Cosmological Extension of the Electrostatic Mass Emergence (EME) Theory

Revision note (v13.0). The scope paragraph has been rewritten. It described the theory as a "local, terrestrial model" based on "the Earth's toroidal field" with a "strict no space/universe mechanisms" rule, which contradicted the rest of the document. The cross-reference to a Lagrangian "derived in Section 5 of the main report" is replaced by the Level 1 action in the main document. The modified Friedmann equations are kept; $G$ in them is the metric-sector constant. The dark-matter and dark-energy analogues are marked speculative and not computed. The earlier "unique signature" and "falsifiable CMB damping tail" are withdrawn: no CMB, matter power spectrum, growth or expansion-history calculation has been done. The linear growth formula $G_{\rm eff}(k,a)$ is stated. See section 1 of the main document. "EME" is the historical name of MCE.

1. Scope and Assumptions

MCE (historically EME) is stated in the main document as three levels. Level 0 is the metric sector, the Einstein–Hilbert action with Newton's constant $G$ taken from experiment. Level 1 is a screened scalar $\phi$ with a species-dependent conformal coupling to matter. Level 2, an emergent origin of the couplings, is an open programme. Nothing in this structure restricts the theory to a terrestrial setting. The toroidal Earth (Toroidal Field Framework) is one optional global boundary condition for the same local equations, and it plays no role in cosmology.

This document applies the Level 1 action on a Friedmann–Lemaître–Robertson–Walker (FLRW) background. It sets up the equations that a cosmological calculation would need, and it states what has and has not been computed. Everything below the level of the equations is Open: no cosmological observable has been calculated for MCE parameters.

To compare with large-scale data one averages the scalar field over cosmological volumes, which gives an effective stress-energy tensor $T_{\mu\nu}^{\text{EME(cosmo)}}$ that can be coupled to the FLRW metric. This coarse-graining is a deliberate, auditable step for comparison with $\Lambda$CDM.

2. Effective Cosmological Stress-Energy Tensor

2.1. Justification for coarse-graining

The scalar has a range $\lambda(\rho)=\hbar/(m_{\rm eff}(\rho)c)$ that depends on the local density (see the screened scalar document). For the illustrative benchmark in that document ($n=1$, $\Lambda=2.4$ meV, $\beta=0.05$, not fitted to data) the range at the cosmic mean matter density is about $3.7\times10^{19}$ m (about 1 kpc). Averaging over a volume $V$ with $\lambda^3\ll V\ll H^{-3}$ then smooths the local non-linear structure and gives a homogeneous and isotropic effective fluid. The Hubble radius is $H^{-1}\sim10^{26}$ m. The earlier text used the fixed length $\lambda_c\sim10^{-6}$ m for this step, and that length is retired.

2.2. The action and the effective fluid

We begin with the Level 1 action given in the main document:

with $M_{\rm Pl}^2=1/(8\pi G)$. The total energy-momentum tensor is $T_{\mu\nu}^{\text{Total}}=T_{\mu\nu}^{M}+T_{\mu\nu}^{\text{EME}}$, and the scalar part is averaged over a large comoving volume:

The dominant contribution comes from the scalar field. After coarse-graining, the effective density and pressure are

where $a$ is the scale factor. The v12 expressions also contained an interaction term $\langle\kappa\phi T\rangle$. In the Einstein-frame form used here the coupling sits in the matter sector: the energy density of non-relativistic matter scales as $a^{-3}e^{\beta\phi/M_{\rm Pl}}$, and the exchange of energy between matter and the scalar appears in the continuity equations. The scalar equation of motion on the background is $\ddot\phi+3H\dot\phi=-V'(\phi)-(\beta/M_{\rm Pl})\,\rho_m e^{\beta\phi/M_{\rm Pl}}$, with $\rho_m$ the conserved matter density (a standard result for conformally coupled scalars; see Khoury and Weltman, Physical Review D 69, 044026, 2004). Its static linearised limit is electrostatics in a screening medium (Section 2.1 of the screened scalar document).

3. Modified Friedmann Equations

The standard Friedmann equation, $H^2=(8\pi G/3)\rho_{\rm Total}$, is modified by the inclusion of the scalar's effective density $\rho_{\text{EME}}$:

where $\rho_b$ is the baryonic matter density, $\rho_r$ the radiation density and $\rho_\Lambda$ a cosmological-constant density. The acceleration equation is

$G$ in these equations is the Newton constant of the metric sector (Level 0). It is taken from experiment, and the equations hold for any total energy-momentum tensor. The statement in v12 that the scalar, rather than curvature, generates the force of gravity is withdrawn. If the potential $V(\phi)$ supplies the late-time acceleration, the separate term $\rho_\Lambda$ should be dropped to avoid counting the same contribution twice.

4. Dark-Matter and Dark-Energy Analogues (Speculative, Not Computed)

The coarse-grained scalar fluid is a candidate for the missing components of the cosmic inventory only in the sense that it has an energy density and pressure that can enter the equations of section 3. No calculation has been done that shows it supplies either component. Both subsections are Open and speculative.

4.1. Dark-matter analogue

If the scalar is non-relativistic and $V(\phi)$ is negligible at late times, the fluid can behave like pressureless matter. The continuity equation is

and for $w_{\text{EME}}=p_{\text{EME}}/\rho_{\text{EME}}\approx0$ it gives $\rho_{\text{EME}}(a)=\rho_{\text{EME},0}\,a^{-3}$.

Two statements from v12 are withdrawn, and one assumption is flagged.

• Withdrawn: that "the effective charge field of baryonic matter itself generates the required extra gravitational pull". For the legacy value of the coupling, $2\beta_0^2=5.7\times10^{-3}$, the unscreened scalar force is 0.57% of the Newtonian force. That is far smaller than the gap between the baryon density and the cosmic dark-matter density, which is a factor of several. Screening reduces the scalar force further.
• Withdrawn: that the model has a "unique signature" in a non-zero, scale-dependent sound speed $c_s^2(k,a)$. A non-zero sound speed is a property of many dark-sector models. No $c_s^2(k,a)$ has been derived for MCE.
• Assumption: that the scalar is non-relativistic with a negligible potential at late times. The benchmark potential, $V=\Lambda^{4+n}/\phi^n$, has no minimum in vacuum, so the field rolls instead of oscillating. A matter-like fluid is therefore not what the benchmark gives without further assumptions, and whether any Level 1 potential gives one has not been shown.

4.2. Dark-energy analogue

If the scalar is dominated by its potential energy, then $p_{\text{EME}}\approx-\rho_{\text{EME}}$ and $\rho_{\text{EME}}\approx\langle V(\phi)\rangle$, as for any slowly rolling scalar. The statement in v12 that this "suggests that the quantum vacuum polarisation that gives rise to the effective charge density is also the source of cosmic acceleration" is withdrawn. It has no calculation behind it. In the screened scalar document the scale $\Lambda=2.4$ meV is chosen as a reference equal to the dark-energy scale, and it is not derived. The cosmological-constant problem is not solved by MCE: the vacuum-energy toy model reduces a discrepancy of about 123 orders of magnitude to about 40, which is not a solution (see the main document).

5. Observational Signatures: What Has and Has Not Been Computed

No CMB, matter power spectrum, growth or expansion-history calculation has been done for MCE. The statements in v12 that the model has a "unique equation of state and scale-dependent coupling" and "makes specific, falsifiable predictions regarding the CMB and the growth of structure" are withdrawn.

5.1. CMB anisotropies

The earlier text asserted a shift in the third and higher acoustic peaks and a "suppression of the power in the damping tail", described as "a highly falsifiable signature". Neither was calculated, and both are withdrawn. A CMB calculation would require:

1. The background evolution of $\phi$ with its matter coupling, for specified $(\beta_0,C,\Lambda,n)$.
2. The perturbation equations for the scalar and the modified Boltzmann hierarchy for the species that couple to it.
3. An implementation in a Boltzmann code (CLASS or CAMB) and a comparison with data.

None of these has been done. Until it is, there is no statement about the CMB.

5.2. Growth of structure

For a scalar-tensor model with a screened scalar, the effective gravitational coupling in linear perturbation theory is scale dependent:

This is a standard scalar-tensor result, not a derivation specific to MCE. It has not been evaluated for MCE parameters. Two limits follow from the formula. For $k\ll a\,m(a)$, $G_{\rm eff}\to G$: scales larger than the scalar range feel no extra force. For $k\gg a\,m(a)$, $G_{\rm eff}\to G(1+2\beta^2)$, which for the legacy value $2\beta_0^2=5.7\times10^{-3}$ is an enhancement of 0.57% before any screening.

For a conformally coupled scalar with real $\beta$, $G_{\rm eff}\ge G$. The linear-theory effect is therefore an enhancement of growth on scales below the scalar range, not the suppression of $P(k)$ at high $k$ that the v12 text claimed. A suppression from a pressure or sound-speed effect (a Jeans length) is a separate mechanism that has not been derived. The v12 claim of a growth index $\gamma(a,k)$ that deviates from the $\Lambda$CDM value of about 0.55 is withdrawn for the same reason.

The Euclid, DESI and DES-type forecast tables in the v12 documents were withdrawn because their inputs were wrong. One input carried a unit error of about 23 orders of magnitude in the wavenumber $k_c$, and another used $\kappa^2C\sim10^{-21}$, which cannot produce per cent effects. Replacing them requires the method above, with the field equation solved for the screening in each environment. The forecast is not yet computed.

5.3. What would constrain the model

Once computed, a cosmological observable excludes a region of $(\beta_0,\Lambda,n)$ in the same way as the laboratory tests: the parameters for which the calculated signal exceeds the observed bound are excluded. Local tests (MICROSCOPE, the Eöt-Wash torsion balance, the Cassini bound, atom interferometry near a source mass) already restrict that space, as set out in section 7 of the screened scalar document.

6. Conclusion

This document gives the equations for applying the Level 1 scalar sector on an FLRW background: the coarse-grained fluid, the modified Friedmann and acceleration equations, and the linear growth coupling $G_{\rm eff}(k,a)$. It does not give a result. The dark-matter and dark-energy analogues are speculative. The linear-theory formula suggests a growth enhancement of order $2\beta^2$ below the scalar range rather than a suppression. No CMB or $P(k)$ calculation has been done, and the earlier claims of a unique signature are withdrawn. The cosmological extension remains an open programme, separate from the laboratory tests that define the testable part of MCE.

References

• Khoury, J., Weltman, A. (2004). Chameleon cosmology. Physical Review D 69, 044026.
• Khoury, J., Weltman, A. (2004). Chameleon fields: awaiting surprises for tests of gravity in space. Physical Review Letters 93, 171104.

Appendix O: Critic's Checklist and Adversarial Rebuttals

Revision note (v13.0). This appendix has been rewritten. The v12 rebuttals depended on four items that are now withdrawn: a scalar mass of $10^{10}$ eV, the screening function $S_\rho=1-\tanh(\rho/\rho_c)$ with a fixed coherence length $\lambda_c=1\ \mu$m, a coupling $\kappa$ "fixed by matching $G$", and the headline value $(6.0\pm0.7)\times10^{-9}$. Also withdrawn are the claims that MCE explains rotation curves, the Bullet Cluster and the CMB, that it "predicted" MICROSCOPE or GW170817, that the cosmological-constant problem falls "from 120 to 4 orders", and that the micrometre test is within reach of current technology. At the legacy point the absolute near-source signal is about $3\times10^{-19}$ m/s². Earth and Sun screening has a first-pass estimate; the full calculation remains open. The answers below use the three-level architecture in section 1 of the main document and the screened scalar sector. Where an answer needs a calculation that has not been done, it says so.

This appendix states the strongest objections that experts in General Relativity, quantum field theory, experimental gravity and cosmology are likely to raise. Each objection is phrased as a critic would phrase it, followed by the current answer. The appendix does not claim that MCE survives every objection. It records which objections have an answer today and which are open.

Status of each answer

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Section 1: Theoretical Physics Objections

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Critique 1.1 — "Why not string theory's non-locality? MCE's non-local operator is ad hoc compared to established frameworks."

Critic's position: String theory provides a UV-complete, non-local framework for quantum gravity in which amplitudes are entire functions of the momenta. MCE's exponential regulator $e^{-\square/\Lambda^2}$ imitates this without a consistent UV completion. Why should the non-locality be taken seriously?

Response (v13.0). The criticism is largely correct.

• The exponential form factor is a choice. It is not derived from string theory or from any UV completion. Earlier versions called it "the unique mathematically minimal choice" and stated that string field theory uses the same regulator. Neither statement is supported, and both are withdrawn. Other entire functions of $\square$ also add no poles to the propagator, and ghost-free infinite-derivative gravity has been studied with several of them (Biswas et al. 2012; Tomboulis 1997; both cited in earlier versions, to be verified). Exponential factors of this kind appear in some string field theory and p-adic string models, which motivated that literature (to be verified). That is a resemblance in form, not a derivation.
• The earlier comparison on scale and testability is withdrawn. It said that MCE predicts laboratory-scale effects "from its non-locality" and that string theory has no confirmed prediction. In v13.0 the Level 1 action is local, $\tfrac12(\partial\phi)^2+V(\phi)$ with a species-dependent conformal coupling. The composition-dependent force comes from the screened scalar, not from the non-local operator. The frequency-dependent response in the MHz to GHz range was a heuristic without a derivation and is not part of the theory.
• The non-local operator is therefore not needed for any Level 1 prediction in the screened scalar document. It is retained in the causality note and in the UV-completion programme.
• The causality result is conditional. The factor $e^{p^2/\Lambda^2}$ grows at timelike momenta, so the retarded-Green's-function argument needs review by someone who works on non-local field theory. The associated length is $\hbar c/\Lambda=1.97\times10^{-17}$ m for $\Lambda=10^{10}$ eV, and $\Lambda$ is a free scale that is no longer tied to a scalar mass.

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Critique 1.2 — "The cosmological constant problem isn't solved. The toy model leaves a large residual, and that is still a problem."

Critic's position: A residual discrepancy is not a solution. The problem has been moved, not solved. (The v12 text claimed a reduction "from 120 to 4 orders of magnitude" and the critic was answering that figure.)

Response (v13.0). The critic is right, and the earlier arithmetic was wrong. The residual is about 40 orders of magnitude, not 4.

• The v12 toy model gives a residual vacuum energy $m_e^2\Lambda^2/(16\pi^2)$. For $m_e=0.511$ MeV and $\Lambda=10^{10}$ eV this is $1.65\times10^{29}$ eV$^4$. The observed dark-energy scale is $(2.3\ \text{meV})^4=2.8\times10^{-11}$ eV$^4$. The residual is therefore about 40 orders of magnitude too large (39.8), not 4.
• The naive Planck-scale estimate, $M_{\rm P}^4$ with $M_{\rm P}=1.22\times10^{28}$ eV, is about 123 orders of magnitude too large (122.9). With the reduced Planck mass, $2.44\times10^{27}$ eV, it is 120 orders, which is probably the origin of the earlier "120".
• The toy model changes the discrepancy from about 123 to about 40 orders. That is not a solution. The $\mathbb{Z}_2$ cancellation of the bulk term was assumed, not derived. The earlier statements that the residual is "calculable" and that baryogenesis might remove it at the scale $m_e$ had no calculation behind them and are withdrawn.
• The cosmological-constant problem is not addressed by MCE. See section 6 of the main document.

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Critique 1.3 — "MCE is just Brans–Dicke theory with extra steps. A scalar coupled to the trace $T$ is a scalar-tensor theory, and those are constrained by solar-system tests."

Critic's position: Scalar-tensor theories (Brans–Dicke, Damour–Esposito-Farèse) contain a scalar that couples to the trace of the energy-momentum tensor. Brans–Dicke is constrained to $\omega>40\,000$ by solar-system tests. MCE is a variant and is already excluded.

Response (v13.0). The critic is right that the bounds apply. The earlier answer, which described MCE as a "type-II scalar-tensor theory" that evades them, is withdrawn.

• Structure. Level 1 is a scalar-tensor theory written in the Einstein frame. The metric has the Einstein–Hilbert action with Newton's constant $G$ from experiment, and matter species $i$ move on the metric $A_i^2(\phi)\,g_{\mu\nu}$ with $A_i=e^{\beta_i\phi/M_{\rm Pl}}$. Brans–Dicke theory is the special case of a massless scalar with one universal coupling.
• What the bounds say. The bound $\omega>40\,000$ is equivalent to $\lvert\gamma-1\rvert\lesssim2.5\times10^{-5}$, the same size as the Cassini result $\gamma-1=(2.1\pm2.3)\times10^{-5}$ (Bertotti, Iess and Tortora, Nature 425, 374, 2003).
• Size of the effect without screening. In the weak-field, massless limit the scalar adds a force of $2\beta^2$ times the Newtonian force (this is the ratio $F_\phi/F_N=2\beta_i\beta_j$ in the screened scalar document) and shifts the post-Newtonian parameter by $\gamma-1=-4\beta^2/(1+2\beta^2)$. For the legacy value $2\beta_0^2=5.7\times10^{-3}$ ($\beta_0\approx0.053$) this is $\gamma-1\approx-1.1\times10^{-2}$, about 500 times the Cassini uncertainty. An unscreened scalar with this coupling is excluded.
• Range assumed in that estimate. The massless estimate assumes that the scalar range in interplanetary space exceeds an astronomical unit (about 93 million miles). That range has not been evaluated for the benchmark potential.
• The role of screening. The solar-system force must be removed by the screening mechanism. In thin-shell screening a dense body sources the field only from a shell of thickness $\Delta R$, and the exterior force is reduced by about $3\,\Delta R/R$ (Khoury and Weltman 2004). For the legacy benchmark the Earth requirement is $3\,\Delta R_\oplus/R_\oplus\lesssim1.9\times10^{-7}$. A first-pass estimate for uniform spheres ($n=1$, $\Lambda=2.4$ meV, $\beta_0=0.053$) 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 an interplanetary density of $10^{-20}$ kg/m³, so the benchmark passes by a factor of about 1.5 to 4. For the Sun the same estimate gives $3\,\Delta R_\odot/R_\odot\approx4\times10^{-11}$. For $n=1$ the factor scales as $\Lambda^{5/2}\beta^{-3/2}\rho_{\rm ambient}^{-1/2}$. The benchmark fails the Earth requirement by about 20 times at $\Lambda=10$ meV and by about 8 times at $\beta_0=0.01$. The full calculation (density profile, atmosphere, ambient field, Moon, the Sun's field at the Earth, and Cassini) is still open. See section 4 of the screened scalar document.
• What makes this MCE, rather than a conventional screened scalar-tensor theory with a phenomenological composition coupling? Level 1 on its own is a chameleon-type theory with an isospin-dependent matter coupling. Environment-dependent mass and thin-shell screening are the standard mechanism of that class. They are not a distinction. The static linearised equation $\nabla^2\delta\phi-m_{\rm eff}^2(\rho)\,\delta\phi=\beta_i\rho_i/M_{\rm Pl}$ is electrostatics in a screening medium: like scalar charges attract, and the thin shell is the conductor analogy (section 1.1 of the main document). The claims that would make the theory more than that coupling are the three in section 1.2, and none is established. (1) The scalar charge originates in a mass-induced vacuum-polarisation asymmetry. Unproven (Level 2). (2) The composition structure is $\delta\propto Z/A-\tfrac12$, with $C$ from hadronic physics. The form is postulated and $C=0.03$ is a benchmark. (3) $G$ itself emerges from the same mechanism. Unproven. Until at least the first two are derived, MCE is a screened scalar fifth-force model with a particular composition coupling. None of this is a reason why the solar-system bounds do not apply.

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Critique 1.4 — "MOND already explains galaxy rotation curves with a single parameter. MCE is more complex and less parsimonious."

Critic's position: MOND fits hundreds of rotation curves with one acceleration scale $a_0\approx1.2\times10^{-10}$ m/s². MCE needs more parameters. Why add complexity?

Response (v13.0). The comparison in earlier versions is withdrawn. Those versions claimed that MCE "explains rotation curves, the Bullet Cluster, CMB structure, WEP adherence, gravitational waves and binary pulsar decay from a single Lagrangian". Most of that claim was unsupported.

• Rotation curves. Level 1 does not explain them. Where the scalar is unscreened, its force is of order $2\beta_0^2$ times the Newtonian force, about 0.6% for the legacy value of $\beta_0$. That is an order-of-magnitude estimate, not a fit. It is far smaller than the discrepancy that rotation curves show in the outer parts of galaxies. No rotation-curve calculation has been done.
• Bullet Cluster. Level 1 does not explain it. A conformally coupled scalar does not bend light beyond General Relativity, so it cannot produce a lensing mass offset from the baryons. The earlier explanation by density screening is withdrawn.
• CMB. Not computed (see 2.1).
• Dark-sector extension. The cosmological extension sketches a coarse-grained scalar fluid as a dark-matter and dark-energy analogue. It is a speculative programme and has no calculation behind it.
• Parameters. Level 1 has four free parameters, $(\beta_0,C,\Lambda,n)$. None is fixed by matching $G$, and $C$ has not been derived from lattice QCD or any other source. The earlier statements that $\kappa$ is fixed by $G$ and that $C_{\rm QFT}$ is constrained to 12% by lattice QCD are withdrawn.
• Comparison. MOND and MCE Level 1 do not address the same data. Level 1 makes no rotation-curve claim, so parsimony against MOND on rotation curves does not arise.

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Critique 1.5 — "GW170817 showed that gravitational waves travel at the speed of light to about one part in $10^{15}$. Scalar-tensor theories generically predict $v_{\rm GW}\neq c$."

Critic's position: The near-simultaneous detection of GW170817 and GRB 170817A constrains the difference between the speed of gravity and the speed of light. Many scalar-tensor theories are excluded by it. Is MCE?

Response (v13.0). Level 1 is consistent with the bound, and the reason is structural.

• The tensor speed equals $c$ because the metric sector is General Relativity: the Einstein–Hilbert action with no modification of the tensor kinetic term. The scalar couples to matter through the conformal factor $A_i^2(\phi)$. It does not couple to the Riemann tensor or to derivatives of the metric, which is what changes the tensor speed in the scalar-tensor theories that GW170817 excluded. This is an inherited result, not a derivation by MCE.
• The bound is a fractional difference between the speed of gravity and the speed of light between about $-3\times10^{-15}$ and $+7\times10^{-16}$ (Abbott et al., Astrophysical Journal Letters 848, L13, 2017). The earlier statement of the objection quoted a single value of $5\times10^{-16}$, which is not the published range.
• The earlier argument that the scalar mode is suppressed because $m_\phi\approx10^{10}$ eV is withdrawn. The scalar mass is not $10^{10}$ eV. It is the environment-dependent $m_{\rm eff}(\rho)$ of the screened scalar document.
• A scalar field that couples to matter can add a breathing-mode polarisation and dipole radiation from systems whose scalar charges differ. The size depends on the screened scalar charge and on the range of the scalar, neither of which has been computed. The statement that the scalar contribution is negligible is therefore not made here.

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Critique 1.6 — "MICROSCOPE constrains the Weak Equivalence Principle to about $10^{-15}$. Your suppression argument is unfalsifiable, because you can always tune it to make the effect disappear."

Critic's position: Whenever a composition test is null, MCE can say that the screening is strong enough or the separation large enough. The theory is unfalsifiable.

Response (v13.0). The criticism is partly right, and the earlier answer is withdrawn. The earlier answer said that MCE "predicted MICROSCOPE", that the coherence length is "constrained, not tunable" to between 1 and 10 μm, and that a null atom-interferometry result would falsify MCE "with no escape route". None of these statements holds.

• MICROSCOPE. The final result is $\eta({\rm Ti,Pt})=[-1.5\pm2.3\,({\rm stat})\pm1.5\,({\rm syst})]\times10^{-15}$, about $2.7\times10^{-15}$ if the statistical and systematic errors are combined in quadrature (Touboul et al., Physical Review Letters 129, 121102, 2022). It was published in 2022, before these documents were written. MCE did not predict it. Consistency with it is a retrodiction.
• What MICROSCOPE does to the model. It fixes a requirement. With the legacy benchmark, an unscreened Earth would give $\Delta a/a\approx1.4\times10^{-8}$ between titanium and platinum. The measured bound requires the Earth's scalar field to be suppressed by at least $5\times10^{6}$, so $3\,\Delta R_\oplus/R_\oplus\lesssim1.9\times10^{-7}$. This is the same order as the condition $\Delta R_\oplus/R_\oplus<10^{-7}$ quoted by Khoury and Weltman. A first-pass estimate meets this requirement by a factor of about 1.5 to 4 (critique 1.3). The full calculation is open.
• Is the model unfalsifiable? A single null result does not end every screened-scalar model, because the model has free parameters $(\beta_0,C,\Lambda,n)$. A family of models is falsifiable in the following sense. Each experiment at a stated sensitivity excludes a region of parameter space, and the surviving region shrinks. Existing examples: Jaffe et al. (Nature Physics 13, 938, 2017; author correction Nature Physics 19, 1946, 2023), with a 0.19 kg tungsten source, found an anomalous acceleration $(41\pm24)$ nm/s² with a one-tailed bound below 81 nm/s² (95%), which excludes $M<1.7\times10^{-3}M_{\rm Pl}$ for $\Lambda=2.4$ meV and $n=1$ ($M=M_{\rm Pl}/\beta$). The legacy benchmark, $\beta_0\approx0.053$ or $M\approx19\,M_{\rm Pl}$, lies far from that excluded corner.
• The test at about 1 μm. The size of the signal depends on the screening and range factor $f(r,\rho,\text{geometry})$, which has to be obtained by solving the field equation for the experimental geometry and has not been computed. The earlier claim of $\Delta a/a\approx6\times10^{-9}$ "with no escape route" is withdrawn. The value $\approx6$ to $7\times10^{-9}$ is a legacy reference point for $2\beta_0^2=5.7\times10^{-3}$, $C=0.03$ and $f=e^{-1}$. Referred to the Newtonian pull of a 1 cm aerogel source, that point is an absolute signal of about $3\times10^{-19}$ m/s² (critique 3.2). It is not within reach of current atom interferometry.
• The coherence length. The fixed $\lambda_c=1\ \mu$m and the claim that it is bounded between 1 and 10 μm are withdrawn. The range is $\lambda(\rho)=\hbar/(m_{\rm eff}(\rho)c)$ and depends on the environment. The thermal wavelength $\hbar c/k_BT=7.63\ \mu$m at 300 K is a possible origin of a micrometre scale and is not a derivation.

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Section 2: Cosmological Objections

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Critique 2.1 — "The CMB acoustic peaks require the right amount of cold dark matter. MCE's dark fluid cannot reproduce the third peak without fine-tuning."

Critic's position: The relative heights of the CMB acoustic peaks constrain the baryon-to-dark-matter ratio precisely. An alternative dark-matter model with a different equation of state shifts the peaks visibly in Planck data.

Response (v13.0). Not computed. No CMB calculation has been done for MCE.

• Level 1, as defined in the main document, has no dark-matter component. The cosmological extension describes a coarse-grained scalar fluid as a speculative dark-matter analogue.
• The earlier claims that this fluid has a non-zero, scale-dependent sound speed, shifts the third peak and suppresses the damping tail are withdrawn as predictions. They were statements of what a calculation would need to show, and no Boltzmann-code calculation (CLASS or CAMB) has been performed.
• The scalar equation of motion in an expanding background, the perturbation equations and the screening in the early universe would all have to be implemented before a comparison with Planck or ACT can be made. Until then, the objection stands unanswered.

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Critique 2.2 — "Large-scale structure simulations (Millennium, IllustrisTNG) were tuned to ΛCDM. Claiming MCE fits them is circular."

Critic's position: All existing simulations assume ΛCDM dark matter and were calibrated against observations on that basis. Agreement with them is built in.

Response (v13.0). The objection is correct, and the earlier response is withdrawn. No simulation has been run.

• The earlier statements that MCE produces "softer density cores" and "10–20% fewer sub-haloes", and that it "resolves" the core–cusp and missing-satellite tensions "naturally", had no calculation behind them and are withdrawn.
• The earlier modified Poisson kernel with a fixed $k_c=1/\lambda_c$ is retired together with the fixed coherence length.
• A fair test needs a solver for the nonlinear scalar equation with environment-dependent screening, run from the same initial conditions as a ΛCDM reference, and compared with observations rather than with ΛCDM output.

In linear perturbation theory the screened scalar changes the effective gravitational coupling to

a standard scalar-tensor result. It has not been evaluated for MCE parameters.

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Section 3: Experimental Objections

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Critique 3.1 — "At $r=1\ \mu$m the Casimir force is orders of magnitude larger than gravity. How can you claim to measure a $10^{-9}$ effect over a Casimir background?"

Critic's position: At sub-micron separations the Casimir force dominates any gravitational or pseudo-gravitational effect by many orders of magnitude.

Response (v13.0). The objection stands as a systematics problem, and the earlier numerical rebuttal is withdrawn.

• Size of the background. For two perfect conductors the Casimir pressure is $P=\pi^2\hbar c/(240\,d^4)$. This gives $1.30\times10^{-3}$ Pa at $d=1\ \mu$m, $2.08\times10^{-2}$ Pa at $0.5\ \mu$m and $1.30\times10^{-7}$ Pa at $10\ \mu$m. These are pressures, not forces, so the force depends on the area. The perfect-conductor formula gives a reference scale only. The difference for real materials has not been computed here.
• Withdrawn claim. The earlier text said the composition-dependent part of the Casimir force is $10^{-3}$ to $10^{-4}$ of the total and gives a differential acceleration of $\Delta a_{\rm Casimir}/a\approx10^{-12}$, "three orders of magnitude below the MCE signal". That value was not calculated. The material-dependent (Lifshitz) difference between aluminium and gold has not been computed anywhere in this document set, and no fractional value is stated as a result.
• What the design relies on. A differential measurement between two compositions of identical geometry removes the common Casimir term. Lateral modulation of the source separates a force with one dependence on position from another. How much of the composition-dependent Casimir–Polder and patch-potential force survives the differencing and the modulation is not yet quantified.
• Where the analysis is. See the revised Casimir note and the systematics discussion in the experimental design document.

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Critique 3.2 — "Atom interferometry has never measured a $10^{-9}$ acceleration difference between two different materials. Your claimed sensitivity is aspirational."

Critic's position: State-of-the-art atom interferometers measure accelerations to about $10^{-12}\,g$ per shot. Reaching $\Delta a/a\sim10^{-9}$ near a source of controlled composition at about 1 μm, with all systematics controlled, has not been done.

Response (v13.0). That configuration has not been done. It is a proposed experiment. The earlier statement that "current technology is sufficient" is withdrawn. The v12 comparison set a fractional signal against a sensitivity quoted as a fraction of $g$, which is not the same quantity. The verified numbers are as follows.

• What "$a$" is. For an atom near a local source, $a$ in $\Delta a/a$ is the Newtonian pull of that source. For a 1 cm slab of aerogel at 10 kg/m³ that pull is $2\pi G\sigma=4.2\times10^{-11}$ m/s². At the legacy point, $7.0\times10^{-9}\times4.2\times10^{-11}=2.9\times10^{-19}$ m/s², about $3\times10^{-19}$ m/s². The Asenbaum per-shot resolution is $1.4\times10^{-11}\,g=1.4\times10^{-10}$ m/s². A $5\sigma$ detection of the legacy signal would need about $5\times10^{18}$ shots, about $3\times10^{12}$ years at 15 s per shot. Even for $2\beta_0^2=1$ the signal is $1.4\times10^{-16}$ m/s² and the time is about $10^{7}$ years. The micrometre test at the legacy point is not within reach of current atom interferometry.
• If $a$ is local $g$. The observable is then the Earth-sourced composition test, which MICROSCOPE and the torsion balances already bound. It is not a new near-source measurement.
• What remains uncomputed. The factor $f(r,\rho,\text{geometry})$ for a real source is still open, so the absolute signal above uses the legacy factor $f=e^{-1}$ and the source's Newtonian pull. It is not a solved field profile.
• Where the difficulty lies. The earlier text treated positioning the atoms about 1 μm from the target and making aerogel targets homogeneous to better than 0.1% as achievable with existing techniques. No source was given, and both claims are withdrawn. Patch potentials, magnetic fields, gravity gradients and Casimir–Polder forces are the systematic limits, discussed in the experimental design document. The same comparison is in section 3 of the main document.

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Critique 3.3 — "Binary-pulsar timing constrains dipole radiation from scalar fields. MCE's scalar should emit dipole radiation from the binary."

Critic's position: A long-range scalar that couples to matter radiates dipole radiation from binaries whose bodies have different scalar charges. Excess dipole radiation would speed up the orbital decay beyond the agreement with the General Relativity quadrupole formula seen in PSR B1913+16 and other systems.

Response (v13.0). The earlier answer is withdrawn. It argued that a scalar mass of $10^{10}$ eV suppresses radiation by a factor $e^{-10^{25}}$, and that the non-local operator acts as a high-pass filter. The scalar mass is not $10^{10}$ eV, and the high-pass argument was not derived.

• Inherited part. The tensor-quadrupole orbital decay comes from the metric sector, which is General Relativity.
• Additional channel. Scalar dipole emission depends on the difference in scalar charge between the two bodies and on whether the scalar range exceeds the orbital separation. For the PSR B1913+16 orbit the radius is of order $10^{9}$ m.
• Screening argument (qualitative). Strongly self-gravitating bodies such as neutron stars are expected to be strongly screened, which would reduce their scalar charge and with it the dipole emission. This is an expectation from the thin-shell mechanism. It has not been turned into a calculation for the benchmark potential, and it is not quoted as a result.
• Bounds. The double-pulsar tests (Kramer et al., Phys. Rev. X 11, 041050, 2021) measure gravitational-wave damping to a fractional precision of $6\times10^{-5}$, in agreement with General Relativity at $1.3\times10^{-4}$ (95%), and set a dipole-radiation parameter $B_D\lesssim4\times10^{-10}$ (95%). The Hulse–Taylor system agrees with the quadrupole formula to a ratio of $0.9983\pm0.0016$ (Weisberg and Huang, 2016). These are strong constraints on scalar-tensor couplings through strongly self-gravitating bodies. Mapping them onto $\beta_0$ and the screening of neutron stars has not been done.
• Status. Open until the neutron-star scalar charge and the range of the scalar in the interstellar medium are computed.

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Section 4: Philosophical and Paradigm Objections

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Critique 4.1 — "Your theory requires a quantum vacuum polarisation that has never been directly detected as a gravitational source. You are assuming the conclusion."

Critic's position: Vacuum polarisation is real (the Lamb shift, the Casimir effect). Its role as a gravitational source has not been established. MCE rests on an unverified assumption.

Response (v13.0). Correct. The link between vacuum polarisation and the scalar coupling is a postulate, and it belongs to Level 2, the open emergent-coupling programme. The precedent is Sakharov's induced gravity (1967; Visser 2002), in which the Einstein–Hilbert term arises from vacuum fluctuations of matter fields. In that approach $1/G$ is set by a cutoff near the Planck scale or a very large number of fields. A cutoff near $10^{10}$ eV, as used in v12, gives an induced $1/G$ roughly 36 orders of magnitude too small. The derivations that Level 2 needs are not available.

• Level 1 does not depend on this postulate. It takes a scalar with a conformal matter coupling and tests it.
• Two earlier statements are withdrawn: that a confirmed signal at about 1 μm would establish the vacuum-polarisation origin of gravity, and that the contribution is "suppressed to below $10^{-4343}$ at macroscopic scales". The first confuses detection of a composition-dependent force with identification of its origin. The second came from the withdrawn fixed-length exponential. A detection would support a screened scalar force. It would not by itself identify vacuum polarisation as the cause.

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Critique 4.2 — "Why believe a gravity theory that was developed without a toroidal Earth model and then retrofitted to be compatible with one?"

Critic's position: MCE was developed as an effective field theory consistent with a spherical Earth. The toroidal framework was added afterwards to accommodate a non-mainstream view. This is retrofitting.

Response (v13.0). The chronology of development does not decide the physics. What decides it is whether the toroidal application survives tests, and for the nominal parameters it does not.

• What stands. The local field equations contain no global topology, so they do not by themselves favour a sphere, a torus or a disc (Appendix J). That is a statement about the equations. It does not imply that every geometry agrees with data.
• What fails. The nominal toroidal parameters, $R_T/r_T=6.4$, give a quadrupole moment $J_2\approx0.38$ against the measured $1.08263\times10^{-3}$, a factor of about 350 too large. The earlier statement that this ratio "matches Earth's radius ratio" is wrong, because the Earth's equatorial-to-polar radius ratio is 1.0034. See Global Geometry Hypothesis Tests.
• Withdrawn. The earlier text said that finding the predicted toroidal harmonics in GRACE-FO data would confirm the toroidal framework. It also said that odd-degree harmonics are forbidden in any spherical model, so that their presence would indicate a torus. The real Earth has measured nonzero odd harmonics (for example $J_3=-2.53\times10^{-6}$), so any toroidal signal would have to appear as a residual after the full standard model. The pole-asymmetry test used a null of zero where standard geodesy already gives about 45 m. The toroidal framework stays listed as a falsifiable branch until a parameter set that reproduces the measured $J_2$ is proposed.

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Critique 4.3 — "How is MCE different from an epicycle system, adding mechanisms to explain each new observation without being falsified?"

Critic's position: Each time MCE meets a problem (equivalence-principle tests, binary pulsars, gravitational waves, Casimir forces) it adds a new suppression mechanism.

Response (v13.0). The objection has force against v12, which added a fixed length, a density function and a mass term separately. The v13.0 structure answers part of it. It does not answer all of it.

1. One action. Level 1 is one action with four free parameters, $(\beta_0,C,\Lambda,n)$. The thin-shell suppression and the environment-dependent range follow from the scalar field equation, not from separate functions fitted to each experiment. A first-pass thin-shell estimate for the Earth and the Sun is given in critique 1.3. The full calculation, the neutron-star scalar charge and the laboratory geometry have not yet been computed, so it is not yet shown that one choice of parameters is consistent with all of them.
2. Retrodiction, not prediction. MICROSCOPE (2022), GW170817 (2017) and antihydrogen free fall (ALPHA, 2023) all predate these documents (2026). The statements that "MCE predicted" them, and that "predictions precede tests", are withdrawn. Consistency with them is a retrodiction. In the case of MICROSCOPE it is a constraint that the model has to meet, and the Earth's thin-shell condition $3\,\Delta R_\oplus/R_\oplus\lesssim1.9\times10^{-7}$ comes from that data. It is not a prediction. The first-pass estimate meets it with a thin margin.
3. A test that can exclude parameters. A composition test near a low-density source can exclude parts of $(2\beta_0^2C,\Lambda,n)$ once $f(r,\rho,\text{geometry})$ is computed. A null result at stated sensitivity excludes a region. It does not by itself end every screened-scalar model, and the documents no longer say that it does.

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Critique 4.4 — "Non-local gravity theories have UV completion problems. How does MCE's non-local operator avoid the issues of other non-local models?"

Critic's position: Operators such as $K(\square)=e^{-\square/\Lambda^2}/(\square+m^2)$ face (a) the ghost problem in perturbative quantisation, (b) a possible breakdown of causality near $E\sim\Lambda$, and (c) the absence of a UV completion.

Response (v13.0). The objection is largely valid, and the earlier answer overstated each point.

• Ghosts. The operator $K$ has one pole at $\square=-m^2$ and the exponential adds none, which is the basis of the ghost-freedom claim. That statement concerns the pole structure. It does not settle unitarity of the full theory.
• Causality. The factor $e^{p^2/\Lambda^2}$ grows at timelike momenta, so the retarded-Green's-function argument is conditional and needs review by an expert in non-local field theory. The non-locality length is $\hbar c/\Lambda=1.97\times10^{-17}$ m for $\Lambda=10^{10}$ eV. The earlier value of $2\times10^{-26}$ m used the time $\hbar/E$ as a length. The label "Lee–Wick" does not apply to entire-function form factors.
• UV. MCE is an effective theory with a cutoff $\Lambda_{\rm EFT}$, a free scale that is no longer tied to a scalar mass. The finite-loop statement relies on the Euclidean suppression $e^{-k^2/\Lambda^2}$, whose status depends on the causality point above. The one-loop beta functions in Appendix L have been revised: the running of $\kappa$ is withdrawn. The earlier analogy with QCD was wrong, because QCD is asymptotically free and has no Landau pole.
• Non-local $F(R)$ and CMB data. The earlier text cited a 2026 non-local $F(R)$ model as fitting ACT, Planck and BICEP data, with a specific tensor-to-scalar ratio $r_{0.05}=0.036\pm0.004$, and concluded that MCE is "observationally consistent with 2026 CMB data". No source was given for these numbers, and the statement is withdrawn. Even if some non-local $F(R)$ model fits CMB data, that would not support MCE: the Level 1 action is local, and no CMB calculation has been done for MCE.
• Status. Open. Required: independent review of the causality argument, and a UV completion or an explicit statement that none is claimed.

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Summary: What Would Exclude Regions of Parameter Space

The wording of the previous version, which listed conditions under which MCE would be "definitively falsified", is withdrawn. A null result at a stated sensitivity excludes a region of the parameter space $(2\beta_0^2C,\ \Lambda,\ n)$. It does not by itself end every screened-scalar model. The table lists what each test constrains.

The earlier table rows on lattice-QCD updates of $m_d-m_u$ and on a "CMB damping tail" falsification condition are removed. The first rested on a proportionality between $C$ and $m_d-m_u$ that has not been derived. The second rested on a CMB calculation that does not exist.

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

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• 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).
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• 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.
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