Quantum-Mechanical Foundation: QVP Postulate and Length Scales

Revision note (v13.0). Sections 1.2, 2 and 3 are rewritten because they rested on withdrawn results: the "EFT matching" value of $\kappa$ , the universal micrometre coherence length with its Lindblad bridge, and the $(6.0\pm0.7)\times10^{-9}$ forecast. The mass-induced vacuum-polarisation (QVP) postulate is kept as the Level 2 motivation of MCE. It is a postulate and is not derived here. No quantity in this document is derived from first principles at v13.0. Replaced by: the parameterised signal of the screened scalar sector (Level 1) and the environment-dependent range $\lambda(\rho)$ . See the architecture section of the main document and the screened scalar sector.

0. Status of the items in this document

Item	Status	Where
QVP source law $\rho_{\rm eff}=A_{\rm QVP}\,\rho_{\rm mass}$ , coefficient $A_{\rm QVP}$	Postulated (Level 2 motivation); coefficient Open	Section 1.1
Newton's constant $G$	Inherited (Level 0, taken from experiment)	Section 1.2
Scalar-sector coupling $\beta_0$	Open (free parameter, bounded by experiment)	Section 1.2
$\kappa=1.623\times10^{-10}$ C/kg and " $\kappa$ fixed by matching to $G$ "	Withdrawn	Section 1.2
$\hbar c/(m_ec^2)=3.86\times10^{-13}$ m	Derived (arithmetic only)	Section 2.1
$\hbar c/(k_BT)=7.63\ \mu$ m at 300 K	Derived (arithmetic only; no link to the gravity mechanism)	Section 2.2
Fixed $\lambda_c=1\ \mu$ m, band $[1,10]\ \mu$ m, Lindblad bridge	Withdrawn	Section 2.3
Environment-dependent range $\lambda(\rho)$	Postulated (follows from the Level 1 action)	Section 2.4
Benchmark $(6.0\pm0.7)\times10^{-9}$	Withdrawn as a prediction; legacy reference point Estimated	Section 2.5

1. The Effective Charge Density Concept

1.1. Mass-Induced Asymmetry in Quantum Vacuum Polarisation (QVP)

The original MCE idea is that an effective charge density $\rho_{\rm eff}$ arises from a mass-induced asymmetry in the quantum vacuum polarisation (QVP). Standard QVP (for example the Uehling potential) is symmetric. Virtual particle-antiparticle pairs such as $e^+e^-$ screen the bare electric charge, and the sign of the effect does not depend on the mass of the source. The MCE postulate is that mass breaks this symmetry and produces a net scalar charge proportional to the mass density $\rho_{\rm mass}$ .

Status: Postulated. This is the Level 2 motivation in the v13.0 status ladder. It is not derived in this corpus, and nothing at Level 1 depends on it: the screened scalar sector takes the species couplings $\beta_i$ as inputs.

Verbal picture used in earlier versions (heuristic, not a calculation). The picture was that the mass of a particle measures its coupling to the Higgs field, that this coupling modifies the local zero-point-field energy density, and that the modification acts as a mass-dependent chemical potential which biases virtual-pair creation and annihilation. Two corrections apply. First, the Higgs coupling accounts for the electron mass, but roughly 99% of the proton mass comes from QCD binding energy, so a picture based on the Higgs coupling cannot be the universal origin of a coupling proportional to mass. Second, no calculation of the proposed bias exists in this corpus. The picture is kept as the motivation for the postulate and is not a derivation.

Source-law target. The postulate takes the form

\rho_{\rm eff}(x) = A_{\rm QVP}\,\rho_{\rm mass}(x) + \mathcal{O}\!\left(\frac{\partial^2}{m_*^2}\right),

where $A_{\rm QVP}$ would be extracted from a regulated vacuum-polarisation diagram in a mass-bearing background. That calculation has not been done (Open). In Level 1 variables the scalar $\phi$ is sourced by $\beta_i\rho_i/M_{\rm Pl}$ for species $i$ , so $A_{\rm QVP}$ corresponds to $\beta_0/M_{\rm Pl}$ up to the species factors $1+\delta_i$ . A computation of $A_{\rm QVP}$ would therefore fix $\beta_0$ . It is part of deliverable (i) in the UV completion roadmap.

Electrostatic backbone (Level 1). The same source term is the right-hand side of the static linearised equation in Section 2.1 of the screened scalar sector,

\nabla^2\delta\phi-m_{\rm eff}^2(\rho)\,\delta\phi=\frac{\beta_i\rho_i}{M_{\rm Pl}}.

That equation is electrostatics in a screening medium. Like scalar charges attract, which is the opposite of ordinary electrostatics, and thin-shell screening is the conductor analogy: the interior of a dense body stays at the minimum, and the exterior field comes from a surface shell. The vector sector is not part of the core. The analogy stops because the scalar couples to mass density, through the trace, and not to a current. Level 2 applies the same electrostatic idea to the vacuum. That step is the QVP postulate above. It is not a derivation, and Level 1 does not depend on it.

1.2. Withdrawn: the "EFT matching" derivation of $\kappa$

The v12 text claimed that

\kappa = \frac{1}{\sqrt{4\pi\epsilon_0}}\sqrt{\frac{G}{c^2}} \approx 1.623\times10^{-10}\ {\rm C/kg},

and that $\kappa$ is therefore fixed once MCE reproduces $G$ . Both statements are withdrawn. Recomputed with SI constants:

Quantity	Expression	Value	Status
Value claimed in v12	none given that reproduces it	$1.623\times10^{-10}$ C/kg	Withdrawn (not reproduced by either expression below)
Written v12 formula	$(4\pi\epsilon_0)^{-1/2}(G/c^2)^{1/2}$	$2.58\times10^{-9}$ (numerical value only)	Withdrawn
Coulomb-type match	$q/m=\sqrt{4\pi\epsilon_0 G}$ , the charge-to-mass ratio at which Coulomb repulsion equals Newtonian attraction	$8.62\times10^{-11}$ C/kg	Derived (arithmetic; reference value only, not an MCE quantity)

Three further points follow from the table.

The claimed value is 1.88 times the Coulomb-type value and 0.063 times the written-formula value. Neither expression gives it.
The written formula does not have the units C/kg. By direct substitution its units are $\mathrm{m^2\,s^{-1}\,C^{-1}}$ , so the number $2.58\times10^{-9}$ cannot be quoted in C/kg. Only the Coulomb-type expression is dimensionally a charge-to-mass ratio.
A coupling constant defined by the requirement that the theory reproduces $G$ only relabels $G$ . It adds no prediction and removes no free parameter.

What replaces it (v13.0).

$G$ is an input from the GR sector (Level 0: the Einstein–Hilbert action with $G$ taken from experiment).
The scalar-sector coupling $\beta_0$ is a free parameter, bounded by experiment. The ratio of the scalar force to the Newtonian force between species $i$ and $j$ is $2\beta_i\beta_j$ before screening and range factors.
The symbol $\kappa$ may be kept as notation for $\beta_0/M_{\rm Pl}$ , a coupling with units of inverse mass, where $M_{\rm Pl}=(8\pi G)^{-1/2}$ is the reduced Planck mass ( $2.435\times10^{27}$ eV). Here $G$ enters only as the conversion between $\beta_0$ and a dimensional coupling. For the legacy value $\beta_0\approx0.053$ (an inference from the old forecast, see Section 2.5), $\beta_0/M_{\rm Pl}\approx2.2\times10^{-29}\ {\rm eV}^{-1}$ .

Clarification on circularity (updated). The v12 text argued that the appearance of $G$ in the formula for $\kappa$ is not circular, because GR and MCE both take $G$ from experiment. That point about $G$ is correct and is now the architecture: $G$ is an input at Level 0 and MCE does not predict it. The conclusion drawn from it in v12 is withdrawn. A matching condition on $G$ fixes nothing in the scalar sector. The v12 claim that MCE explains why gravity has the inverse-square form, why it is universally attractive and why it shows material-dependent violations of the weak equivalence principle is also withdrawn. The inverse-square form and the universality of attraction belong to the metric sector (Level 0). The scalar sector adds a composition-dependent, screened force on top of it. A non-circular route to the size of the gravitational coupling exists only at Level 2, where $1/G$ would be computed from a cutoff and a field content (induced gravity). That calculation is open.

2. Length Scales: Arithmetic and Status

2.1. The electron reduced Compton wavelength

The arithmetic retained from v12 is

\frac{\hbar c}{m_ec^2}=\frac{(1.0546\times10^{-34}\ {\rm J\,s})(2.998\times10^{8}\ {\rm m/s})}{(0.511\times10^{6}\ {\rm eV})(1.602\times10^{-19}\ {\rm J/eV})}=\frac{3.16\times10^{-26}\ {\rm J\,m}}{8.19\times10^{-14}\ {\rm J}}=3.86\times10^{-13}\ {\rm m}.

This is the reduced Compton wavelength of the electron. The v12 text defined a QVP coherence length $\lambda_c=\hbar c/(\alpha_{\rm EME}E_{\rm ZPF})$ with $\alpha_{\rm EME}\approx1$ and $E_{\rm ZPF}=m_ec^2$ , and read the number above as the coherence length of the mass-induced QVP. That reading is withdrawn. The choice $E_{\rm ZPF}=m_ec^2$ was an assumption, and $\alpha_{\rm EME}=\kappa^2/(4\pi\epsilon_0G/c^2)$ is not dimensionless as written and was not shown to equal 1.

2.2. The thermal wavelength

At 300 K, $k_BT=0.02585$ eV and

\frac{\hbar c}{k_BT}=\frac{197.3\ {\rm eV\,nm}}{0.02585\ {\rm eV}}=7.63\times10^{3}\ {\rm nm}=7.63\ \mu{\rm m}.

The v12 "environmental bridge" was $\lambda_c^{\rm eff}=\lambda_c^{\rm fund}\,(E_{\rm ZPF}/k_BT)$ with $\lambda_c^{\rm fund}=\hbar c/E_{\rm ZPF}$ . Substituting gives

\lambda_c^{\rm eff}=\frac{\hbar c}{E_{\rm ZPF}}\cdot\frac{E_{\rm ZPF}}{k_BT}=\frac{\hbar c}{k_BT}.

The electron mass cancels. The result does not depend on $E_{\rm ZPF}$ , and so does not depend on the choice $E_{\rm ZPF}=m_ec^2$ . The v12 value of $7.4\ \mu$ m is replaced by the correct value, 7.63 $\mu$ m. The ratio $m_ec^2/k_BT=1.98\times10^{7}$ , which the v12 text described as a shift of "seven orders of magnitude consistent with the Lindblad master equation", is the ratio of two energies and appears only because $\lambda_c^{\rm eff}$ was defined by multiplying by it. It carries no dynamical content.

Reading. The micrometre band is a thermal-wavelength estimate for room temperature. It is not derived from the gravity mechanism. It may be mentioned only as a possible origin of a micrometre scale.

2.3. Withdrawn: the universal $\lambda_c=1\ \mu$ m, the band $[1,10]\ \mu$ m and the Lindblad bridge

The following v12 statements are withdrawn.

The universal factor $S_r=e^{-r/\lambda_c}$ with a fixed $\lambda_c=1\ \mu$ m, and the working band $\lambda_c^{\rm eff}\in[1,10]\ \mu$ m. Neither follows from Section 2.2.
The statement that the decoherence rate obeys $\Gamma\propto\kappa^2T$ , that the effective mass is $m_{\rm eff}\propto\Gamma$ , and that a Lindblad master equation produces the bridge from $3.86\times10^{-13}$ m to the micrometre band. No Lindblad operators, no proportionality constants and no calculation were ever given. These statements are heuristic and are removed. The Lindblad bridge is withdrawn from the research programme as well (deliverable (iv) of the roadmap).
The claim that the benchmark $\lambda_c=1\ \mu$ m is the "conservative lower edge" because it gives the strongest macroscopic suppression. With $\lambda_c$ withdrawn, the statement has no content.

2.4. What replaces it: the environment-dependent range $\lambda(\rho)$

In the Level 1 action the scalar has a mass that depends on the surrounding density. For $V(\phi)=\Lambda^{4+n}/\phi^{n}$ the effective potential is $V_{\rm eff}=V+\rho\,e^{\beta\phi/M_{\rm Pl}}$ , with

\phi_{\min}(\rho)=\left(\frac{n\,\Lambda^{4+n}M_{\rm Pl}}{\beta\,\rho}\right)^{\frac{1}{n+1}},\qquad m_{\rm eff}^2(\rho)=n(n+1)\Lambda^{4+n}\phi_{\min}^{-(n+2)}+\frac{\beta^2\rho}{M_{\rm Pl}^2},\qquad \lambda(\rho)=\frac{\hbar}{m_{\rm eff}c}.

The range is long in a vacuum chamber and short in dense matter. The free parameters are $(\beta_0, C, \Lambda, n)$ and no combination of them is fixed by matching $G$ . Values of $\lambda(\rho)$ for illustrative parameters, the thin-shell screening that applies to dense bodies, and the open calculations are given in the screened scalar sector. The density-dependent suppression used in v12 forecasts is discussed in Density Screening: Phenomenological Profile and Thin-Shell Replacement.

2.5. The forecast: parameterised signal and legacy reference point

The v12 headline $\Delta a/a=(6.0\pm0.7)\times10^{-9}$ and its band $(6.0\text{–}14.8)\times10^{-9}$ are withdrawn as predictions. In v13.0 the differential acceleration between two materials is the parameterised signal

\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=\frac{m_n-m_p}{m_p}=1.378\times10^{-3}.

Here $f$ collects screening and range effects. It must be computed from the scalar field equation in the experimental geometry. The simplest estimate is $f\approx e^{-r/\lambda(\rho)}$ . $f$ has not been computed for any real geometry (not yet computed).

At the benchmark $C=0.03$ :

$C\varepsilon=0.03\times1.378\times10^{-3}=4.13\times10^{-5}$ .
For aluminium and gold, $\Delta(Z/A)=0.0807$ , so $\Delta\delta_{\rm Al\text{-}Au}=4.13\times10^{-5}\times0.0807=3.34\times10^{-6}$ .

Legacy reference point (Estimated). The v12 unsuppressed value $1.9\times10^{-8}$ equals $2\beta_0^2\,\Delta\delta_{\rm Al\text{-}Au}$ for $2\beta_0^2=5.7\times10^{-3}$ (so $\beta_0\approx0.053$ ). The factor $5.7\times10^{-3}$ was not identified in the v12 documents. Reading it as $2\beta_0^2$ is an inference made in v13.0, and it is labelled as such. With $f=e^{-1}$ :

1.9\times10^{-8}\times e^{-1}=6.99\times10^{-9}.

The v12 headline of $6.0\times10^{-9}$ is this value multiplied by a further factor of 0.86 attributed to QCD running of $C$ . That factor is possibly counted twice, and the " $\pm0.7$ " propagated a lattice-QCD uncertainty onto a coefficient $C$ that is not derived. The headline is therefore not a prediction with error bars. It is retained only as a legacy reference point of the parameter surface, about $6$ – $7\times10^{-9}$ for $2\beta_0^2=5.7\times10^{-3}$ , $C=0.03$ and $f=e^{-1}$ . The band $(6.0\text{–}14.8)\times10^{-9}$ came from the withdrawn range $\lambda_c\in[1,10]\ \mu$ m and is withdrawn with it.

Screening is required for this reference point. A scalar with $2\beta_0^2=5.7\times10^{-3}$ is far above the Cassini-type bounds if it is unscreened in the solar system. The reference point is meaningful only where screening removes the solar-system scalar force. A first-pass thin-shell estimate for the benchmark passes the Earth requirement from MICROSCOPE, $3\,\Delta R_\oplus/R_\oplus\lesssim1.9\times10^{-7}$ , by a factor of about 1.5 to 4, and passes the Sun by many orders. The full calculation is open. The estimate and the open items are in Sections 4 and 8 of the screened scalar sector.

Absolute size at the legacy point. For an atom near a local source, $a$ in $\Delta a/a$ is the Newtonian pull of that source. A 1 cm aerogel slab at 10 kg/m³ gives $2\pi G\sigma=4.2\times10^{-11}$ m/s², so the legacy fractional value $7.0\times10^{-9}$ is an absolute signal of $2.9\times10^{-19}$ m/s², about $3\times10^{-19}$ m/s². That is not within reach of current atom interferometry. The comparison is in section 3 of the main document.

3. Conclusion

The QVP source law is a Level 2 postulate. Its coefficient $A_{\rm QVP}$ is not computed, and it is the quantity that would fix $\beta_0$ if induced gravity could be made to work. At Level 1, $G$ is an input from the metric sector and $\beta_0$ is a free parameter bounded by experiment. The Level 1 field equation is the electrostatic backbone in Section 2.1 of the screened scalar document: electrostatics in a screening medium, with like scalar charges attracting. The v12 values of $\kappa$ and of the coherence length, and the Lindblad bridge connecting them, are withdrawn. The micrometre scale is at most a thermal-wavelength estimate, $\hbar c/k_BT=7.63\ \mu$ m at 300 K, which is independent of the electron mass. The testable output is the parameterised signal $2\beta_0^2\,\Delta\delta\,f$ , with $f$ still to be computed from the field equation in the experimental geometry. Referred to the source's own pull, the legacy point is an absolute signal of about $3\times10^{-19}$ m/s², which is not a claim of present experimental reach.