Experimental Design and Numerical Simulation Frameworks for MCE Theory

Revision note (v13.0). This document is rewritten. The v12 numbering was broken (2.1, then 7.1.1, 7.2, then 2.2) and several sections used withdrawn mechanisms: the fixed length $\lambda_c=1\ \mu$ m and the profile $S_\rho=1-\tanh(\rho/\rho_c)$ . The signal is now $\Delta a/a\simeq2\beta_0^2\,\Delta\delta\,f$ from the screened scalar sector. Withdrawn: the benchmark $(6.0\pm0.7)\times10^{-9}$ as a prediction (it is a legacy reference point), the Casimir step $10^{-3}\times10^{-9}=10^{-12}$ , the MICROSCOPE/STEP suppression table, the GADGET Yukawa-kernel and Bullet Cluster sections, the GR-mimicry table, the GRACE-FO laser-ranging table and "toroidal coupling" forecast, and the superconducting-cage prediction (kept only as an untested speculation, Section 7). Added: the absolute signal for the source's Newtonian pull (Section 4.3). The case $a=g$ is the Earth-sourced test. The Earth's thin-shell factor is a first-pass estimate; the full calculation is open. See the architecture section of the main document. "EME" is the historical name of MCE.

1. Goal and observable

The aim is to measure whether the acceleration of a test body towards a source depends on isospin composition, as the screened scalar sector of MCE implies. The observable is the fractional differential acceleration

\frac{\Delta a}{a}\simeq 2\beta_0^2\,\Delta\delta\;f(r,\rho,\text{geometry}),\qquad \Delta\delta=C\,\varepsilon\,\Delta\!\left(\frac{Z}{A}\right),\qquad \varepsilon=1.378\times10^{-3}.

$2\beta_0^2$ is the ratio of the scalar force to the Newtonian force between species of the same unshifted coupling, before screening and range factors.
$\Delta\delta$ is the difference in the isospin coupling shift between the two compositions. $C$ is a free coefficient, with benchmark $0.03$ , and is not derived.
$f$ is the screening and range factor. It has to be obtained by solving the nonlinear scalar field equation in the geometry of the experiment. The simplest estimate is $f\approx e^{-r/\lambda(\rho)}$ . It has not been computed for any real geometry.
$a$ is the Newtonian pull of the local source (Section 4.3). The case in which $a$ is the local gravitational acceleration is the Earth-sourced test of Section 5.

The composition differences used in this document are:

Pair	$\Delta(Z/A)$	$\Delta\delta$ at $C=0.03$
Aluminium and gold	0.0807	$3.34\times10^{-6}$
Titanium and platinum	0.0598	$2.47\times10^{-6}$
Beryllium and titanium	0.0158	$6.52\times10^{-7}$
Rubidium-85 and rubidium-87	0.0100	$4.14\times10^{-7}$

Item	Status
Isospin form of $\delta(Z,A)$	Postulated
Value of $C$	Postulated (benchmark, not derived)
Screening and range factor $f$ in a real geometry	Open
Earth's thin-shell factor at the benchmark	Estimated (first pass); full calculation Open
Perihelion advance, light deflection, Shapiro delay, gravitational-wave speed	Inherited from the metric sector, not tests of the scalar sector

The last row replaces the v12 table that presented the classical tests of general relativity as predictions of the MCE scalar and vector fields. A scalar sourced by the trace of the energy-momentum tensor does not couple to light, so those results belong to Level 0 of the status ladder.

2. Atom-interferometry protocol near a low-density source

2.1 Configuration

Sources. Two low-density aerogel targets of identical geometry and different dopant, for example aluminium-doped and gold-doped silica aerogel, with a density of order 10 kg/m³. A single target with alternating aluminium-doped and gold-doped regions is the alternative for lateral modulation (Section 2.3). The dopant mass fraction, the porosity and the geometry are not specified in the document set.
Atoms. Cold $^{87}$ Rb, in an ultra-high-vacuum chamber. The composition difference then sits in the source, and the same atoms probe both compositions. A species difference ( $^{85}$ Rb and $^{87}$ Rb, $\Delta\delta=4.14\times10^{-7}$ ) is a separate, smaller lever.
Standoff. About 1 μm between the atoms and the target surface. The v12 value of 100 μm for the drop height, quoted with a suppression of $e^{-100}$ , is withdrawn together with the fixed $\lambda_c$ . Atoms cannot fall freely for a long interrogation time while staying 1 μm from a surface, so a realisation would hold or guide the atoms, for example in an optical lattice. No design is specified here.
Measurement. A differential atom interferometer measures the phase $\Delta\Phi\propto a\,T^2$ , with $T$ the interrogation time.

The reason for a low-density target in v12 was that $S_\rho\approx1$ . In the present framework the benefit has to be shown by the field solution. A low-density target also has an effective dielectric response closer to that of vacuum, so its Casimir–Polder coupling is smaller than that of a solid metal. This remains a practical reason for the choice, although the reduction has not been calculated.

2.2 Separation scan

The separation is scanned over about 0.5 to 10 μm, the range proposed in v12, so that the dependence of the signal on distance is measured and not assumed. The scan range is a design choice, not a prediction. The expected distance dependence of the scalar term comes from the field solution (Section 2.5). The Casimir–Polder term has a different dependence on distance (see the Casimir note).

2.3 Lateral modulation

The source is translated laterally so that the atoms see the two compositions alternately. The signal is then read at the modulation frequency, which separates it from slowly varying, composition-independent terms. Two cautions apply.

The composition-dependent Casimir–Polder term and the composition-dependent scalar term both change at the modulation frequency, so modulation separates them from common-mode terms, not from each other.
Source motion at the modulation frequency can introduce vibration at the same frequency (Section 3).

2.4 Control runs

Both halves of the source with the same dopant (a null composition).
Undoped aerogel against doped aerogel.
Swapped positions of the two dopants.
Different atomic internal states and, where possible, different species, because the Casimir–Polder acceleration depends on the atom's polarisability while the scalar acceleration depends on its coupling $\beta_i$ .

2.5 Numerical framework for $f$

The factor $f$ requires the static solution of the field equation obtained from the effective potential of the screened scalar document:

\nabla^2\phi=\frac{\partial V_{\rm eff}}{\partial\phi}=-\frac{n\Lambda^{4+n}}{\phi^{\,n+1}}+\frac{\beta\,\rho(\mathbf{x})}{M_{\rm Pl}}\,e^{\beta\phi/M_{\rm Pl}},

with the boundary condition $\phi\to\phi_{\min}$ of the residual gas far from the apparatus. The vacuum-chamber walls are part of the geometry, because the range in the vacuum can exceed the chamber size. The acceleration of an atom of species $i$ is $-(\beta_i/M_{\rm Pl})\nabla\phi$ . A solver specification and validation checks are given in Appendix N. No solution exists yet, so no value of $f$ is claimed.

3. Systematics

A composition-independent effect cancels in the difference between the two source compositions. An effect that depends on composition, even slightly, does not cancel. The table lists the effects that do.

Systematic	Mechanism	Mitigation	Quantification
Casimir–Polder force	A neutral atom is attracted to the surface with a strength set by the atom's polarisability and the target's dielectric response, which differs between aluminium-doped and gold-doped aerogel	Measure the optical properties of the actual targets, model with Lifshitz theory, modulate the source, scan the separation, vary the atomic state	The aluminium–gold differential is not computed. See the Casimir note
Electrostatic patch potentials	Surface potentials that vary over the surface produce electric fields and field gradients, and a polarisable atom is pulled towards stronger field. An insulating aerogel can also hold surface charge	Kelvin-probe maps of the target surfaces, non-magnetic conductive coatings (which change the Casimir–Polder interaction and the field geometry and must be included in the model), discharge procedures, null runs	Not yet computed
Magnetic fields	Atoms in states with a nonzero magnetic quantum number feel a force proportional to the field gradient. Ferromagnetic impurities in the dopant or the aerogel produce local gradients	Use the magnetically insensitive state, multi-layer shielding, screen the target materials for magnetic impurities, map the field with the atoms	Not yet computed
Gravity gradients	The Earth's vertical gradient, $2g/R_\oplus=3.1\times10^{-6}\ {\rm s^{-2}}$ , and gradients from nearby laboratory masses act differently on atoms at different heights. The Newtonian pull of the targets themselves also differs if their masses differ	Gradient compensation, mapping and subtraction of nearby masses, reversal of the target orientation	Earth gradient computed. Mass-mismatch requirement in Section 4.4
Vibration	Vibration of the platform or retro-reflecting mirror enters the interferometer phase. Source motion at the modulation frequency can put vibration into the signal band	Active isolation, a common reference mirror for the differential measurement, recording and correlating the vibration, runs with the source stationary	Not yet computed
Target homogeneity	Differences in density, dopant fraction, porosity or surface roughness between the two compositions change the Newtonian pull, the Casimir–Polder force and the patch fields independently of the scalar	Fabricate both compositions from the same aerogel batch, characterise the density and dopant distribution, swap the dopants between regions	Requirement in Section 4.4

Temperature also matters at these separations, because the thermal wavelength $\hbar c/k_BT=7.63\ \mu$ m at 300 K is of the same order as the largest standoff in the scan. The temperature of the apparatus must be recorded and used in the Casimir–Polder model.

The v12 tables assigned fractional sizes to individual systematics (for example $10^{-5}$ for stray fields before mitigation, $10^{-10}$ for magnetic gradients and $10^{-14}$ for thermal noise) with no derivation or source. They are withdrawn. The sizes have to be computed or measured for the actual apparatus.

4. Sensitivity budget

Only the numbers in the verified list are used for external experiments. All other numbers below are computed from the stated inputs.

4.1 Signal at the reference points

For aluminium and gold, with $C=0.03$ :

\Delta\delta_{\rm Al\text{-}Au}=0.03\times1.378\times10^{-3}\times0.0807=3.34\times10^{-6}.

The v12 unsuppressed value $1.9\times10^{-8}$ is reproduced by $2\beta_0^2=5.7\times10^{-3}$ , so $\beta_0\approx0.053$ . This value is an inference from the old number. It is not derived. Then

2\beta_0^2\,\Delta\delta_{\rm Al\text{-}Au}=5.7\times10^{-3}\times3.34\times10^{-6}=1.9\times10^{-8}\qquad(f=1).

The legacy reference point uses $f=e^{-1}$ :

1.9\times10^{-8}\times e^{-1}=1.9\times10^{-8}\times0.368=7.0\times10^{-9}.

The v12 headline value $6.0\times10^{-9}$ is $7.0\times10^{-9}$ multiplied by a further factor of 0.86 from QCD running, which may be counted twice. The reference point is therefore quoted as $6$ to $7\times10^{-9}$ , with no error bar, and is a point on the parameter surface, not a prediction.

The choice $f=e^{-1}$ corresponds to $r=\lambda$ , a range of 1 μm in the gap. For the illustrative parameters of the screened scalar document the range in a vacuum chamber is about $7\times10^{7}$ m, so the range factor alone gives $f\approx1$ (Appendix N). The value $e^{-1}$ is kept because the v12 numbers are expressed in it.

4.2 Required sensitivity in $\Delta a/a$

Let $\sigma$ be the total uncertainty (statistical and systematic) on the fractional differential acceleration.

Reference point	Signal	$\sigma$ for a $5\sigma$ detection	$\sigma$ for a $3\sigma$ detection	$\sigma$ at which a null result excludes the signal at 95% (one-sided)
Legacy, with factor 0.86	$6.0\times10^{-9}$	$1.2\times10^{-9}$	$2.0\times10^{-9}$	$3.6\times10^{-9}$
Legacy, $f=e^{-1}$	$7.0\times10^{-9}$	$1.4\times10^{-9}$	$2.3\times10^{-9}$	$4.3\times10^{-9}$
Unscreened, $f=1$	$1.9\times10^{-8}$	$3.8\times10^{-9}$	$6.3\times10^{-9}$	$1.2\times10^{-8}$

The columns are signal $/5$ , signal $/3$ and signal $/1.645$ . For example $7.0\times10^{-9}/5=1.4\times10^{-9}$ .

A bound $X$ on $\Delta a/a$ for aluminium and gold is a bound on the product of the parameters:

2\beta_0^2\,C\,f<\frac{X}{\varepsilon\,\Delta(Z/A)}=\frac{X}{1.378\times10^{-3}\times0.0807}=\frac{X}{1.11\times10^{-4}}.

For $X=10^{-9}$ this gives $2\beta_0^2Cf<9.0\times10^{-6}$ . The benchmark has $2\beta_0^2C=5.7\times10^{-3}\times0.03=1.7\times10^{-4}$ .

4.3 What $\sigma$ means in absolute acceleration

For an atom near a local source, the reference acceleration in $\Delta a/a$ is the Newtonian pull $a_N$ of that source. Section 3 of the main document fixes this choice. The Casimir note calls it reading (ii). The scalar force and the Newtonian force come from the same body, so

\Delta a=\left(2\beta_0^2\,\Delta\delta\,f\right)a_N .

The case $a=g$ is the Earth-sourced composition test already bounded by MICROSCOPE, the torsion balances and the Asenbaum measurement (Section 5). The v12 comparison of a fractional signal with a sensitivity quoted as a fraction of $g$ mixed the two quantities. That comparison is withdrawn, including the conclusion that a $5\sigma$ result would take less than one shot.

For a slab of density $\rho_s$ and thickness $t$ , with lateral size much larger than $t$ and than the standoff, $a_N=2\pi G\rho_s t$ , independent of the standoff. The dopant mass fraction $x\le1$ multiplies $\Delta\delta$ further, and is not included. The legacy fractional signal used below is $7.0\times10^{-9}$ ( $f=e^{-1}$ ).

The shot budget uses the per-shot resolution of Asenbaum et al. (2020), up to $1.4\times10^{-11}\,g=1.4\times10^{-10}$ m/s², and 15 s per shot, as in the main document. The figure $5.4\times10^{-11}$ per $\sqrt{\rm Hz}$ is that experiment's sensitivity, recorded in Section 5, and the budget below is the shot count. For the 1 cm aerogel slab the legacy signal is $2.9\times10^{-19}$ m/s². A $5\sigma$ detection needs about $5\times10^{18}$ shots, about $3\times10^{12}$ years. With $2\beta_0^2=1$ and $f\approx1$ (the vacuum range of the illustrative parameters is far longer than the standoff), the same slab gives $\Delta\delta\,a_N=1.4\times10^{-16}$ m/s², about $2.5\times10^{13}$ shots and about $1\times10^{7}$ years. The 10 cm and solid rows scale the legacy comparison by $a_N^{-2}$ .

Source (illustrative geometry)	$a_N$ (m/s²)	Signal $\Delta a$ (m/s²)	Shots for $5\sigma$	Time at 15 s per shot
Aerogel, 10 kg/m³, $t=1$ cm (legacy point)	$4.2\times10^{-11}$	$2.9\times10^{-19}$	about $5\times10^{18}$	about $3\times10^{12}$ years
Aerogel, 10 kg/m³, $t=10$ cm	$4.2\times10^{-10}$	$2.9\times10^{-18}$	about $5\times10^{16}$	about $3\times10^{10}$ years
Solid at 19,320 kg/m³, $t=1$ cm (not low density)	$8.1\times10^{-8}$	$5.7\times10^{-16}$	about $1.3\times10^{12}$	about $8\times10^{5}$ years
Aerogel, 10 kg/m³, $t=1$ cm, $2\beta_0^2=1$ , $f\approx1$	$4.2\times10^{-11}$	$1.4\times10^{-16}$	about $2.5\times10^{13}$	about $1\times10^{7}$ years

The thicknesses are illustrative. The count assumes white noise and no systematic floor, so it understates the real requirement. It is a comparison with a published free-fall resolution. Section 2.1 does not specify an apparatus that holds atoms 1 μm from a surface, and this table does not assign that resolution to such an apparatus.

The legacy reference point is out of reach of the verified atom-interferometer resolution by many orders of magnitude for a laboratory-scale source. The v12 statement that the required sensitivity is "well within the reach of current technology" used a single-shot figure of $10^{-12}$ that is not in the verified list and was not referred to the source's pull. That statement is withdrawn. The same withdrawal covers the claim that the micrometre test is within reach of current atom interferometry.

4.4 Budget terms

The total uncertainty combines the statistical term and the systematic terms in quadrature:

\sigma^2=\sigma_{\rm stat}^2+\sigma_{\rm CP}^2+\sigma_{\rm patch}^2+\sigma_{B}^2+\sigma_{\rm grav}^2+\sigma_{\rm vib}^2+\sigma_{\rm hom}^2 .

Each systematic term has to be reduced below the target $\sigma$ of Section 4.2. Two requirements follow directly.

Target homogeneity and mass matching. The Newtonian pull of the two compositions differs by $(\Delta m/m)\,a_N$ , where $\Delta m/m$ is the fractional difference in mass per unit area seen by the atoms. For this to stay below the $5\sigma$ target of $1.4\times10^{-9}$ in $\Delta a/a$ , the fractional mass difference must be matched, or known and corrected, to better than $1.4\times10^{-9}$ . This follows the definition of $a$ as the source's Newtonian pull.
Casimir–Polder. The differential Casimir–Polder acceleration has to be below the same target multiplied by $a_N$ . The Casimir note gives the scale of the problem under reading (ii) and states what must be computed.

The remaining terms are not yet computed.

5. Roles of MICROSCOPE, torsion balances and satellite tests

The verified results are:

Experiment	Result
MICROSCOPE final result (Touboul et al., 2022)	$\eta({\rm Ti,Pt})=[-1.5\pm2.3\,({\rm stat})\pm1.5\,({\rm syst})]\times10^{-15}$ , about $2.7\times10^{-15}$ with the errors combined in quadrature. Reference pair Pt–Pt: $[0.0\pm1.1\,({\rm stat})\pm2.3\,({\rm syst})]\times10^{-15}$
Eöt-Wash rotating torsion balance (Schlamminger et al., 2008)	$\eta({\rm Be,Ti})=(0.3\pm1.8)\times10^{-13}$
Atom interferometer, $^{85}$ Rb and $^{87}$ Rb (Asenbaum et al., 2020)	$\eta=[1.6\pm1.8\,({\rm stat})\pm3.4\,({\rm syst})]\times10^{-12}$
Eöt-Wash short-range test (Lee et al., 2020)	Newtonian gravity fits between 52 μm and 3.0 mm. A gravitational-strength Yukawa interaction must have range below 38.6 μm (95% confidence)

These results were published before the present documents and are consistent retrodictions only.

5.1 MICROSCOPE

The test masses are small and move in the exterior field of the Earth. The scalar acceleration of each is sourced by the Earth, and the Earth is a dense body of radius 3,959 miles. In the screened scalar framework its external field is reduced by about $3\,\Delta R_\oplus/R_\oplus$ relative to an unscreened body. The differential acceleration of titanium and platinum is then

\frac{\Delta a}{a}\simeq 2\beta_0^2\,\Delta\delta_{\rm Ti\text{-}Pt}\;\frac{3\,\Delta R_\oplus}{R_\oplus}\,f .

At the benchmark, $5.7\times10^{-3}\times2.47\times10^{-6}=1.4\times10^{-8}$ before screening. Against $2.7\times10^{-15}$ this requires a suppression of at least $5\times10^{6}$ , so $3\,\Delta R_\oplus/R_\oplus\lesssim1.9\times10^{-7}$ and $\Delta R_\oplus/R_\oplus\lesssim6.5\times10^{-8}$ . This is the same order as the Earth condition $\Delta R_\oplus/R_\oplus<10^{-7}$ quoted by Khoury and Weltman (2004). It is an estimate with an uncertainty scale and not a confidence limit.

MICROSCOPE therefore constrains the thin-shell condition of the Earth, which depends on the potential, the coupling and the cosmological field value. It does not constrain the micrometre test directly, because the source is different (the Earth against an aerogel target), the environment is different (an orbit against a chamber with a surface 1 μm from the atoms) and the field solution is different. The two tests depend on the same parameters $(\beta_0,C,\Lambda,n)$ through different functions, the Earth's shell thickness and the factor $f$ of the laboratory geometry. A MICROSCOPE result consistent with zero is satisfied if the Earth is screened, whatever $f$ is in the laboratory, and a null micrometre result says nothing about the Earth's shell. 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. Against $\lesssim1.9\times10^{-7}$ the benchmark passes by a factor of about 1.5 to 4. The full calculation (density profile, atmosphere, ambient field, Moon and Sun) is open; see Section 4 of the screened scalar document.

5.2 Torsion balances

Eöt-Wash rotating balance, beryllium and titanium. The unscreened difference is $5.7\times10^{-3}\times6.52\times10^{-7}=3.7\times10^{-9}$ . Against the uncertainty $1.8\times10^{-13}$ the required suppression is at least $2\times10^{4}$ , about 250 times less demanding than MICROSCOPE for the same benchmark. It is the same type of constraint, on the screening of the Earth.
Short-range test (Lee et al.). This is a test of the force law, not a differential composition test. The scalar force between two bodies is $2\beta_i\beta_j$ times the Newtonian force, so the benchmark has strength $2\beta_0^2=5.7\times10^{-3}$ , 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, because the range depends on the density and on the screening of the bodies, so the field solution is needed to apply the result.
Patterned attractors. The v12 proposal of a torsion pendulum with attractor segments of high and low $\delta(Z,A)$ is kept as an option. The v12 fitting form $F=(Gm_1m_2/r^2)[1+\alpha e^{-r/\lambda}]$ with $\alpha$ and $\lambda$ from the old $\lambda^\pm$ is withdrawn. The fit has to use the field solution.

5.3 Satellite tests

MICROSCOPE is the only satellite composition test with a verified result in this document set. The proposed STEP mission has no verified target sensitivity here, and the v12 value is not used. The GRACE and GRACE-FO gravity-field models, including HUST-Grace2026s, measure the static gravity field of the Earth. They test the global geometry (see Appendix P and the global geometry tests) and do not test composition dependence. The v12 laser-ranging noise figure was quoted in picometres per root hertz, whereas the published requirement is 80 nm/√Hz, and the v12 signal-to-noise table built on it is withdrawn.

6. Falsification logic

A null result at total uncertainty $\sigma$ excludes a region of parameter space and does not end every screened-scalar model. For aluminium and gold, a null result excludes at 95% (one-sided)

2\beta_0^2\,C\,f>\frac{1.645\,\sigma}{\varepsilon\,\Delta(Z/A)}=1.5\times10^{4}\,\sigma .

The excluded region in $(2\beta_0^2C,\Lambda,n)$ follows once $f(\Lambda,n,\text{geometry})$ is computed. The result does not exclude:

models with $f\ll1$ in the apparatus, because the range in the gap is short or the source is screened;
smaller $\beta_0$ or $C$ ;
other potentials, and other couplings of the scalar.

Three further statements apply.

Structure of the composition dependence. The signal in any pair should scale with $\Delta(Z/A)$ at fixed geometry and $f$ . The ratios are independent of $C$ : aluminium–gold to beryllium–titanium is $0.0807/0.0158=5.1$ , and titanium–platinum to aluminium–gold is $0.0598/0.0807=0.74$ . Signals that do not follow these ratios contradict the isospin form of $\delta$ , whatever the size of $C$ .
Joint use with the Earth-sourced bounds. A micrometre null excludes a region defined by $f$ . MICROSCOPE excludes a region defined by the Earth's shell. The union of the two is the excluded region, and neither implies the other.
A positive result. A signal is attributed to the scalar only if it survives the control runs of Section 2.4, follows the separation scan expected from the field solution, reverses with the composition and follows the $\Delta(Z/A)$ ratios above. The v12 claim that this experiment is the only one capable of establishing or falsifying MCE is withdrawn.

7. Material removed from the v12 version

v12 item	Disposition
Benchmark $(6.0\pm0.7)\times10^{-9}$ and envelope $(6.0\text{–}14.8)\times10^{-9}$ for $\lambda_c\in[1,10]\ \mu$ m	Withdrawn as predictions. Legacy reference point of $6$ to $7\times10^{-9}$ (Section 4.1)
Spatial suppression $e^{-100}$ at 100 μm and the fixed $\lambda_c$	Retired. Replaced by the range $\lambda(\rho)$
Single-shot sensitivity $10^{-12}$ , averaged to $10^{-15}$ over $10^{6}$ drops, "decisive, high-priority"	Withdrawn. Not sourced, and not referred to the source's pull (Section 4.3)
Differential Casimir estimate $10^{-3}\times10^{-9}=10^{-12}$	Withdrawn. Not a derivation
Quantitative systematic sizes in the v12 tables	Withdrawn (Section 3)
MICROSCOPE/STEP suppression table with $e^{-10^4}$ and $2\times10^{-8}$ entries	Withdrawn. Replaced by the thin-shell requirement of Section 5.1
Connection to GR tests ("scalar-vector-tensor" table, black-hole interior)	Retired. These results are inherited from the metric sector
Galactic-dynamics framework with the Bullet Cluster as the primary test	Withdrawn. A conformally coupled scalar does not bend light beyond GR, so it does not produce a lensing mass offset from the baryons
Modified GADGET-4 Poisson solver with kernel $-4\pi G/(k^2+k_c^2)$	Retired. See Appendix N
Boltzmann-code framework with an "EME effective fluid"	Replaced by the growth-of-structure method in Appendix P, not yet computed
Laser-ranging table and "toroidal coupling" forecast ( $3\times10^{-13}$ m/s²)	Withdrawn. The signal had no derivation and the noise figure was wrong by a factor of 1,000
Superconducting Faraday-cage test	Untested speculation (below)
Closing statement that the test is "decisive" and the theory "ready for empirical engagement"	Withdrawn

Superconducting-cage test. The v12 text proposed that the quantum component of the field couples to the zero-point field, that Cooper pairs alter this coupling, and that a gravimeter inside a niobium or YBCO shield would show a change on cooling through $T_c$ , at a sensitivity requirement of $\Delta g/g\lesssim10^{-14}$ . Matter couples to the scalar only through $A_i^2(\phi)\,g_{\mu\nu}$ in the screened scalar action, and no term links the superconducting transition to the coupling. The proposal is an untested speculation. No signal size has been derived, and the quoted sensitivity is not part of the programme.