Appendix P: Data Integration and Forecasts

MCE Theory v13.0 — October 2026

Revision note (v13.0). Rewritten. The table of "actual bounds" now contains only verified results. Removed: the MAGIS-100 "bound" and every row built on it (the instrument is still being installed at Fermilab and has produced no results), the "Stanford 2022 below $7\times10^{-9}$" and "Eöt-Wash 2023" entries (no source), the Euclid, DESI and MACS J0025 forecast tables with their signal-to-noise numbers, and the "survival windows" arithmetic, which contradicted itself. The forecast inputs were wrong: a wavenumber of $6.3\times10^{6}\ {\rm m^{-1}}$ was read as $h$/Mpc, an error of about 22.5 orders of magnitude, and $\kappa^2C\sim10^{-21}$ cannot give percent effects. They are replaced by a statement of what each real experiment constrains, a growth-of-structure method marked "not yet computed", and a protocol for the global-geometry tests with HUST-Grace2026s. Earth and Sun screening is the first-pass estimate; the full calculation is open. Hamilton, Sabulsky, Kramer, and Weisberg and Huang are entered from the verified list. See the architecture section of the main document and the screened scalar sector.

Overview

This appendix compares the parameter space of the screened scalar sector (Level 1 of the status ladder) with published measurements. It has three parts:

1. A table of verified bounds, and what each constrains in the language of $(2\beta_0^2C,\ \Lambda,\ n)$ (Sections 1 and 2).
2. The method for a growth-of-structure forecast, marked "not yet computed" (Section 3).
3. A protocol for the global-geometry tests with the HUST-Grace2026s gravity-field model (Section 4).

Falsification is stated as exclusion of a region of parameter space (Section 5).

1. Verified bounds

The Jaffe correction replaced an earlier, larger exclusion, and the corrected figures are used. All of these results were published before the present documents, so they are consistent retrodictions, not tests of earlier predictions.

MAGIS-100 is not a data source. It is under construction at Fermilab. The laser laboratory was completed in January 2026, installation is due to finish in late 2027 and commissioning in 2028. There are no results.

2. What each experiment constrains

2.1 Dictionary

Three kinds of measurement bear on the parameters.

• Earth-sourced composition tests (MICROSCOPE, rotating torsion balance, Asenbaum). The scalar acceleration of each test mass is sourced by the Earth, whose external field is reduced by the thin-shell factor $s_\oplus=3\,\Delta R_\oplus/R_\oplus$. For a pair with composition difference $\Delta(Z/A)$,

  $$\frac{\Delta a}{a}\simeq\left(2\beta_0^2\,C\,s_\oplus\right)\varepsilon\,\Delta\!\left(\frac{Z}{A}\right)f,\qquad \varepsilon=1.378\times10^{-3},$$

  so a bound $X$ on $\eta$ is a bound on the product $2\beta_0^2C\,s_\oplus f<X/[\varepsilon\,\Delta(Z/A)]$. The thin-shell factor $s_\oplus$ depends on $(\Lambda,n,\beta_0)$ and on the cosmological field value. A first-pass estimate for the benchmark is given in Section 2.2. The full calculation is open.

• Source-mass tests (Hamilton, Jaffe). The composition difference between the tungsten source and the caesium atoms enters only at the $10^{-6}$ level, so these tests bound the composition-independent coupling $\beta_0$ (as $M=M_{\rm Pl}/\beta$), $\Lambda$ and $n$. They do not bound $C$ directly.

• Force-law tests (Lee). The scalar force between bodies is $2\beta_i\beta_j$ times the Newtonian force, so the strength is $\alpha=2\beta_0^2$ before screening, with a range $\lambda(\rho)$ that depends on density.

2.2 Arithmetic at the legacy benchmark

The benchmark has $2\beta_0^2=5.7\times10^{-3}$ and $C=0.03$, so $2\beta_0^2C=1.7\times10^{-4}$. The value $2\beta_0^2=5.7\times10^{-3}$ is an inference from the old number $1.9\times10^{-8}$ (screened scalar document, Section 6). The uncertainty scales below are not confidence limits. A formal 95% limit is larger than the scale by a factor of a few, and is not computed here.

The scale for Asenbaum is the statistical and systematic errors combined in quadrature, $\sqrt{1.8^2+3.4^2}\times10^{-12}=3.8\times10^{-12}$. The last column divides the bound by $2\beta_0^2C=1.7\times10^{-4}$. It is a requirement on the product $s_\oplus f$, obtained by dividing the uncertainty scale by the benchmark $2\beta_0^2C$. It is not a computed value of $s_\oplus$.

A first-pass estimate for uniform spheres ($n=1$, $\Lambda=2.4$ meV, $\beta_0=0.053$; screened scalar document, Section 4) 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 an interplanetary density of $10^{-20}$ kg/m³. Against the MICROSCOPE requirement $\lesssim1.9\times10^{-7}$ the benchmark passes by a factor of about 1.5 to 4. The Sun's first-pass factor is about $4\times10^{-11}$, which passes by many orders of magnitude. For $n=1$ the factor scales as $\Lambda^{5/2}\beta^{-3/2}\rho_{\rm ambient}^{-1/2}$. It exceeds the Earth requirement by a factor of about 20 at $\Lambda=10$ meV and by a factor of about 8 at $\beta_0=0.01$. The full calculation (density profile, atmosphere, ambient field value, Moon, the Sun's field at the Earth, and the Cassini comparison) is open.

For the other experiments:

• Hamilton (2015). One-tailed 95% limit $a<5.5\ \mu$m/s². In the paper's words, this excludes chameleons at $\Lambda=2.4$ meV for $M<2.3\times10^{-5}M_{\rm Pl}$ (most conservative case). With $M=M_{\rm Pl}/\beta$ that is $\beta>4.3\times10^{4}$. The benchmark $\beta_0\approx0.053$ is about $8\times10^{5}$ times smaller.
• Jaffe (2017, author correction 2023). The corrected exclusion $M<1.7\times10^{-3}M_{\rm Pl}$ at $\Lambda=2.4$ meV and $n=1$ is $\beta>1/(1.7\times10^{-3})=590$, that is $2\beta^2>6.9\times10^{5}$. The benchmark has $\beta_0\approx0.053$ and $2\beta_0^2=5.7\times10^{-3}$, which is $1.1\times10^{4}$ times smaller in $\beta$ and $1.2\times10^{8}$ times smaller in $2\beta^2$. The benchmark is far from this excluded corner. The exclusion applies to that source and chamber and to the stated $\Lambda$ and $n$. The uncorrected arXiv figures are superseded and are not used.
• Sabulsky (2019). $a_\phi=-77\pm201$ nm/s², with a 90% limit $<183$ nm/s². No exclusion in $M$ is taken from this paper: the screened scalar document records that the paper's chameleon wording is ambiguous.
• Kramer (2021) and Weisberg and Huang (2016). The tensor damping results, $1.3\times10^{-4}$ (95%) for the double pulsar and the Hulse–Taylor ratio $0.9983\pm0.0016$, belong to the metric sector. Kramer's dipole bound $B_D\lesssim4\times10^{-10}$ (95%) constrains scalar-tensor couplings of strongly self-gravitating bodies. The mapping onto $\beta_0$ has not been done.
• Lee. The benchmark strength $5.7\times10^{-3}$ is about 175 times weaker than gravity. The quoted limit of 38.6 μm is for gravitational strength. The limit at strength $5.7\times10^{-3}$ has to be read from the exclusion curve of the paper, which is to be verified. The scalar force is not a Yukawa force of fixed range, so applying the result needs the field solution for the bodies of the torsion balance.
• Cassini (Bertotti, Iess and Tortora, Nature 425, 374, 2003): $\gamma-1=(2.1\pm2.3)\times10^{-5}$. This is the solar-system counterpart of the Earth condition. The first-pass solar factor is about $4\times10^{-11}$. Comparing that estimate with the Cassini bound is part of the full calculation, which is open.

2.3 Reading

The verified bounds are of three kinds: Earth-sourced composition tests, a strongly coupled corner, and a force-law test. At the benchmark the Earth-sourced tests leave the point in place when $s_\oplus f$ meets the requirements in Section 2.2. The first-pass estimate does so for the Earth, by a factor of about 1.5 to 4, and for the Sun by many orders of magnitude. The full calculation is open. The Earth-sourced tests bound $s_\oplus$ and the product $2\beta_0^2C\,s_\oplus f$, not $2\beta_0^2C$ alone. The GW170817 result (Abbott et al., 2017), a fractional difference between the speed of gravity and light between about $-3\times10^{-15}$ and $+7\times10^{-16}$, bounds the tensor speed, which belongs to the metric sector. It is not used to bound the scalar parameters.

3. Growth of structure: forecast method (not yet computed)

No forecast is given. The method is as follows.

1. Background. Solve for the field $\phi(a)$ and the mass $m(a)$ in the cosmic mean density for the chosen $(\beta,\Lambda,n)$. For the illustrative parameters of the screened scalar document ($n=1$, $\Lambda=2.4$ meV, $\beta=0.05$) the cosmic-mean row gives $m=5.4\times10^{-27}$ eV and a range of $3.7\times10^{19}$ m (about 1.2 kpc).

2. Effective coupling. In linear perturbation theory, for a screened scalar in a scalar-tensor theory, the effective gravitational coupling is a standard scalar-tensor result:

   $$G_{\rm eff}(k,a)=G\left[1+\frac{2\beta^2k^2}{k^2+a^2m^2(a)}\right],$$

   with $k$ the comoving wavenumber. It tends to $G$ for $k\ll am$ and to $G(1+2\beta^2)$ for $k\gg am$.

3. Growth. Insert $G_{\rm eff}(k,a)$ in the equation for the matter overdensity, $\ddot\delta+2H\dot\delta=4\pi G_{\rm eff}\,\bar\rho_m\,\delta$, and solve with the background of step 1.

4. Observables. The matter power spectrum $P(k,z)$ and the growth rate times the clustering amplitude, $f\sigma_8(z)$. Weak-lensing observables follow from the matter distribution. A conformally coupled scalar adds no deflection of light beyond general relativity for a given metric.

5. Nonlinear and screened regions. The linear formula holds where the scalar is unscreened and the field perturbation is small. Dense regions need the nonlinear solver and N-body treatment described in Appendix N.

6. Parameters constrained. Cosmic matter has no isospin composition difference at leading order, so the growth forecast constrains $\beta$, $\Lambda$ and $n$, not $C$.

7. Data. Cosmic microwave background, weak-lensing and galaxy-clustering surveys (Planck, DES, KiDS, DESI, Euclid). Their specifications are not quoted, because none has been verified.

Two scale checks (Estimated, not forecasts):

• The formula limits the change in the coupling to $G_{\rm eff}/G-1\le2\beta^2$, which is $5.7\times10^{-3}$ for $2\beta_0^2=5.7\times10^{-3}$. Growth integrates the modification over time, so the effect on $P(k)$ can be larger than the fractional change in $G$ by a factor of several. The size of that effect for a scale-dependent coupling is not computed here.
• For the illustrative parameters the transition scale is $am=5.4\times10^{-27}\ {\rm eV}/\hbar c=2.7\times10^{-20}\ {\rm m^{-1}}=8.4\times10^{2}\ {\rm Mpc^{-1}}$ at $a=1$. The extra term is about $2\beta^2(k/am)^2=5.7\times10^{-3}\times(k/844\ {\rm Mpc^{-1}})^2$, which is $8\times10^{-9}$ at $k=1\ {\rm Mpc^{-1}}$ and $2\times10^{-7}$ at $k=5\ {\rm Mpc^{-1}}$. The scales of the v12 table ($0.1$ to $5\ h$/Mpc) are far below $am$ for these parameters, so the effect on them is negligible there. Other values of $\Lambda$ and $n$ change $m(a)$ and have not been scanned.

The v12 table, which gave $+3.2\%$ to $+9.1\%$ in $P(k)$ at $k=1$ to $5\ h$/Mpc, is withdrawn.

The v12 inputs were wrong. The wavenumber $k_c=2\pi/\lambda_c=6.3\times10^{6}\ {\rm m^{-1}}$ for $\lambda_c=1\ \mu$m was quoted as $6\times10^{6}\ h$/Mpc. In ${\rm Mpc^{-1}}$ it is $1.9\times10^{29}$, so the quoted value was too small by a factor of $3.2\times10^{22}$. The coefficient $\kappa^2C$ with the withdrawn $\kappa=1.6\times10^{-10}$ C/kg is $8\times10^{-22}$, which cannot give percent effects. The CMB damping-tail and lensing statements built on the same inputs are withdrawn.

4. Global-geometry tests with HUST-Grace2026s

The geometry hypotheses (flat disc, torus) are set out in Global Geometry Hypothesis Tests for MCE, together with the standard-geodesy benchmarks they have to meet. This section states what the gravity-field model can and cannot supply.

The model. HUST-Grace2026s is a real static gravity-field model (ESSD preprint essd-2026-53; 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 given to degree and order 180. At that degree the shortest resolved half-wavelength is $\pi R_\oplus/180$, about 69 miles.

Null hypothesis. The pole geoid asymmetry is not zero in standard geodesy. The odd zonal harmonic $J_3=-2.53\times10^{-6}$ already gives a north–south geoid difference of about 45 m. The v12 null expectation of $0\pm3$ mm was wrong. A geometry hypothesis has to predict the residual after the full standard model, not a departure from zero.

Protocol.

1. Obtain the spherical-harmonic coefficients from ICGEM under the DOI above.
2. Synthesise geoid heights and gravity anomalies to degree 180, with the reference system and permanent-tide convention stated in the model documentation.
3. Write the alternative hypothesis in the same form: predicted coefficients (for example $J_2$ and $J_3$ of a torus with stated parameters) or predicted maps.
4. Compare the residuals against the formal errors given in the paper.

Limits. The noise floors attached to this model in v12, and used in grace_anomaly_sim.py, are not from the source paper. None is given here, and a forecast needs the paper's own error estimates. A static field model cannot test the rotation or reference-frame hypotheses. The v12 "geomagnetic coupling" signal has no derivation, and with the formula as written it gives $2.5\times10^{-25}$ against the $2\times10^{-14}$ that was quoted, so it is withdrawn. No signal-to-noise figure is stated.

5. Falsification as parameter-space exclusion

A null result at a stated sensitivity excludes a region of $(2\beta_0^2C,\ \Lambda,\ n)$. It does not by itself end every screened-scalar model, and it does not affect Level 0. The v12 statement that four simultaneous nulls would falsify MCE at $5\sigma$ is withdrawn.

• Earth-sourced tests exclude values of $2\beta_0^2C\,s_\oplus f$ above the bounds in Section 2.2. The first-pass estimate covers one benchmark point. The excluded region in $(\Lambda,n)$ follows once $s_\oplus(\Lambda,n,\beta_0)$ is computed across that space. The full calculation is open.
• Near-source composition tests with total uncertainty $\sigma$ in $\Delta a/a$ exclude, for aluminium and gold, $2\beta_0^2C\,f>1.5\times10^{4}\,\sigma$ at 95% (one-sided). The relation between this fractional bound and the absolute acceleration sensitivity is set out in the experimental design.
• Source-mass tests exclude the strongly coupled corner (Section 2.2).
• Not excluded: models with a short range or strong screening in the apparatus, smaller $\beta_0$ or $C$, other potentials and other couplings.

6. Material removed from v12

7. Status

References

• Touboul, P., et al. (2022). MICROSCOPE mission: final results of the test of the equivalence principle. Physical Review Letters 129, 121102.
• Schlamminger, S., et al. (2008). Test of the equivalence principle using a rotating torsion balance. Physical Review Letters 100, 041101.
• Asenbaum, P., Overstreet, C., Kim, M., Curti, J., Kasevich, M. A. (2020). Atom-interferometric test of the equivalence principle at the $10^{-12}$ level. Physical Review Letters 125, 191101.
• Hamilton, P., et al. (2015). Atom-interferometry constraints on dark energy. Science 349, 849.
• Jaffe, M., et al. (2017). Testing sub-gravitational forces on atoms from a miniature in-vacuum source mass. Nature Physics 13, 938. Author correction, Nature Physics 19, 1946 (2023).
• Lee, J. G., et al. (2020). New test of the gravitational $1/r^2$ law at separations down to 52 μm. Physical Review Letters 124, 101101.
• Sabulsky, D. O., et al. (2019). Experiment to detect dark energy forces using atom interferometry. Physical Review Letters 123, 061102.
• Kramer, M., et al. (2021). Strong-field gravity tests with the double pulsar. Physical Review X 11, 041050.
• Weisberg, J. M., Huang, Y. (2016). Relativistic measurements from timing the binary pulsar PSR B1913+16. Astrophysical Journal 829, 55.
• Khoury, J., Weltman, A. (2004). Chameleon fields: awaiting surprises for tests of gravity in space. Physical Review Letters 93, 171104. Chameleon cosmology, Physical Review D 69, 044026.
• Bertotti, B., Iess, L., Tortora, P. (2003). A test of general relativity using radio links with the Cassini spacecraft. Nature 425, 374.
• Abbott, B. P., et al. (2017). Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A. Astrophysical Journal Letters 848, L13.
• HUST-Grace2026s: ESSD preprint essd-2026-53; ICGEM DOI 10.5880/icgem.2026.001.