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 The v12 rebuttals depended on four items that are now withdrawn: a scalar mass of $10^{10}$ eV, the screening function $S_\rho=1-\tanh(\rho/\rho_c)$ with a fixed coherence length $\lambda_c=1\ \mu$m, a coupling $\kappa$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 Also withdrawn are the claims that MCE explains rotation curves, the Bullet Cluster and the CMB, that it "predicted" MICROSCOPE or GW170817, that the cosmological-constant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 Neither statement is supported, and both are withdrawn.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.