The BMV Cube Law: A Mass-Independent Prediction for the Quantum Gravity Tabletop Test

Paper M: The BMV Cube Law from the Quantum-Geometric Duality Series

Two Clocks in the Same Experiment

The Bose-Marletto-Vedral proposal is the cleanest tabletop test of the quantum nature of gravity yet devised. Place two massive particles, each in a spatial superposition, near enough that their mutual Newtonian attraction can entangle the branches. If the entanglement appears, gravity cannot have been a classical mediator: by the LOCC argument, classical channels cannot generate entanglement between systems that were not already entangled. The QGEM collaboration has turned this into a concrete design with nanodiamonds, NV-center interferometers, and target masses around $10^{-17}$ kg.

The standard treatment runs a single clock. The Newtonian phase between the four branches grows linearly in time. After

$\tau_{\mathrm{BMV}} \sim \frac{\hbar\,d^{3}}{G m^{2} \Delta^{2}}$

the phase reaches $\pi/2$ and the entanglement witness fires. Throughout this interval, gravity is treated as a passive carrier of phase: it does not decohere the source masses and does not fluctuate appreciably.

Quantum-Geometric Duality runs a second clock alongside the first.

The Per-Particle Decoherence Rate

In the constrained Feynman-Vernon picture of Paper K, the linearised Wheeler-DeWitt constraint forces each branch of a matter superposition to carry its own dressed coherent state of the gravitational field. The overlap between these dressed states decays. For a single mass $m$ displaced over a distance $\Delta$, the post-saturation decoherence rate is

$\Gamma = \frac{G m^{2}}{\hbar\,\Delta}\,,$

at order $G^{1}$. This is the Diosi-Penrose energy-over-$\hbar$ form, with one reinterpretation: the relevant length scale is the arm separation $\Delta$, not the inter-particle distance $d$. A mass in superposition does not know about its neighbour; it knows about its own two locations.

The BMV phase grows at a rate set by the inter-particle geometry. The QGD overlap decays at a rate set by the arm separation. They compete, and the visibility at the moment the witness fires is the product.

The Cube Law

The algebra is short. The visibility of the off-diagonal density-matrix element after time $\tau$ is $V = \exp(-2\Gamma\tau)$, where the factor of two counts the two particles each shedding coherence. With $\Gamma \propto 1/\Delta$ and $\tau_{\mathrm{BMV}} \propto d^{3}/\Delta^{2}$:

$\Gamma\,\tau_{\mathrm{BMV}} \;\sim\; \frac{G m^{2}}{\hbar\,\Delta} \cdot \frac{\hbar\,d^{3}}{G m^{2}\,\Delta^{2}} \;=\; \left(\frac{d}{\Delta}\right)^{3}\,.$

Mass and Newton's constant cancel exactly. What remains is purely geometric:

$V(\tau_{\mathrm{BMV}}) \;=\; \exp\!\Bigl[-\,c_{g}\,(d/\Delta)^{3}\Bigr]\,,$

where $c_{g}$ depends only on arm orientation. Parallel to the inter-particle axis, $c_{g} = \pi/2$. Perpendicular (the QGEM "diamond" geometry), $c_{g} = \pi$ — twice the suppression at the same $\Delta/d$, because the perpendicular phase formula is half as efficient and the experiment must wait twice as long for the witness to fire.

This is an unusually rigid prediction. Three features deserve attention.

Three Features That Make It Distinctive

Mass independence. Every other proposed gravitational decoherence mechanism carries explicit mass scaling. The Diosi-Penrose rate is $\propto m^{2}$. CSL collapse rates depend on local mass density. Pikovski-style time-dilation decoherence depends on composition. Only QGD's combination of $\Gamma \propto m^{2}$ with $\tau_{\mathrm{BMV}} \propto 1/m^{2}$ produces a clean cancellation. The exponent is geometric.

Parameter freedom. There is no analogue of the CSL collapse rate $\lambda$ or the Diosi smearing length $\sigma_{0}$ to fit. The prefactors $\pi/2$ and $\pi$ are fixed by the arm geometry alone. Up to the $\mathcal{O}(1)$ choice of witness threshold, the prediction is rigid.

Angular structure with a magic angle. The witness phase is, at leading order in $\Delta/d$, a Newtonian dipole-dipole interaction with Legendre angular dependence $|1 - 3\cos^{2}\theta|$:

$V_{\mathrm{lead}}(\theta) \;=\; \exp\!\biggl[-\,\frac{\pi\,(d/\Delta)^{3}}{|1 - 3\cos^{2}\theta|}\biggr]\,.$

This vanishes at the unique angle

$\cos^{2}\theta_{m} = \tfrac{1}{3}\,, \qquad \theta_{m} = \arccos(1/\sqrt{3}) \approx 54.74^{\circ}\,,$

the same magic angle that nuclear magnetic resonance has used for sixty years to suppress dipolar couplings. At $\theta_{m}$ the dipole BMV phase is identically zero. The witness must wait for the quadrupole term, which scales as $(\Delta/d)^{4}$ rather than $(\Delta/d)^{2}$, and the visibility law changes character:

$V_{\mathrm{lead}}^{\mathrm{magic}} \;=\; \exp\!\Bigl[-\,\tfrac{9\pi}{7}\,(d/\Delta)^{5}\Bigr]\,.$

A quintic, not a cubic. No collapse model can produce this: collapse rates depend on the local matter distribution and are blind to the orientation of the superposition arms relative to a distant partner. The $(d/\Delta)^{5}$ scaling at the magic angle is a direct fingerprint of the geometric structure of the Newtonian phase that QGD is competing against. A single apparatus that can rotate its arms therefore sweeps three qualitatively distinct QGD signatures with the same hardware.

What QGEM Should See

For the QGEM-class design point $\Delta = d/2$, all three QGD predictions collapse to essentially zero residual visibility:

Standard quantum gravity predicts $V \approx 1$ in all three. The gap is unambiguous. The discrimination-feasible window — where QGD predicts $V$ between roughly $0.05$ and $0.95$, neither washed out nor undetectable — sits in different regions for each geometry. The perpendicular layout offers the widest window, $\Delta/d \in [1.3,\,31.6]$, with no singularity anywhere. The parallel-axis layout is narrower, $\Delta/d \in [0.85,\,0.95]$, constrained from above by the singularity at $\Delta = d$ where the arms touch the partner particle. The magic-angle window is comparable to the perpendicular but traces out the quintic functional form, which decays more slowly in $\Delta/d$ and produces a visibly different curve as the experiment scans.

What Falsifies What

The experimental logic is binary at the design point.

If QGEM observes the standard BMV entanglement amplitude — $V \approx 1$ at $\Delta/d \sim 1$ in any of the three orientations — the constrained-IF picture is falsified, and gravity is confirmed as the passive phase-mediator of the standard treatment.

If QGEM observes cube-law suppression with the geometry-dependent prefactor and no mass dependence across a sweep, QGD is selected over Diosi-Penrose, CSL, and Pikovski in a single run. None of those models can reproduce the mass cancellation.

If the magic-angle measurement additionally shows quintic suppression with prefactor $9\pi/7$, no decoherence mechanism whose rate is independent of arm orientation survives — a constraint that excludes every collapse model on the market.

The Honest Caveats

The derivation lives entirely within linearised gravity. At QGEM-class masses the linearisation parameter $G m / (c^{2} \Delta)$ is around $10^{-38}$, so this is not the worry. The post-saturation rate applies for $\tau > d/c$; for QGEM this threshold is microscopic ($d/c \sim 10^{-12}$ s) while $\tau_{\mathrm{BMV}} \sim 1$ s, so the saturated regime dominates by twelve orders of magnitude.

The numerical prefactors carry the same factor-of-two ambiguity inherited from Paper K's modular-Hamiltonian identification. The cube exponent itself is rigid; so is the magic-angle quintic and the mass cancellation. These are the load-bearing pieces.

Environmental decoherence — residual gas, blackbody radiation, internal degrees of freedom of the nanodiamond — acts in parallel and adds to the QGD suppression. The QGEM background budget is comparable to the QGD prediction at $\Delta/d \sim 1$ in the parallel layout, which is precisely why the parameter sweep matters: the cube law's specific functional form is what distinguishes it from a flat environmental background.

A Sharp Question for the Decade

QGEM hardware is under construction. Nanoparticle ground-state cooling is demonstrated; two-particle entanglement remains the major gap, with timelines pointing to the early 2030s for first attempts at the full protocol. Whatever the experiment sees, the parameter sweep across $\Delta/d$ and arm orientation will produce a curve of only a few possible shapes.

If it is flat at $V = 1$, gravity passively mediates phase and the Wheeler-DeWitt constraint does not enforce the dressed-state structure on initial conditions. If it is a cube law with the right prefactors, the constrained influence functional is the correct way to integrate out gravity in the lab. If the magic-angle null is also present, the orientation-blindness of collapse models is ruled out as a class.

Three specific shapes. The experiment will choose between them.

This is Paper M of the Quantum-Geometric Duality series, deriving the mass-independent cube-law and magic-angle quintic-law predictions for the BMV/QGEM entanglement witness from the constrained Feynman-Vernon influence functional of Paper K.