropic to leading order	A $\cos(2\theta_T)$ pattern with $\Delta g/g\sim(r_T/R_T)^2(\lambda_c/r_T)^2$	Withdrawn: it used the fixed coherence length $\lambda_c$ , which is retired, and was not derived
Pole-to-pole asymmetry	Not zero. Standard geodesy gives a geoid-height difference of about 45 m between the poles (odd zonal harmonic $J_3=-2.53\times10^{-6}$ )	$\Delta g_{\rm poles}/g\sim10^{-8}$ , "a unique TF signature absent in any spherical model"	Withdrawn: the null was wrong and the value was not derived

The v12 pole-asymmetry null of 0 ± 3 mm treated the standard value as zero. A model of the Earth as an oblate spheroid has no pole asymmetry, but the real Earth does, because of $J_3$ and the higher odd harmonics. The EGM2008 geoid heights are about +14.9 m at the North Pole and −30.1 m at the South Pole (Pavlis et al. 2012, quoted from the Global Geometry document and not re-derived here). The $J_3$ term alone gives about 32 m. A torus prediction of 0.72 m would have to be found as a residual after the full standard model, not against zero.

2.4. Reconciling the direction of "down"

A common objection to toroidal geometry is: "If the Earth is a torus, why does 'down' point under one's feet and not towards the centreline of the torus?"

The answer in MCE is the same as in Newtonian gravity. "Down" is the direction of the gradient of the Newtonian potential of the metric sector at the observer's position, not the gradient of the scalar $\phi$ , and not a geometric centre. On the surface of a torus that gradient points into the torus body. On a sphere all the local vertical directions converge on one point. On a torus they converge towards the tube's centreline on the outer surface and diverge from it on the inner surface. This difference is what produces the multipole structure of section 2.3, and the measured multipoles are not those of a ring with $R_T/r_T=6.4$ .

3. The Standard Heliocentric (SH) Framework

3.1. The classical tests are Level 0 results

The classical tests of gravity are results of the metric sector of the theory (Level 0): the Einstein–Hilbert action with Newton's constant $G$ taken from experiment. They are inherited from General Relativity and are not derived from the MCE scalar. A pure scalar sourced by the trace $T$ does not couple to light ( $T=0$ for radiation), and pure scalar gravity gives the wrong perihelion advance and light deflection. The v12 table attributed these results to "the $\phi$ field gradient" and to "scalar potential"; that attribution is withdrawn.

Classical test	Result	Status
Mercury perihelion precession	43 arcsec/century	Inherited from General Relativity
Light deflection by the Sun	1.75 arcsec	Inherited
Gravitational redshift	$\Delta\nu/\nu=\Delta\Phi_N/c^2$	Inherited
Shapiro time delay	Spacetime curvature	Inherited
Keplerian orbits	Geodesics of the weak-field metric	Inherited
Binary-pulsar tensor-quadrupole decay	Tensor radiation from the metric sector	Inherited
Speed of tensor gravitational waves	Equal to $c$ , because the tensor kinetic term is unmodified	Inherited
Scalar contribution in the solar system	Negligible only if thin-shell screening holds	Estimated: first-pass estimate passes with a thin margin; full calculation open

Level 1 is a scalar-tensor theory in the Einstein frame, so the Cassini bound $\gamma-1=(2.1\pm2.3)\times10^{-5}$ (Bertotti, Iess and Tortora 2003) applies. It is satisfied only if the screening mechanism removes the solar-system scalar force. A first-pass estimate for uniform spheres, with $n=1$ , $\Lambda=2.4$ meV and $\beta_0=0.053$ , gives $3\,\Delta R_\oplus/R_\oplus=1.2\times10^{-7}$ at a galactic ambient density of $1.7\times10^{-21}$ kg/m³ and $5.1\times10^{-8}$ at $10^{-20}$ kg/m³, against the MICROSCOPE requirement $\lesssim1.9\times10^{-7}$ . The Earth passes by a factor of about 1.5 to 4, and the Sun gives $3\,\Delta R/R=4.0\times10^{-11}$ . The full calculation (density profile, atmosphere, ambient field value, Moon, the Sun's field at the Earth, and Cassini) is open.

3.2. Where the Newtonian potential comes from

In the Sun-centred application, the Newtonian potential $\Phi=-GM_\odot/r$ and the Keplerian orbits follow from the metric sector with $G$ taken from experiment. The v12 derivation through $\nabla^2\phi_\odot=-\kappa\rho_\odot c^2$ and the matching condition $\kappa^2/(4\pi)=G/c^2$ is withdrawn. It fixed $\kappa$ by $G$ , which only relabels $G$ , and the value of $\kappa$ did not follow from its own formula.

The statement in v12 that "the heliocentric model is a consequence of MCE, not an assumption" is withdrawn. The field equations are geometry-neutral (section 1): they do not select a configuration of the solar system.

3.3. Reader alignment guide

For readers approaching from a Sun-centred observational standpoint: the measured solar-system effects (orbital periods, spacecraft trajectories, lensing of background stars, gravitational redshift in atomic clocks) are reproduced by General Relativity in the weak field. MCE inherits that agreement through its metric sector. The additional MCE scalar force is constrained by those same data: it must be screened. The question of why gravity exists at all is not answered by MCE at Levels 0 and 1; Level 2 (an emergent origin of $G$ ) is an open programme.

For readers approaching from a toroidal or other alternative standpoint: the local physics is identical. A toroidal mass distribution produces a "down" direction at every surface point. The global multipole structure is where the framework meets data, and the nominal parameters fail there (section 2.3).

4. The Geomagnetic Coupling in the Toroidal Context

4.1. Geomagnetic toroidal field and a possible coupling

The v12 text proposed a coupling between the geomagnetic field and a vector field $A_\mu^{\rm EME}$ through a term

\mathcal{L}_{\text{geo-QVP}} = \xi \, F_{\mu\nu}^{\text{EM}} \, F^{\mu\nu}_{\text{EME}},

with $\xi$ a small coupling of dimension mass $^{-2}$ . The vector field is withdrawn from the core (section 1.1), so this coupling is speculative and optional. It is not part of Level 1.

4.2. Quantitative estimate (withdrawn)

The v12 estimate used $B_T\sim10^{-3}$ T, $\xi\sim G/c^4$ and $\kappa=1.623\times10^{-10}$ C/kg in

\frac{\Delta g}{g} \sim \frac{G B_T^2}{\mu_0 c^4 \kappa^2},

and reported $\approx2\times10^{-14}$ . Evaluated as written with $c^4$ , the expression gives $2.5\times10^{-25}$ . The v12 numerical denominator used $8.99\times10^{16}$ , which is $c^2$ , not $c^4$ ; with $c^2$ the expression gives $2.2\times10^{-8}$ . Neither value is $2\times10^{-14}$ . The expression is also not dimensionless when $\kappa$ is in C/kg. Since $\kappa$ is withdrawn, the estimate is withdrawn, and no replacement has been computed. The v12 statements that the effect is "at the sensitivity frontier of GRACE-FO" and that a null result constrains $\xi<100\,G/c^4$ are withdrawn.

4.3. Grounding in the experimental literature: Tajmar/Graham comparison

An electromagnetism-gravity coupling is not unprecedented in the experimental literature. The most directly relevant work is that of Tajmar, de Matos and collaborators (2006 to 2011), who searched for anomalous gravitomagnetic fields from rotating superconducting rings. The table repeats the values as quoted in the v12 text. All numbers in the "Measured coupling", "GR prediction" and "Graham" columns are quoted from the v12 text and are to be verified against the cited papers. The "MCE prediction" column is not computed in v13.0: it depended on the withdrawn vector field and on an undefined coefficient $\xi'$ .

Experiment	Configuration	Measured coupling (as quoted in v12; to be verified)	GR prediction (as quoted; to be verified)	MCE prediction
Tajmar et al. 2006 (AIP Conf. Proc.)	Nb ring, 6500 rpm, $T<T_c$	$B_g/\Omega\approx10^{-8}$ m/s² per rad/s	$10^{-26}$ m/s² per rad/s	Not computed
Tajmar et al. 2008 (ESA report)	Nb ring, varied geometry, gyroscope readout	$B_g\approx(3.6\pm0.6)\times10^{-3}\,B_\phi$	Far below $10^{-20}$	Not computed
Graham et al. 2011 (as cited in v12; the reference is to be verified)	Atomic beam in a rotating frame	Null result: $\lvert B_g\rvert<5\times10^{-9}$ m/s² at $2\sigma$	Far below $10^{-20}$	Not computed
Earth, satellite gradiometry	Not yet searched	Not applicable	Zero	The v12 value $2\times10^{-14}$ is withdrawn (section 4.2)

The v12 table stated that the Tajmar effect is " $\sim10^{12}\times$ larger than GR", while the note below it stated an "enhancement factor $\sim10^{18}$ ". With the quoted numbers, $10^{-8}/10^{-26}=10^{18}$ , so the table's $10^{12}$ was inconsistent with its own entries. The ratios to MCE "predictions" in the v12 table are withdrawn with those predictions.

Critical note on the Tajmar anomaly. The reports of a large anomalous gravitomagnetic signal in 2006 to 2008 were not confirmed by independent replication (Hathaway and Cleveland 2009 and Graham et al. 2011, as cited in v12; both references to be verified). The effect is widely attributed to systematic error in the gyroscope readout, such as stray magnetic coupling to the superconducting ring. MCE does not predict the Tajmar anomaly. The v12 statement that the coupling is suppressed by a factor $G/c^4\sim10^{-44}$ relative to electromagnetic couplings was not derived, and it is not repeated.

Parity hints. The Tajmar 2008 report noted a possible dependence on the direction of rotation. This is interesting as an experimental observation. Without an independently confirmed measurement of the anomaly it remains speculative. The v12 estimate of a parity asymmetry of order $10^{-28}$ depended on the withdrawn coupling and is not computed.

What Tajmar-type infrastructure could test. The SQUID-based gravimeters and cryogenic rotation platforms developed for these experiments could test whether a superconducting shield changes a gravity signal. The v12 prediction of a signal below $10^{-14}$ inside a niobium shield was not derived and is withdrawn. Such a test would be a search for an unexplained effect, with no MCE amplitude attached to it.

Analysis protocol (revised). A cross-correlation of gravity-field residuals with the geomagnetic toroidal pattern has no predicted amplitude to compare against, because the estimate of section 4.2 is withdrawn. A correlation could also arise from unrelated causes (crustal and mantle structure correlate with the field in places). The v12 protocol subtracted the "best-fit even-degree harmonic expansion" to leave a residual. That step would leave the real Earth's measured odd harmonics in the residual. A usable protocol needs a null model that contains the measured odd harmonics, the topography and a density model, and a stated amplitude from the theory. Neither exists.

5. Satellite-Gravimetry Tests of the Toroidal Framework

The v12 text presented "specific, numerical, pre-registered predictions" for GOCE and GRACE-FO. They assumed the nominal TF parameters ( $R_T/r_T=6.4$ , "matching Earth's mean radius to semi-minor axis ratio"), $B_T=10^{-3}$ T and $\xi=G/c^4$ . The nominal parameters are excluded by the measured $J_2$ (section 2.3), the geomagnetic estimate is withdrawn (section 4.2), and the columns of "predicted significance" were not computed from any stated noise model. The values are listed below with their status. None is a pre-registered prediction.

5.1. The HUST-Grace2026s model

HUST-Grace2026s is a real static gravity-field model, described in an ESSD preprint (essd-2026-53) and distributed by ICGEM (DOI 10.5880/icgem.2026.001). It uses GRACE data from April 2002 to June 2017 and GRACE-FO data from June 2018 to March 2025, and is complete to degree and order 180 (a half-wavelength of about 110 km, about 69 miles, at the equator). The v12 text also attributed noise floors of $3\times10^{-13}$ m/s² (30°S to 30°N), $6\times10^{-13}$ m/s² (high latitudes) and 3 mm (polar geoid) to this model. Those numbers are not from the source and are withdrawn. The error spectrum should be taken from the ESSD paper and the ICGEM calibrated errors when an analysis is made.

5.2. Status of the v12 forecasts

Observable	v12 value	v13.0 status
Pole-to-pole gravity difference	$8.1\times10^{-6}$ m/s² "above the standard model"	Withdrawn: it came from the excluded nominal parameters and was compared against a null of zero. The relation between this value and the 0.72 m geoid difference quoted beside it was not derived
Geoid harmonic $\delta C_{3,1}$	$\approx2\times10^{-10}$ , "forbidden by symmetry in any spherical model"	Withdrawn: the real Earth has measured odd harmonics. The known $C_{3,1}$ term is of order $10^{-6}$ (to be verified against ICGEM), far above the quoted value
Geomagnetic-gravity cross-correlation	Pearson $r\approx0.03$	Withdrawn: no derivation, and the underlying amplitude estimate is withdrawn
Gravity-gradient anisotropy	$\Delta T_{zz}\approx1.5\times10^{-12}$ s $^{-2}$	Withdrawn: no derivation
Temporal variation correlated with the solar cycle	$\delta g\approx3\times10^{-13}$ m/s²	Withdrawn: no derivation

5.3. Revised test structure

If a revised toroidal parameter set that reproduces the measured $J_2$ is proposed, the tests would take the following form.

Test 1 (pole asymmetry). Compute the geoid height difference between the poles from the toroidal model and compare it with the standard value of about 45 m, which comes from the measured odd zonal harmonics ( $J_3=-2.53\times10^{-6}$ and higher). The signal is a difference between two large numbers and is only meaningful if the model reproduces the measured odd harmonics to the required accuracy. The null hypothesis is the standard geodesy value, not zero.

Test 2 (odd harmonics). Fit the measured odd-degree harmonics with the toroidal model. A model that cannot reproduce them is excluded.

Test 3 (geomagnetic correlation). Only meaningful with a stated amplitude and a null model (section 4.3).

The analysis with an MCE hypothesis has not been performed. The public data (GOCE, GRACE, GRACE-FO, HUST-Grace2026s) exist.

6. Observational Tests Distinguishing TF from SH Within MCE

Both applications are internal to MCE. The question is which global geometry fits the observations.

Test	TF (v12, nominal parameters)	SH / standard geodesy	Status
Quadrupole moment $J_2$	0.376 (348 times the measured value)	$1.08263\times10^{-3}$ , measured	TF nominal parameters: Withdrawn
Pole-to-pole asymmetry	0.72 m, to be found as a residual	About 45 m, measured	TF value not derived: Withdrawn
Odd-degree harmonics	Claimed to be a TF signature	Nonzero and measured	Withdrawn as a discriminator
Geomagnetic-gravity correlation	$r\approx0.03$	No amplitude predicted	Withdrawn
Gravity-gradient anisotropy	$1.5\times10^{-12}$ s $^{-2}$	Not applicable	Withdrawn
Anisotropic "QVP phase shift" varying with the geomagnetic toroidal angle	A phase shift that varies with the angle; no value derived	No toroidal angular dependence	Withdrawn: a heuristic without a derivation, tied to the retired $\lambda_c$ and $\rho_c$

7. Philosophical Position: Empirical Priority Over Geometric Assumption

The global geometry that is correct is the one that agrees best with the totality of high-precision data: satellite gravimetry, seismology, very long baseline interferometry, pulsar timing and cosmological surveys. The existing body of evidence (spacecraft trajectories, lunar laser ranging, satellite geoid maps) is overwhelmingly consistent with a near-spherical (oblate spheroid) Earth. MCE inherits that agreement through its metric sector.

The TF framework made additional claims (pole asymmetry, odd harmonics, geomagnetic-gravity coupling). For the nominal parameters they do not survive the comparison with the measured $J_2$ , the measured odd harmonics and the standard geodesy value of the pole asymmetry. TF remains a falsifiable branch until a parameter set that reproduces the measured $J_2$ is proposed. This is a statement about one application of the equations. It does not constrain Level 1 of MCE.

The v12 text called MCE "the first alternative gravity framework that is both locally equivalent to GR and globally geometry-agnostic". That claim is withdrawn: geometry-neutral field equations are the normal situation for a covariant theory.

8. Summary Table

Property	GR	MCE / SH	MCE / TF
Geometric commitment	Pseudo-Riemannian manifold	Locally pseudo-Riemannian (background metric)	Locally pseudo-Riemannian (background metric)
Global application	Set by boundary conditions and data	Near-spherical Earth, Sun-centred solar system, as in standard practice	Toroidal Earth hypothesis; the nominal parameters are excluded by $J_2$
Keplerian orbits	Yes	Yes: Inherited from the metric sector	Not computed for a toroidal body
Light deflection, perihelion advance, Shapiro delay	Yes	Yes: Inherited (Level 0), not derived from the scalar	Yes locally, Inherited; global consequences not computed
Distinguishing prediction	None (baseline)	Composition-dependent screened scalar force (Level 1), if screening allows	None that survives for the nominal parameters
Test by satellite gravimetry	Reference	Standard geodesy	Yes; the nominal parameters are already excluded by $J_2$

The MCE/TF framework is falsifiable, and for its nominal parameters it is falsified by the measured $J_2$ . It neither requires nor forbids the Sun-centred application. It is a separate, independently testable hypothesis.