1 Introduction

Diósi  and Penrose  proposed that a mass in a superposition of two locations loses coherence at the rate \[\begin{equation} \Gamma=\frac{C\,\EG}{\hbar},\qquad \EG=\frac{G}{2}\iint\frac{\Delta\rho(\mathbf{x})\,\Delta\rho(\mathbf{y})}{|\mathbf{x}-\mathbf{y}|}\,d^3x\,d^3y, \qquad \Delta\rho=\rho_L-\rho_R , \label{eq:dp-rate} \end{equation}\] with \(C\) a number of order one (\(C=1\) in Diósi’s normalization). The rate is first order in \(G\) and does not depend on any environment. Reviews of gravitational decoherence  survey the models that produce it and the experiments that test it.

Where can such a rate come from? Treating the graviton field as an environment gives master equations whose rates are also first order in \(G\), but they decohere superpositions of different energies . For a spatial superposition of one body the two branches carry the same mass, the monopole of the source difference vanishes, and the field channel gives no rate (Section 4). The same holds, in a Gaussian treatment, for any translation-invariant state that obeys the spectrum condition. What remains is a stationary medium that breaks the spectrum condition: a bath at finite temperature. Black-hole horizons decohere superpositions through finite-temperature quanta, and the effect is reproduced by modelling the hole, or ordinary matter, as a system at finite temperature ; those quanta are free and on shell and lack Diósi–Penrose-form noise, so they fall outside the class bounded here. Dissipative Diósi–Penrose models  postulate a bath with that noise explicitly.

This paper shows that, under conditions stated below, a finite-temperature bath cannot be the source. If the mass density couples to the bath through a Hamiltonian, the correlator that carries the noise also carries a dissipative response, and a body moving through the bath feels friction. Theorem 3.1 fixes that friction from two numbers, the noise strength \(C\) and the fluctuation–dissipation temperature \(T_{\mathrm{eff}}\). Binary pulsars measure friction very well. Provided the bath’s noise keeps the Diósi–Penrose form, is white over the drift band and is passive at the wavelengths a neutron star resolves, the orbital decay of the double pulsar PSR J0737\(-\)3039A/B requires \[\begin{equation} T_{\mathrm{eff}}/C>3.3\times10^{16}~\mathrm{K} \label{eq:headline} \end{equation}\] (Corollary 3.3). For dissipative Diósi–Penrose models this raises the lower bound on the collapse temperature by a factor \(3\times10^{29}\).

What survives is Diósi–Penrose noise in the infinite-temperature limit, that is, noise without back-action , together with baths whose kilometre-scale behaviour differs from their laboratory-scale behaviour (Section 3.4). Sections 5 and 6 give the allowed parameter region of the survivor and a prediction that measures its coefficient \(C\) independently of its smearing length.

Section 2 fixes conventions. Section 3 contains the drag theorem, the pulsar bound and its consequences. Section 4 states precisely, as known results, why the rate cannot come from quantized linearized gravity or from a translation-invariant positive-energy state. Section 7 lists the loopholes that remain open.

2 Setting, the rate from noise, and the scaling test

The measured quantity.

A rigid body of mass \(M\) is prepared in a superposition of two static positions a distance \(d\) apart. Both branches carry the same mass. The source is inertial. The two paths are recombined and the fringe visibility \(|z(t)|\), with \(z=2\rho_{LR}\), is recorded as a function of the time \(t\) spent apart. A rate is a constant \(\Gamma\) with \(|z(t)|\propto e^{-\Gamma t}\) at late times.

The rate from noise.

Let the mass density couple to a potential operator, \(H_{\rm int}=\int\rho\,\Phi\,d^3x\). The branches differ through the which-path variable \(B(t)=\int\Delta\rho(\mathbf{x})\,\Phi(\mathbf{x},t)\,d^3x\). For Gaussian fluctuations and pure dephasing, \[\begin{equation} -\ln|z(t)|=\frac{1}{2\pi\hbar^2}\int_{-\infty}^{\infty}\!d\omega\;S_B(\omega)\, \frac{1-\cos\omega t}{\omega^2},\qquad S_B(\omega)=\frac12\int\!d\tau\,e^{i\omega\tau}\bigl\langle\{\delta B(\tau),\delta B(0)\}\bigr\rangle . \label{eq:dephasing} \end{equation}\] Since \(\int(1-\cos\omega t)\,\omega^{-2}d\omega=\pi t\), the late-time slope is \[\begin{equation} \Gamma=\lim_{t\to\infty}\frac{-\ln|z(t)|}{t}=\frac{S_B(0)}{2\hbar^2}. \label{eq:rate-floor} \end{equation}\] A rate requires a zero-frequency floor. If \(S_B(\omega)=\order{\omega^2}\) as \(\omega\to0\), the integrand of Eq. \(\eqref{eq:dephasing}\) is bounded at small \(\omega\) and \(-\ln|z|\) saturates.

Diósi–Penrose noise.

Classical white potential noise with \[\begin{equation} \langle\delta\Phi(\mathbf{x},t)\,\delta\Phi(\mathbf{y},s)\rangle= \frac{C\hbar G}{|\mathbf{x}-\mathbf{y}|}\,\delta(t-s),\qquad S_\Phi(\mathbf{k},\omega)=\frac{4\pi C\hbar G}{k^2}\ \ \text{for all }\omega, \label{eq:dp-noise} \end{equation}\] gives \(S_B(0)=C\hbar G\iint\Delta\rho\,\Delta\rho\,|\mathbf{x}-\mathbf{y}|^{-1}=2C\hbar\EG\), so Eq. \(\eqref{eq:rate-floor}\) reproduces Eq. \(\eqref{eq:dp-rate}\). At \(C=1\) this is the noise behind Diósi’s master equation \(\dot{\hat\rho}=-(G/2\hbar)\iint[\hat M(\mathbf{x}),[\hat M(\mathbf{y}),\hat\rho]]\, |\mathbf{x}-\mathbf{y}|^{-1}\) ; the normalization used with Penrose’s prescription in Refs.  is larger by a factor \(8\pi\), as stated in Ref. , so it corresponds to \(C=8\pi\). We measure \(C\) in Diósi units throughout.

The energy.

\(\EG\) in Eq. \(\eqref{eq:dp-rate}\) contains the same-branch terms; they make \(\EG\geq0\). For uniform spheres of radius \(a\), \[\begin{equation} \EG=GM^2\Bigl(\frac{6}{5a}-\frac1d\Bigr)\qquad(d\geq2a), \label{eq:EG-sphere} \end{equation}\] which saturates at \((6/5)GM^2/a\) for \(d\gg a\) (Appendix 8). The point-mass expression \(GM^2/d\) is the magnitude of the cross term alone; it is not a limit of Eq. \(\eqref{eq:dp-rate}\) for any mass distribution.

The scaling test.

At fixed geometry, write \(F(M,t)=-\ln|z|\) and define \[\begin{equation} R_M=\frac{F(2M,t)}{F(M,t)},\qquad R_t=\frac{F(M,2t)}{F(M,t)} . \label{eq:ratios} \end{equation}\] A rate of the form \(\eqref{eq:dp-rate}\) has \(R_M=4\) and \(R_t=2\): the mass enters once through each coupling in the variance, \(G\) once through the noise strength, and the white spectrum makes the exponent linear in \(t\). We apply this test below.

3 The drag theorem and the double-pulsar bound

3.1 The drag theorem

Let \(H_{\rm int}=\int\rho\,\Phi\,d^3x\) with \(\Phi\) a bath operator. Per unit volume define the symmetrized noise and the dissipative response \[\begin{equation} S(\mathbf{k},\omega)=\frac12\int\!dt\,e^{i\omega t}\bigl\langle\{\Phi_{\mathbf{k}}(t),\Phi_{-\mathbf{k}}\}\bigr\rangle,\qquad \chi''(\mathbf{k},\omega)=\frac{1}{2\hbar}\int\!dt\,e^{i\omega t}\bigl\langle[\Phi_{\mathbf{k}}(t),\Phi_{-\mathbf{k}}]\bigr\rangle . \label{eq:S-chi} \end{equation}\] Fix a wavenumber \(k_{*}\) and a speed \(v_{\max}\), and assume:

1. Reciprocity: the operator \(\rho\) couples to is the one that carries the noise, through a Hamiltonian (or Schwinger–Keldysh) coupling, so \(S\) and \(\chi''\) come from the same correlator.

2. Stationarity: the bath state is stationary, homogeneous and isotropic in a rest frame, at scales \(\geq1/k_{*}\).

3. Diósi–Penrose noise: \(S(k,0)=4\pi C\hbar G/k^2\) for \(k\leq k_{*}\).

4. Whiteness over the drift band: \(\chi''(k,\omega)/\omega\) equals its \(\omega\to0\) value for \(|\omega|\leq k_{*}v_{\max}\), \(k\leq k_{*}\).

5. Finite positive temperature: \(T_{\mathrm{eff}}(k)\equiv S(k,0)/\bigl(2\kB\,\partial_\omega\chi''(k,0)\bigr)\in(0,\infty)\) for \(k\leq k_{*}\); a KMS state at temperature \(T\) has \(T_{\mathrm{eff}}\equiv T\).

6. Passivity: \(\omega\chi''(k,\omega)\geq0\) at every \(k\) and \(\omega\). This holds in every KMS state at \(T>0\); for other states it is a separate assumption.

7. Linear response: the bath responds linearly to the body’s density, so the friction is linear in the velocity for all \(|\mathbf{v}|\leq v_{\max}\).

3.2 The double-pulsar bound

PSR J0737\(-\)3039A/B has \(P_b=0.1022515593\) d and \(\dot P_b=-1.247920(78)\times10^{-12}\); its orbital decay agrees with the quadrupole formula to \(1.3\times10^{-4}\) at \(95\%\) confidence . Any additional decay therefore obeys \[\begin{equation} \Gamma_{\rm orb}<\gamma_{\max}=\frac{1.3\times10^{-4}\,|\dot P_b|}{3P_b}=6.1\times10^{-21}~\mathrm{s}^{-1}. \label{eq:gammamax} \end{equation}\] Both bodies are neutron stars. Model each as a uniform sphere of \(M=1.2\,\Msun\) and \(R_{\mathrm{NS}}=14\) km, light and large and therefore of low density, \(\bar\rho=2.1\times10^{17}\) kg m\(^{-3}\). Each star moves about the centre of mass at \(300\)–\(325\) km s\(^{-1}\), so \(v_{\max}=10^3\) km s\(^{-1}\) also covers a centre-of-mass velocity of several hundred km s\(^{-1}\) relative to the bath. With \(k_{*}=10/R_{\mathrm{NS}}\), (W) and (L) are needed only up to \(k_{*}v_{\max}=714~\mathrm{s}^{-1}\) at kilometre wavelengths.

The bound is an order-of-magnitude statement. Strong-field corrections (compactness \(\approx0.2\)) are of order one; a realistic stellar profile raises \(\rho_{k_{*}}\) (Remark 3.1) and strengthens it. A drag only adds decay, so a one-sided limit from the measured ratio \(\dot P_b^{\rm GW}/\dot P_{b,\rm GR}^{\rm GW} =0.999963(63)\)  is about twice as strong; we quote the two-sided value. The two-sided band also excludes an active bath that pushes bodies: \(|T_{\mathrm{eff}}|/C>3.3\times10^{16}\) K. Free fields at \(T>0\) are not constrained: their spectral weight sits on \(|\omega|=ck\), so \(S(k,0)=0\) and they fail (D). The method, binary-pulsar timing as a probe of friction from a medium, was used by Pani  for dark matter; we found no prior application to decoherence baths.

3.3 Consequences for known models

Dissipative Diósi–Penrose.

Bahrami, Smirne and Bassi introduced dissipation into the DP model to tame its energy increase, and found that it does not reconcile the model with data unless the cutoff is large . The linear-friction model of Ref.  adds friction at a collapse temperature \(T_\beta\) to the kernel \(D_k=4\pi\hbar G\,e^{-R_0^2k^2}/k^2\). Its centre-of-mass damping \(\Gamma=\hbar^2\eta/(2M\kB T_\beta)\), with \(\eta\) the \(k^2\)-weighted DP diffusion , is Eq. \(\eqref{eq:drag}\) at \(C=1\), \(T_{\mathrm{eff}}=T_\beta\). Corollary 3.3 gives \(T_\beta>3.3\times10^{16}\) K for \(R_0\ll R_{\mathrm{NS}}\). Levitated-mechanics experiments exclude \(T_\beta<10^{-13}\) K for \(R_0<1~\mu\)m ; the pulsar bound is stronger by \(3\times10^{29}\). It excludes the parameter point \(R_0=10^{-15}\) m, \(T_\beta\approx3\) K (the temperature of the cosmic microwave background) discussed in Ref.  by \(10^{16}\). At \(T_\beta>3\times10^{16}\) K the dissipative model is plain DP for every laboratory observable, so the X-ray bounds of Section 5 apply to it unchanged.

Classical-channel and classical–quantum gravity.

The models of Kafri, Taylor and Milburn  and of Tilloy and Diósi  treat the Newtonian field as classical and sourced by a measurement record. Their dissipative generalizations thermalize to a finite effective temperature  and fall under Theorem 3.1 through the Einstein relation; with Diósi–Penrose noise they require \(T_{\mathrm{eff}}>3.3\times10^{16}\,C\) K. Without friction the response of the potential to the signal is instantaneous and real, \(\chi''=0\), and there is no drag. The same holds for the Newtonian limit of classical–quantum gravity, where the potential is classical and diffusive and friction terms are possible but not included . In that framework Oppenheim et al. prove a trade-off between the decoherence of the quantum system and the diffusion it induces in the classical field: long coherence times require strong diffusion relative to the coupling . It is the classical–quantum analogue of Remark 3.1(ii). Both relations tie the noise that decoheres matter to a second, independently measurable coefficient: the trade-off to the diffusion of the metric, the Einstein relation to the friction on matter through \(T_{\mathrm{eff}}\). They constrain different things and are compatible. A frictionless classical–quantum model sits at \(T_{\mathrm{eff}}=\infty\), where the drag vanishes; the trade-off then bounds the diffusion of the potential, not a temperature.

Finite-temperature matter and horizons.

A system at finite temperature decoheres a nearby superposition, and the decoherence by a black-hole horizon is reproduced by modelling the hole as such a system ; locally it is carried by very-low-frequency quanta in the laboratory . These quanta are free and on shell, so they fail (D), and horizon decoherence is not in the class Theorem 3.1 bounds. The theorem applies to a medium that fills space with Diósi–Penrose-form noise at kilometre scales; such a medium drags neutron stars, and the pulsar bound excludes it unless its temperature exceeds Eq. \(\eqref{eq:pulsar-bound}\).

3.4 The gap: noise without back-action

Corollary 3.3 leaves open the limit \(T_{\mathrm{eff}}\to\infty\) at fixed noise. In that limit \(\chi''\to0\) while \(S\) stays finite, so the drag vanishes. Operationally it is noise without back-action: a classical potential field acting on matter, whose reduced dynamics is the random-unitary DP master equation with Hermitian Lindblad operators, which is how Diósi wrote the model . For a bath with a finite-dimensional or type-I algebra the limit is its tracial state; a quantum field algebra of type III has no tracial state, and only the limit, or a classical noise field, realizes it. Classical stochastic gravitational fields acting on matter belong to this class . The line between reciprocal and non-reciprocal noise is therefore not the sharp one; the sharp parameter is \(T_{\mathrm{eff}}\). Non-reciprocal DP is the \(T_{\mathrm{eff}}=\infty\) end of the dissipative family, and the pulsars allow only \(|T_{\mathrm{eff}}|/C>3.3\times10^{16}\) K around it. This end, with every bath above that temperature, is escape (G1).

It is not the only escape. An ordinary-temperature bath can still hide where its kilometre-scale behaviour differs from its laboratory-scale behaviour:

1. an infrared cut: the kernel departs from \(1/k^2\) below some \(k_{\rm IR}\gg1/R_{\mathrm{NS}}\). The drag then comes only from the star’s density structure above \(k_{\rm IR}\). For a sharp-edged uniform sphere that fraction is \(0.955/(k_{\rm IR}R_{\mathrm{NS}})\), \(6.8\times10^{-7}\) at \(k_{\rm IR}=100\) m\(^{-1}\); for a real star it depends on the crust and surface profile, which we have not modelled;

2. a slow bath: a correlation time longer than \(R_{\mathrm{NS}}/(10v_{\max})=1.4\) ms at kilometre wavelengths violates (W) and suppresses the drag, while the bath can still be white at laboratory \(k\);

3. a scale-dependent temperature: \(T_{\mathrm{eff}}(k)\) finite at laboratory \(k\) but above \(3.3\times10^{16}\,C\) K at \(k\lesssim10/R_{\mathrm{NS}}\);

4. active modes: a non-KMS state with \(\omega\chi''<0\) at \(k>k_{*}\) violates (P) and can cancel the drag from \(k<k_{*}\).

A bath that co-moves with each body (non-linear entrainment, violating (L) and (S) locally) also escapes, at the price of having no rest frame.

4 Why the rate cannot come from quantized linearized gravity or a translation-invariant positive-energy state

The statements in this section are known results or direct consequences of them. We state them precisely, with their hypotheses, because the drag theorem addresses what they leave open. Throughout, the source is a static, inertial rigid body with equal branch masses, and it enters as an external c-number density.

4.1 The field channel at linearized order

In a state that is KMS at inverse temperature \(\beta\), the fluctuation–dissipation theorem gives \(S_B(\omega)=\hbar J(\omega)\coth(\beta\hbar\omega/2)\), with \(J\) the dissipative spectral density of \(B\), so \[\begin{equation} S_B(0)=\frac{2}{\beta}\lim_{\omega\to0}\frac{J(\omega)}{\omega}: \label{eq:fdt-floor} \end{equation}\] a floor needs an Ohmic response. The vacuum restricted to a Rindler wedge , or for a conformal field to a causal diamond , is KMS for its modular flow; describing the vacuum in modular time adds no excitations. A free massless field obeys \(\Box\hat\phi=0\), so in a translation-invariant state its two-point function is supported on \(\omega=\pm c|\mathbf{k}|\), and weight at \(\omega\to0\) comes only from \(k\to0\). For two equal spheres with form factor \(\tilde\rho(k)\), \[\begin{equation} \bigl\langle|\Delta\tilde\rho_{\mathbf{k}}|^2\bigr\rangle =2\,\tilde\rho(k)^2\Bigl(1-\frac{\sin kd}{kd}\Bigr) \;\xrightarrow{\;kd\to0\;}\;\tfrac13 M^2k^2d^2 . \label{eq:drho} \end{equation}\] For the most favourable model, a propagating field normalized so that its reorganization energy \(\pi^{-1}\int_0^\infty J\,\omega^{-1}d\omega\) equals \(\EG\) (Appendix 8), \[\begin{equation} J(\omega)=\frac{G\omega}{c}\bigl\langle|\Delta\tilde\rho_{\omega/c}|^2\bigr\rangle \;\simeq\;\frac{GM^2d^2}{3c^3}\,\omega^3\qquad(\omega d/c\ll1), \label{eq:J} \end{equation}\] which is super-Ohmic. Only a difference of branch energies makes \(J\) Ohmic. That is the channel of the graviton master equations , whose rates are first order in \(G\) for superpositions of different energies and vanish for the equal-mass spatial superposition considered here. In linearized general relativity the static density has no propagating scalar partner: its potential is fixed by the constraint, an instantaneous real kernel that dissipates nothing, and it fluctuates only through the matter that sources it, at order \(G^2\).

These results agree with direct calculations. In the graviton vacuum, the decoherence functional of a spatial superposition that is opened and closed again approaches a constant set by \((mv/M_{\rm P})^2\) at late times rather than growing linearly . The which-path information radiated by a superposition is set by the time taken to recombine it, not by the time it is held apart, \(N\sim Q^2/T_{\rm rec}^4\) in Planck units for a mass quadrupole \(Q\) , and in the inertial Minkowski vacuum it does not grow with the holding time .

4.2 The clock channel

A clock at \(\mathbf{x}_{\rm obs}\) whose rate depends on the branch through gravitational redshift , as for the observer adjoined in the crossed-product construction , gives a structureless source (\(E=Mc^2\)) a coherence of constant modulus that rotates at \[\begin{equation} \omega_{\rm rel}=\frac{M\,|\Phi_L-\Phi_R|(\mathbf{x}_{\rm obs})}{\hbar} =\frac{GM^2d}{\hbar\,(r^2-d^2/4)}, \label{eq:omega-rel} \end{equation}\] on the axis at distance \(r\) from the midpoint; at \(r=d\) this is \((4/3)\,GM^2/(\hbar d)\). It shifts the fringes and leaves their contrast intact. For any number of clocks with Hamiltonians \(H_k\) at \(\mathbf{x}_k\) in any joint state, the branch-relative evolution is generated by \(\sum_kH_k\,\Delta\Phi(\mathbf{x}_k)/c^2\) with \(\Delta\Phi=GM\varphi(\mathbf{x})\), \(\varphi\) independent of \(G\) and \(M\), so \[\begin{equation} z(t)=\Bigl\langle\exp\Bigl(\frac{i\,GMt}{\hbar c^2}\sum_k H_k\,\varphi(\mathbf{x}_k)\Bigr)\Bigr\rangle . \label{eq:probe} \end{equation}\]

Finite clock-energy variances give \(F\propto G^2M^2t^2\), the time-dilation dephasing of interfering clocks . Cauchy-like energy distributions give \(F\propto GMt\), exponential at intermediate times but linear in \(M\); for a spectrum bounded below the decay cannot remain exponential at late times . A factor \(M^2\) appears only if the probe is a second massive body, the setting of gravitationally mediated entanglement .

4.3 Translation-invariant states with positive energy

The obstruction in Proposition 4.1 does not need linearity, within a Gaussian treatment. Let the potential \(\Phi\) (local or gravitationally dressed) and the bath state \(\Omega\) satisfy:

1. \(\Omega\) is invariant under spacetime translations \(U(a)=e^{iP\cdot a}\);

2. the joint spectrum of \(P^\mu\) lies in the closed forward cone \(\bar V_+\);

3. \(\Phi(x)=U(x)\Phi(0)U(x)^\dagger\), and the connected two-point spectral measure \(\mu\) of \(\Phi\) is locally finite;

4. the connected part has no atom at \(p=0\);

5. \(|\Delta\tilde\rho(\mathbf{k})|^2\leq M^2k^2d^2\), which holds for a rigid body since \(|\Delta\tilde\rho|^2=4|\tilde\rho|^2\sin^2(\mathbf{k}\cdot\mathbf{d}/2)\) and \(|\tilde\rho|\leq M\).

The content is the spectral support of Wightman functions , applied to which-path noise. (B) is an assumption about the bath state; in asymptotically flat gravity one would justify it by positivity of the total energy, which we do not derive here. The proposition does not reach higher cumulants or a quantized source. It covers the modular state of a diamond, because that state is the vacuum. Within the Gaussian treatment, a white floor therefore needs a state that breaks (A) or (B), for example one with a rest frame. Relatedly, Diósi identifies a hidden defect in a proposed Lorentz-invariant GKLS master equation and argues that no such Markovian equation should exist . The escapes are the broken assumptions:

• (B) fails for every thermal, stationary non-equilibrium or infinite-temperature state: the generator of time translations in its GNS representation has spectrum \(\mathbb{R}\). This is the class of Section 3. Relativistic collapse models escape the same way: a mediating field at each spacetime point , a scalar field whose energy spectrum takes all real values , or classical noise with a Lorentz-invariant correlation , which as a classical field need not obey (B).

• Killing horizons replace the positive generator of laboratory time by a boost generator with spectrum \(\mathbb{R}\). They decohere at rates set by horizon data that vanish for an inertial laboratory without a horizon .

• (C) fails only for an infrared accumulation of spectral weight at the tip of the cone; for a static displaced source we expect none, which is argued, not shown.

• (D) fails in the source’s own Gauss-law sector, where boosts are spontaneously broken. We conjecture that the corrections are of order \(GM\) relative to the leading term, \(G^2M^3\) and higher, but have not shown it.

• Non-unitary dynamics (Penrose-type objective reduction ) lies outside the bath framework.

5 The survivor and its allowed region

There is no friction and no detailed-balance factor, so every bound on DP noise applies directly. Two parameters remain, \(C\) and \(R_0\). The analysis does not fix \(C\); it is a free parameter of order one.

Why a density smearing.

The smearing must give the momentum diffusion a finite \(k^2\) moment. Smearing the full density by \(e^{-k^2R_0^2/2}\) per factor gives the kernel \((4\pi/k^2)\,e^{-k^2R_0^2}\) and, for any constituent of mass \(m\), nucleus or electron, \[\begin{equation} \frac{d\langle p^2\rangle}{dt}=4\pi C\hbar G\,m^2\!\int\!\frac{d^3k}{(2\pi)^3}\,e^{-k^2R_0^2} =\frac{C\hbar G\,m^2}{2\sqrt\pi\,R_0^3},\qquad \frac{P}{M}=\frac{1}{2m^2}\frac{d\langle p^2\rangle}{dt}=\frac{CG\hbar}{4\sqrt\pi\,R_0^3}, \label{eq:heating} \end{equation}\] a heating rate per unit mass independent of \(m\). A cutoff of the kernel instead, \((1-e^{-r/R_0})/r\) with Fourier transform \(4\pi/[k^2(1+k^2R_0^2)]\), falls only as \(k^{-4}\); for constituents of size \(r_N\) its \(k^2\) moment is \(2R_0/r_N\) times the Gaussian one and diverges for point electrons. The same divergence appears if only nuclei are smeared. We therefore smear the full mass density with a Gaussian, the form used in the published bounds.

Converting the bounds.

Spontaneous radiation and heating are both proportional to \(C/R_0^3\). A lower bound \(R_{\rm pub}\) published at normalization \(C_{\rm pub}\) becomes \[\begin{equation} R_0>R_{\rm pub}\,(C/C_{\rm pub})^{1/3}. \label{eq:convert} \end{equation}\] Table 1 lists the bounds at \(C=1\). The normalization of each was checked against an independent statement: Eq. \(\eqref{eq:heating}\) reproduces the heating figures of Ref. , and the XENONnT statement that it improves on MAJORANA by a factor of five  holds only for the corrected MAJORANA value  read at \(C=8\pi\), where the ratio is \(5.2\).

At the edge of the region the heating is \(P/M=G\hbar/[4\sqrt\pi\,(4.5~\text{\AA})^3]=1.1\times10^{-17}\) W kg\(^{-1}\), independent of \(C\) and six orders below the cryostat floor. The leading power law \(R_0^{-3}\) is used down to atomic scales; refined atomic emission rates  would move Eq. \(\eqref{eq:region}\) by the cube root of the change in rate. Two candidate upper bounds do not apply. Inverse-square-law tests constrain a smearing of the Newtonian force; here the mean field is that of general relativity and only the noise is smeared. A requirement that the noise also make macroscopic bodies classical within a fixed time  would cap \(R_0\); the survivor is a decoherence clause, not a solution of the measurement problem, and we do not impose it. Figure 1 shows the region.

6 The sharp prediction: a measurement of \(C\)

Every non-interferometric bound measures \(C/R_0^3\) and cannot separate \(C\) from the free \(R_0\). An interferometric rate can, in one regime, once the other decoherence channels are accounted for.

Physical content. Once the spheres no longer overlap, the rate is set by the self-energy of each branch’s density, \(\frac65GM^2/a\), minus the cross term \(GM^2/d\); the distance enters only through that cross term. The separation dependence is \(\EG(d)/\EG(\infty)=1-5a/6d\): \(0.583\) at \(d=2a\), within \(10\%\) of saturation only for \(d\geq8.33\,a\). Below \(d=2a\) the overlapping-sphere formula of Appendix 8 applies and tends to \(GM^2d^2/(2a^3)\). In the allowed region \(\eqref{eq:region}\) \(R_0\) exceeds the interatomic spacing of about \(2.5\) Å for \(C>0.17\), so the continuum density applies; granularity then adds a relative correction of order \(\bar m a/(MR_0)\), with \(\bar m\) the mean nuclear mass, which is below \(10^{-11}\) for the benchmark below.

Benchmarks.

For \(1~\mu\)g of silica (\(\rho=2200\) kg m\(^{-3}\), \(a=47.7~\mu\)m) at \(d=1\) mm, \(\EG=24.2\,GM^2/d\) and \[\begin{equation} \tau=\frac{\hbar}{C\EG}=\frac{65.4~\mathrm{ps}}{C}, \label{eq:tau-silica} \end{equation}\] rising to \(108~\mathrm{ps}/C\) at \(d=2a\) and falling to \(62.8~\mathrm{ps}/C\) as \(d\to\infty\). The smearing correction is \((5/2)(R_0/a)^2=2\times10^{-10}\) at \(R_0=4.5\) Å. A space platform of the MAQRO class , with free evolution of order \(100\) s, reaches the saturated regime: a sphere of \(a=150\) nm and \(\rho=5000\) kg m\(^{-3}\) has \(\tau=39.5~\mathrm{s}/C\) for \(d\gg a\) and \(67.8~\mathrm{s}/C\) at \(d=2a\). A silica sphere needs \(a\geq173~\mathrm{nm}\;C^{-1/5}\) for \(\Gamma\times100~\mathrm{s}\geq1\) at \(d\gg a\). Figure 2 shows the rate against radius.

What a measurement determines.

In the saturated regime \(C=\hbar\Gamma_{\rm grav}/\EG\), with \(\EG\) known from \(M\), \(a\) and \(d\), where \(\Gamma_{\rm grav}\) is the measured visibility decay rate after the known decoherence channels (residual-gas collisions, black-body emission, scattering and absorption, trap and recoil noise) have been subtracted. With those subtracted, one measurement determines \(C\). A rate below the prediction means either a smaller \(C\) or \(R_0\gtrsim a\); the two are separated by the radius dependence, \(\Gamma\propto a^5\rho^2\) at fixed \(C\) in the saturated regime against a suppression once \(R_0\) approaches \(a\). Distinguishing two values of \(C\) needs \(\Gamma_{\rm grav}\) to the fractional precision of their difference. A residual rate of second order in \(G\), or none, excludes the survivor at the tested \((C,R_0)\).

7 Discussion

7.1 What remains open

• The survivor (Definition 5.1): noise without back-action, or a bath at \(|T_{\mathrm{eff}}|/C>3.3\times10^{16}\) K (escape G1). Neither quantized linearized gravity, nor a translation-invariant positive-energy state in a Gaussian treatment, nor a finite-temperature bath obeying the hypotheses of Theorem 3.1 at kilometre scales supplies it.

• Baths that differ at kilometre scales (G2–G5): an infrared cut of the kernel, a correlation time longer than \(1.4\) ms, a temperature that rises toward long wavelengths, or active modes above \(k_{*}\). The infrared cut is bounded only through the high-\(k\) structure of neutron stars, which we have not modelled; closing these escapes needs that structure, or bodies resolved at other scales.

• Horizons: real rates set by horizon data , absent for an inertial laboratory without a horizon.

• Beyond the Gaussian treatment: higher cumulants, a quantized source, non-tempered infrared sectors and the Gauss-law sector of the source (Section 4) are not excluded by any argument here.

• Objective reduction in Penrose’s sense : outside the bath framework.

Corollary 3.3 rests on a uniform-sphere neutron-star model and on treating the star as a rigid body with linear friction up to \(10^3\) km s\(^{-1}\); both are conservative within the condition of Remark 3.1(iv), and strong-field corrections are of order one. We have found no earlier bound on decoherence-bath friction from orbital decay, but searched only arXiv and the general literature.

7.2 Consequences

1. Dissipative Diósi–Penrose models need \(T_\beta>3.3\times10^{16}\) K, a bound \(3\times10^{29}\) above the laboratory one; at such temperatures they coincide with the non-dissipative model.

2. A rate of first order in \(G\) observed for an isolated, inertial, equal-mass spatial superposition would not confirm a derivation from quantized gravity or from a thermal medium. By Corollary 3.3 and Section 4 it would signal noise without back-action, one of the escapes G2–G5, a horizon, physics beyond the Gaussian treatment (higher cumulants, a quantized source, non-tempered infrared or Gauss-law sectors), or non-unitary reduction.

3. In the saturated regime of Proposition 6.1, once the other decoherence channels are subtracted, one visibility measurement gives \(C=\hbar\Gamma_{\rm grav}/\EG\). For \(1~\mu\)g of silica the prediction is \(\tau=65\) ps\(/C\); a MAQRO-class platform with \(100\) s of free evolution reaches the regime with spheres of \(150\)–\(175\) nm radius.

4. Radiation and heating searches can raise the floor \(\eqref{eq:region}\) on \(R_0\) but cannot exclude the survivor, since no noise observable bounds \(R_0\) from above.

7.3 Emergent-gravity programmes

Programmes in which the Einstein equation is an equilibrium condition of diamond entanglement , such as the quantum-geometric correspondence that motivated this work, sometimes look to the coarse-grained geometry for a decoherence mechanism. The results above constrain that hope. If the geometry is described by the vacuum, Proposition 4.3 gives no Gaussian white floor; if it is described as a medium at finite temperature, the medium drags and Corollary 3.3 requires \(T_{\mathrm{eff}}/C>3.3\times10^{16}\) K. A rate linear in \(G\) in such a programme is therefore an additional postulate of the Diósi type, with \(C\) not fixed, unless the medium uses one of the escapes G2–G5.

8 The energy \(\EG\) of two uniform spheres

With \(\int d^3k\,(2\pi)^{-3}\,4\pi k^{-2}f(\mathbf{k})=(2/\pi)\int_0^\infty dk\,\langle f\rangle\) for any angle average \(\langle f\rangle\), Eq. \(\eqref{eq:dp-rate}\) reads \[\begin{equation} \EG=\frac{G}{\pi}\int_0^\infty dk\,\bigl\langle|\Delta\tilde\rho_{\mathbf{k}}|^2\bigr\rangle , \label{eq:EG-fourier} \end{equation}\] with the angle average \(\eqref{eq:drho}\) and the form factor of a uniform sphere, \(\tilde\rho(k)=3M(\sin ka-ka\cos ka)/(ka)^3\). The spectral density \(\eqref{eq:J}\) has reorganization energy \(\pi^{-1}\int_0^\infty J\,\omega^{-1}d\omega\) equal to \(\eqref{eq:EG-fourier}\), which fixes its normalization; its small-\(\omega\) expansion gives the \(\omega^3\) law. In position space, with \(\iint\rho\rho\,|\mathbf{x}-\mathbf{y}|^{-1} =(6/5)M^2/a\) for each ball (twice the self-energy \(\frac35GM^2/a\) divided by \(G\)) and \(\iint\rho_L\rho_R\,|\mathbf{x}-\mathbf{y}|^{-1}=M^2/d\) for disjoint balls (Newton’s shell theorem), \[\begin{equation} \EG=\frac G2\Bigl[2\cdot\frac{6M^2}{5a}-\frac{2M^2}{d}\Bigr] =GM^2\Bigl(\frac{6}{5a}-\frac1d\Bigr)\qquad(d\geq2a). \end{equation}\] For overlapping balls, \(s=d/a<2\), the cross term follows from the overlap integral of two uniform balls, and \[\begin{equation} \EG=\frac{GM^2}{a}\Bigl(\frac{s^2}{2}-\frac{3s^3}{16}+\frac{s^5}{160}\Bigr) \;\xrightarrow{\;s\to0\;}\;\frac{GM^2d^2}{2a^3}. \label{eq:EG-overlap} \end{equation}\] Both branches agree at \(s=2\), where \(\EG=0.7\,GM^2/a\). Quadrature of \(\eqref{eq:EG-fourier}\) reproduces both forms to better than one part in \(10^6\) at \(s=0.3\), \(1.2\), \(3\) and \(10\).

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