Field Roles and Material Dependence Justification

Revision note (v13.0). Section 1 is rewritten around the v13.0 field content (metric and one conformally coupled scalar). Withdrawn: the claim that a second, vector field is needed to resolve "like charges repel", the claim that buoyancy has an electromagnetic foundation in MCE, the description of the isospin structure as "derived", and the coefficient $2.36\times10^{-7}$. The correct coefficient is $C\varepsilon=4.13\times10^{-5}$ at $C=0.03$, and Section 2.3 recomputes the material values. The form $\delta(Z,A)=C\varepsilon\,(Z/A-\tfrac12)$ is kept as a postulated, falsifiable structure. See the architecture section of the main document and the screened scalar sector.

1. Field Content at v13.0

1.1. Withdrawn: two fields needed to resolve "like charges repel"

The v12 text stated that MCE needs both a scalar and a vector field "to resolve the like-charges-repel paradox". It assigned the scalar to attraction and an MCE vector $A_\mu^{\rm EME}$, coupled to the electromagnetic current, to the repulsion between like electric charges and to consistency with Maxwell's equations. This is withdrawn. Ordinary electromagnetism already makes like charges repel and already satisfies Maxwell's equations. There is no paradox for a second field to resolve.

A vector field coupled to baryon number would produce a force that is repulsive between like charges and would add to the scalar force. It would be a fifth force constrained by torsion-balance tests of the equivalence principle and of the inverse-square law. The relevant bounds are not collected in this document. The vector is optional and is not needed by any Level 1 result.

The v12 text also stated that the sum of the scalar and vector interactions lets MCE "explain both attraction (gravity) and repulsion (buoyancy, electrostatic repulsion) from a unified electromagnetic foundation". That statement is withdrawn for three reasons.

• Gravity at Level 0 comes from the metric sector. The scalar is an additional force and not the whole of gravity.
• Electrostatic repulsion is ordinary electromagnetism.
• Buoyancy is the net force from the pressure gradient in a fluid in a gravitational field. The pressure itself, like every contact force between atoms, is electromagnetic in origin in standard physics, but that gives MCE no special foundation for buoyancy, and no MCE derivation of buoyancy exists. The claim is removed.

1.2. The scalar couples to the trace of the stress tensor

Matter species $i$ moves in the metric $A_i^2(\phi)\,g_{\mu\nu}$. At linear order in $\phi$ the interaction is

where $T_i=T^\mu{}_\mu$ is the trace of the stress tensor of species $i$ (the overall sign depends on the metric signature). The symbol $\kappa$ used in v12 corresponds to $\beta_0/M_{\rm Pl}$. A massive particle has rest energy $m_iA_i(\phi)$, which is why the effective potential contains the term $\rho\,e^{\beta\phi/M_{\rm Pl}}$.

• Why the trace. The coupling to $T$ is the one produced by rescaling the metric that matter sees (a conformal coupling). If all species share one $\beta$ it is universal and does not violate the weak equivalence principle. The v12 statement that $T$ is "the only Lorentz-invariant scalar that can be constructed from the energy-momentum tensor" is withdrawn, because $T^{\mu\nu}T_{\mu\nu}$ and couplings to curvature are also scalars. The conformal coupling is a choice of the simplest form.
• Non-relativistic limit. For non-relativistic matter $|T|\to\rho c^2$, so the coupling is proportional to the mass density and the force on species $i$ is proportional to $m_i\beta_i$.
• Photons. The Maxwell stress tensor is traceless in four dimensions, so $T=0$ and the scalar does not couple to light at tree level. A scalar sourced by the trace therefore cannot supply light bending, the Shapiro delay or the perihelion advance. Those results come from Level 0.
• Antimatter. The trace has the same sign for matter and antimatter, so the scalar couples to both with the same sign. This is an argument from CPT invariance of the trace coupling and not a computed result.
• Contrast with the vector. An electromagnetic field couples to the conserved current $J^\mu$. The scalar couples to $T$. The two are different objects, and the scalar replaces nothing in electromagnetism.

The scalar force between species $i$ and $j$ is attractive when $\beta_i\beta_j>0$, and its ratio to the Newtonian force is $2\beta_i\beta_j$ before screening and range factors.

2. Material Dependence Through $\delta(Z,A)$

2.1. The form of the species coupling

The species couplings are $\beta_i=\beta_0\,[1+\delta_i]$ with

$C$ is a dimensionless coefficient with a benchmark of $C\approx0.03$. It is not derived. Because $N=A-Z$,

so $\delta$ is proportional to half the fractional proton excess, and it is negative for neutron-rich nuclei.

Status. The isospin form is Postulated. It is motivated by the neutron–proton mass difference, and it is clear and falsifiable: it fixes which material pairs give a large signal. The normalisation $C$ is Open. The v12 text called the structure "derived" and said the normalisation was "anchored by hadronic matching plus lattice-QCD input". No loop or lattice calculation of $C$ exists in this corpus, and both descriptions are withdrawn. The symbol $C_{\rm QFT}$ used in v12 is replaced by $C$.

Reference point. $\delta$ vanishes at $Z/A=\tfrac12$ by construction, that is for nuclei with equal numbers of protons and neutrons. This is a property of the formula and is not a derived cancellation of QVP terms. Silicon is close to this point but not on it (Section 2.3).

Limitation. The formula assumes that the dominant composition dependence is isospin. A full treatment would include nuclear binding energy and the electron contribution to the mass of a neutral atom. Those terms are not computed here.

2.2. Correction of the coefficient

The v12 text quoted the coefficient of $(Z/A-0.5)$ as $2.36\times10^{-7}$. That value is withdrawn. The correct coefficient is

which is 175 times larger. The old number is consistent with $2\beta_0^2\,C\varepsilon$ for $2\beta_0^2=5.7\times10^{-3}$, since $5.7\times10^{-3}\times4.135\times10^{-5}=2.36\times10^{-7}$. That reading is an inference. The factor $5.7\times10^{-3}$ was not stated in v12. The same factor connects the old unsuppressed value $1.9\times10^{-8}$ to $2\beta_0^2\,\Delta\delta_{\rm Al\text{-}Au}$ (Section 2.4).

2.3. Recomputed material values

Inputs are $\varepsilon=(939.565-938.272)/938.272=1.3784\times10^{-3}$, so $C\varepsilon=0.03\times1.3784\times10^{-3}=4.1353\times10^{-5}$, and the standard atomic weights $A$ below. $A$ is used as a proxy for the mean nucleon number of the natural isotope mixture.

Arithmetic used for each row:

• Al: $13/26.982=0.481803$; $0.481803-0.5=-0.018197$; $4.1353\times10^{-5}\times(-0.018197)=-7.52\times10^{-7}$.
• Si: $14/28.0855=0.498478$; $-0.001522$; $4.1353\times10^{-5}\times(-0.001522)=-6.29\times10^{-8}$.
• Ti: $22/47.867=0.459607$; $-0.040393$; $4.1353\times10^{-5}\times(-0.040393)=-1.67\times10^{-6}$.
• Pt: $78/195.08=0.399836$; $-0.100164$; $4.1353\times10^{-5}\times(-0.100164)=-4.14\times10^{-6}$.
• Au: $79/196.97=0.401076$; $-0.098924$; $4.1353\times10^{-5}\times(-0.098924)=-4.09\times10^{-6}$.

The pair differences that determine a test are:

For aluminium and gold, $\Delta\delta=4.1353\times10^{-5}\times0.0807=3.34\times10^{-6}$.

Cautions on the table:

• The silicon value is sensitive to isotopic composition. Silicon-28 has $Z=N=14$ and $Z/A=\tfrac12$ exactly, so $\delta=0$ for pure silicon-28. The tabulated $-6.29\times10^{-8}$ is for natural abundance.
• The table is for pure elements. The MICROSCOPE test masses were alloys (a platinum–rhodium alloy and a titanium alloy), so the titanium and platinum row is indicative only.
• The table gives $\delta$ and $\Delta\delta$. It does not give a signal. The signal is $\Delta a/a\simeq2\beta_0^2\,\Delta\delta\,f$, where $f$ is the screening and range factor for the experimental geometry (not yet computed).

2.4. Reading the old numbers

• The v12 unsuppressed difference of $1.9\times10^{-8}$ is $2\beta_0^2\,\Delta\delta_{\rm Al\text{-}Au}=5.7\times10^{-3}\times3.34\times10^{-6}=1.9\times10^{-8}$ (so $\beta_0\approx0.053$). This identification of $2\beta_0^2$ is an inference.
• The pair titanium and platinum is the MICROSCOPE pair. The 2022 result is $\eta({\rm Ti,Pt})=[-1.5\pm2.3\,({\rm stat})\pm1.5\,({\rm syst})]\times10^{-15}$ (Touboul et al., Physical Review Letters 129, 121102). With the legacy benchmark, an unscreened Earth would give $\Delta a/a\approx1.4\times10^{-8}$ between titanium and platinum. A first-pass thin-shell estimate gives $3\,\Delta R_\oplus/R_\oplus=1.2\times10^{-7}$ for a galactic ambient density, against the requirement $\lesssim1.9\times10^{-7}$, so the benchmark passes by a factor of about 1.5 to 4. The full Earth field solution is open. Section 4 of the screened scalar sector gives the estimate.
• Cassini-type bounds on a scalar in the solar system apply unless screening removes the solar-system force. That is the role of the screening mechanism. The same first-pass estimate gives $3\,\Delta R_\odot/R_\odot\approx4\times10^{-11}$ for the Sun. The full calculation, including the Cassini comparison, is open.

3. What Remains