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 the mass density smeared over a length \(R_0\) and \(C\) a number of order one (\(C=1\) in Diósi’s normalization). The rate is first order in \(G\) and needs no environment. It is within reach of a free-fall experiment: a sphere of radius \(150\) nm and density \(5000\) kg m\(^{-3}\) held in a superposition of positions \(1~\mu\)m apart for \(100\) s accumulates \(\Gamma t=2.2\,C\), independently of \(R_0\) for \(R_0\ll150\) nm . Suppose the fringes fade at this rate once every other channel is subtracted. What has been learned about the symmetry of spacetime?

A companion paper  answered part of the question. A bath that supplies the rate through a Hamiltonian coupling drags moving bodies, and the orbital decay of the double pulsar PSR J0737\(-\)3039A/B bounds its fluctuation–dissipation temperature at kilometre scales by \(T_{\mathrm{eff}}/C>3.3\times10^{16}\) K; noise without back-action is the member of that class without a temperature parameter. This paper asks how the noise can act relativistically. Under stated hypotheses on matter and on the noise, among them a finite rate of vacuum excitation, the law by which the noise acts on matter cannot be Lorentz invariant; for a noise without feedback whose spectrum reaches timelike momenta its Lorentz stabilizer is compact, so it has a rest frame. A noise with timelike spectral support excites the vacuum without bound at second order; a noise with purely spacelike support does so at fourth order for time-ordered dynamics, and X- and \(\gamma\)-ray data exclude it independently at \(C>2\times10^{-8}\). A law that reads its time direction from the state instead of carrying it is a dynamical aether (Theorem 2.2).

The obstruction to covariant collapse dynamics is old. Ghirardi, Grassi and Pearle found that relativistic dynamical reduction excites the vacuum ; Pearle avoided the excitation at lowest order with a noise of spacelike, tachyonic support  and later with a quasi-relativistic model whose collapse operator annihilates the vacuum and whose time ordering is preferred . Myrvold showed that a Lorentz-invariant Markovian collapse dynamics with a stable vacuum needs non-standard degrees of freedom . Jones, Gasbarri and Bassi showed that a mass-coupled covariant collapse model cannot avoid both a divergent energy rate and superluminal signalling . Diósi formulated continuous measurement of quantum fields relativistically  and for retarded observables , examined relativistic Gorini–Kossakowski–Lindblad–Sudarshan equations , and found that the local, Lorentz-invariant kernels relativity demands of a hybrid classical–quantum dynamics heat at an infinite rate . On the other side, Bedingham built a covariant, finite collapse model with a pointer field , and Tumulka a frame-free relativistic flash model for non-interacting particles . For gravity, Kafri, Taylor and Milburn modelled Newtonian interaction as a classical channel ; Tilloy and Diósi sourced a classical Newtonian field by a continuous record of the mass density and recovered the Diósi–Penrose rate , and their principle of least decoherence fixes \(C=1\) . Oppenheim and collaborators built a classical–quantum theory of gravity , derived its decoherence–diffusion trade-off , obtained the Diósi–Penrose kernel in its Newtonian limit , and wrote covariant path integrals . Gisin showed that dynamics nonlinear over mixtures can signal .

What is new here is limited to three items. (1) Theorem 4.4 links a causal, exactly Newtonian classical response and complete positivity to an infinite vacuum excitation in every covariant record-and-feedback theory. It uses neither the Markov property nor a non-radiation hypothesis; the boost-orbit divergence it ends in is the standard one . (2) The spacelike-support door is closed twice: by theory at fourth order for time-ordered dynamics, applying Pearle’s higher-order re-excitation  to the Diósi–Penrose normalization, and by data, because covariance ties the noise near the light cone to the noise that fixes the rate, so its spontaneous emission cannot be cut off (Section 5). (3) A refit of the XENONnT electronic-recoil spectrum  with the cut emission spectrum of a frame theory (Appendix 10). Everything else is assembled from the cited work.

Section 2 fixes the setting and states the theorem. Section 3 recalls arm (i). Sections 4–6 prove arm (ii): state-independent laws, the spacelike door, and the state-dependent alternative. Section 7 gives the observable signature of a frame, the X-ray cutoff.

2 Setting and statement

2.1 What a rate measures

A rigid body is prepared in two branches of equal mass at static positions; the source is isolated and inertial. A rate is a constant \(\Gamma\) with fringe visibility \(|z(t)|\propto e^{-\Gamma t}\) at late times. Let \(X\) denote whatever decoheres the body: a classical stochastic potential, a quantum bath, or a classical field driven by a measurement record. \(X\) acts on matter through \(H_{\rm int}=\int\rho\,\Phi\,d^3x\), and \[\begin{equation} S(\mathbf{k},\omega)=\frac12\int\!dt\,e^{i\omega t}\, \bigl\langle\{\delta\Phi_{\mathbf{k}}(t),\delta\Phi_{-\mathbf{k}}(0)\}\bigr\rangle \label{eq:S} \end{equation}\] is its symmetrized spectrum per unit volume. Signature is \((-+++)\), \(q=(\omega,\mathbf{k})\), \(q^2=k^2-\omega^2/c^2\); \(q\) is spacelike for \(q^2>0\). The mass density is smeared by a normalized Gaussian of width \(R_0\), so that \[\begin{equation} \EG=\frac{G}{2}\int\!\frac{d^3k}{(2\pi)^3}\,\frac{4\pi}{k^2}\,e^{-k^2R_0^2}\,|\Delta\rho_{\mathbf{k}}|^2 . \label{eq:EG-k} \end{equation}\]

Gaussianity is not used. A compound-Poisson noise of phase kicks has the same quadratic coefficient and leaves the Diósi–Penrose class only for large kicks, where its rate saturates at the jump rate and stops scaling with \(\EG\). The Markov property is not used either. A coloured noise gives a rate for \(t\gg\tau_c\); for \(t\ll\tau_c\) the exponent is quadratic in \(t\) and there is no rate. What is used is stationarity, integrable correlations, and \(\Gamma\tau_c\ll1\).

2.2 Hypotheses

1. Rate. For an isolated, inertial, equal-mass two-branch superposition, the late-time decoherence exponent at second order in \(\Delta\rho\) is \(C\EG/\hbar\) with \(C>0\), over at least one decade of laboratory wavenumbers, with \(\Gamma\tau_c\ll1\).

2. Standard matter. Matter is a local relativistic quantum field theory with a unique Poincaré-invariant vacuum \(\Omega\) obeying the spectrum condition. \(X\) couples to a local density \(O(x)\), a component or contraction of \(T^{\mu\nu}\), whose vacuum Källén–Lehmann weight \(\sigma_O(s)\) is positive for almost every \(s>0\), and whose time-ordered two-point amplitude \(\langle n|T\,\tilde O(q_1)\tilde O(q_2)|\Omega\rangle\) to two-particle states is nonzero on a set of spacelike \((q_1,q_2)\) of positive measure.

3. Stable vacuum under the noise. At each order in \(G\) at which it is computed, the probability per unit volume and time that the noise excites the matter vacuum is finite.

4. No further degrees of freedom. Either (a) \(X\) is a classical stochastic field \(\phi\), or a bath in a tracial state, acting through \(\int O\phi\,d^4x\) with time-ordered evolution; or (b) \(X\) is a classical field \(h\) that responds linearly to a continuous record of \(O\) and acts back through \(\int hO\,d^4x\). Nothing else carries energy, momentum or excitations.

5. State-independence. The law of \(X\), meaning its spectrum, its kernels and its record, does not depend on the quantum state of matter.

6. Gaussian record, Newtonian response (case (b)). The record noise and the induced influence functional are Gaussian; the decoherence kernel \(\Lambda(q)\) and the response kernel \(\tilde D(q)\) are translation-invariant; \(\tilde D\) is retarded and exactly Newtonian in the infrared, \(q^2\tilde D(q)\to2\pi G/\hbar\) as \(q^2\to0\) in the normalization of Section 4.3.

For the traceless components of \(T^{\mu\nu}\) the weight \(\sigma_O\) near \(s\to0\) comes from photon (and graviton) pairs. For \(T^\mu{}_\mu\) it comes, above the hadronic and \(e^+e^-\) thresholds, from the trace anomaly; below \(2m_ec^2\) there are no light charged particles and gluons are confined, and the two-photon weight comes from the Euler–Heisenberg operators, suppressed by powers of \(s/m_e^2c^2\) but positive. The amplitude condition in (Q) holds at tree level whenever \(O\) is bilinear in a massive field, as for \(:\!\psi^2\!:\) or \(m\bar\psi\psi\) (Section 5).

2.3 The theorem

Arms (i) and (ii) are independent: neither uses the other, and neither uses the Newtonian mean field except through (G). Arm (i) is Section 3. In (ii), case (a) is Lemma 4.1 for a noise with timelike spectral support and Lemma 5.2 for a Poincaré-invariant one with purely spacelike support; whether a spacelike-only noise invariant under a smaller non-compact group can satisfy (R) is open (Section 5); case (b) is Theorem 4.4. (iii) is Proposition 5.4; it does not use time ordering, so it also covers dynamics that drop it. Section 6 treats laws that violate (L). Case (b) is weaker than case (a): Theorem 4.4 excludes Lorentz-covariant kernels, not kernels invariant under a non-compact subgroup such as \(SO(2,1)\) (Remark 4.2).

3 Arm (i): back-action at kilometre scales

Ref.  proves arm (i); we restate it and add one reading. If the noise of Eq. \(\eqref{eq:dp-spectrum}\) is carried by a bath operator coupled through a Hamiltonian, the same correlator carries the dissipative response \(\chi''(\mathbf{k},\omega)\), and a body moving through the bath is dragged. Under stationarity, whiteness over the drift band, passivity and linear response at wavenumbers \(k\leq k_{*}=10/R_{\rm NS}\), the friction rate is \(\gamma\geq(2\pi/3)(C\hbar G/\kB T_{\mathrm{eff}})\rho_{k_{*}}\), and the orbital decay of PSR J0737\(-\)3039A/B gives \[\begin{equation} |T_{\mathrm{eff}}|/C>T_*=3.3\times10^{16}~\mathrm{K} \label{eq:pulsar} \end{equation}\] (Theorem 3.1 and Corollary 3.3 of Ref. ). The escapes are a kernel cut below kilometre wavenumbers, a bath slower than \(1.4\) ms at those scales, a temperature that rises toward long wavelengths, and active modes; each changes the bath between the laboratory and the neutron star.

The bound is a statement about the dissipative slope alone. The member of the class without a temperature parameter is \(T_{\mathrm{eff}}=\infty\), \(\chi''\equiv0\): a classical noise, a bath in a tracial state, or a classical field sourced instantaneously by a record. These are the cases (a) and (b) of hypothesis (N).

4 Arm (ii): Lorentz invariance is broken

4.1 Noise without feedback

The vector \(u\) is unique when the law is isotropic about it; an anisotropic law with a smaller compact stabilizer still distinguishes frames but may fix several timelike directions. The mechanism of Lemma 4.1, a positive rate density integrated over a non-compact orbit, is the standard obstruction to relativistic collapse ; no novelty is claimed for it. No resummation makes a positive, invariant excitation density of nonzero-momentum states finite. The divergence is the price of the symmetry, not of the floor \(\eqref{eq:dp-spectrum}\): a noise with that zero-frequency spectrum and a frequency profile that falls faster than any power in a frame, for instance \(e^{-\omega^2/\Omega^2}\), excites the vacuum at a finite rate that does not depend on a momentum cutoff, even for components of \(T^{\mu\nu}\), whose weight grows as \(\sigma_O\propto s^2\). A Lorentzian profile does not suffice there: against \(\sigma_O\propto s^2\) its frequency integral diverges. Spectral support on timelike \(q\) is used; a noise without it is Section 5.

4.2 Covariant positivity

Static Newtonian sources sit at spacelike \(q\). A covariant, completely positive classical sector can therefore respond to them only through the trace: it is Nordström-like, gives no light deflection and no gravitomagnetism. For an ultralocal kernel this is the statement of Oppenheim, Russo and Weller-Davies that a positive semidefinite kernel cannot capture the transverse part of the Einstein equations in Lorentzian signature, and their CQ–Nordström theory is the trace case ; the lemma extends it to non-local kernels by the little-group argument. For pure noise without feedback, Lemma 4.3 fixes \(O=-T^\mu{}_\mu\), which for laboratory matter is \(\rho c^2\) up to \(p/\rho c^2\).

4.3 Record and feedback

In case (b) a classical field responds linearly to a record of \(O\), \(h=D\ast(O+\xi)\) with record noise \(\xi\) of spectral density \(N(q)\), and acts back through \(\int hO\). Integrating out \(h\) gives a Gaussian influence functional on matter with decoherence kernel \(\Lambda(q)\) and response \(\tilde D(q)\). We normalize both so that the decoherence exponent of a static superposition is \(t\int d^3k\,(2\pi)^{-3}\Lambda(0,\mathbf{k})|\Delta\rho_{\mathbf{k}}|^2\) and the Newtonian response is \(\tilde D(0,\mathbf{k})=2\pi G/(\hbar k^2)\); the Diósi–Penrose rate is then \(\Lambda(0,\mathbf{k})=C\,\tilde D(0,\mathbf{k})\). The record costs decoherence \(1/(4N)\), and the field noise \(|\tilde D|^2N\) acts back on \(O\), so at every \(q\) \[\begin{equation} \Lambda(q)\;\geq\;\frac{1}{4N(q)}+|\tilde D(q)|^2N(q)\;\geq\;|\tilde D(q)| , \label{eq:tradeoff} \end{equation}\] with equality at \(N=1/(2|\tilde D|)\). This is the measurement imprecision–back-action inequality resolved in frequency; at \(\omega=0\) it is the floor \(C\geq1\) of Tilloy and Diósi  and the decoherence–diffusion trade-off of Oppenheim et al. .

4.4 Non-local couplings and white noise

Two structural facts locate the time direction.

Non-local couplings need an ordering. If \(O\) does not commute with itself at spacelike separation, as for the spacelike projection \(T^{\mathrm{SL}}\) of Section 7 or for a number density \(a^\dagger a\), the second-order averaged generator contains \(\theta(t_x-t_y)\,S(x-y)\,[O(x),O(y)]\) on spacelike pairs. That term depends on the surface used to order the pair, so the Tomonaga–Schwinger evolution is not integrable and a foliation enters the dynamics. We give this as an argument, not a theorem. Pearle’s quasi-relativistic model, whose collapse operator annihilates the vacuum, has a preferred ordering , and his later relativistic model evolves in a parameter that labels a foliation of spacelike hypersurfaces . Non-local couplings violate the locality in (Q), so they evade Lemma 4.1 and Theorem 4.4 only by bringing their own foliation.

White noise defines leaves. A noise white in a time \(\tau\) with a non-local spatial kernel, \(\langle\phi(x)\phi(y)\rangle=\delta(\tau(x)-\tau(y))K(x,y)\), has its correlations supported on the relation \(\tau(x)=\tau(y)\), whose classes are the level sets of \(\tau\). A white Diósi–Penrose noise, whose kernel is \(1/r\) by Lemma 2.1, therefore defines a simultaneity foliation. In Minkowski space a constant \(u\) and the \(u\)-orthogonal hyperplanes are equivalent; in curved spacetime a coloured noise could single out a congruence that is not hypersurface-orthogonal, and that case is open.

5 The spacelike door

Lemma 4.1 leaves one option for noise without feedback whose law keeps a non-compact symmetry: a classical Gaussian field whose spectrum vanishes on timelike \(q\). We treat the Poincaré-invariant case. By Lemma 4.3 it couples through \(O=-T^\mu{}_\mu=\rho c^2(1-3p/\rho c^2)\), the mass density of laboratory matter up to the negligible \(3p/\rho c^2\). In the normalization of Eq. \(\eqref{eq:S}\), \[\begin{equation} S(q)=4\pi C\hbar G\,s(q^2)\,\theta(q^2),\qquad\text{e.g. }\ s(u)=\frac{e^{-uR_0^2}}{u+\mu^2}. \label{eq:spacelike-noise} \end{equation}\] This is Pearle’s tachyonic noise  with the Diósi–Penrose normalization put in by hand.

At second order heating, drag, signalling and the vacuum do not distinguish this noise from a frame noise. At fourth order the vacuum does.

A \(1+1\)-dimensional evaluation of Eq. \(\eqref{eq:fourth}\) for \(:\!\psi^2\!:\) gives a finite positive \(f\), equal at rapidities \(0\), \(1\) and \(2\) to \(10^{-3}\). Pearle found that his tachyonic model re-excites the vacuum beyond lowest order and avoided it by giving up time ordering, which permits spacelike transport ; Lemma 5.2 is that mechanism applied to the Diósi–Penrose door. The door is therefore closed by theory, at order \(C^2\), for time-ordered dynamics. With (S), Lemma 5.2 excludes a Poincaré-invariant spacelike-only law. It does not extend to every smaller symmetry: a noise supported on \(\{q^0=cq_z\}\) is invariant under boosts along \(z\), and its momenta never sum to a timelike vector. That noise fails (R), whose floor is required in every direction; whether (R) restores the conclusion for such subgroups is open. The emission bound below is an independent closure that does not use time ordering.

5.1 An independent closure by X- and \(\gamma\)-ray data

The \(u\) that enter Eq. \(\eqref{eq:emission}\) are the same \(u\) that fix the rate at \(\omega=0\). No covariant form factor can cut the emission: on the light cone a function of \(q^2\) equals its value at \(q^2\to0\), so the near-cone noise at energy \(E\) is not suppressed for any \(E\). A frame theory escapes because its smearing \(e^{-k^2R_0^2}\) is in \(\mathbf{k}\), not in \(q^2\) (Section 7).

Each search fixes a white edge \(R_X\) at \(C=1\), the radius at which the white emission just saturates its limit: \(4.5\) Å for XENONnT , \(0.867\) Å for MAJORANA  and \(0.184\) Å for Gran Sasso , all in Diósi units . By Eq. \(\eqref{eq:emission}\), at each search’s lowest energy the covariant emission exceeds the white emission at \(R_X\) by at least \((2/\sqrt\pi)(E R_X/\hbar c)^3\int s\):

With \(\int s\geq4\), less than the \(2\ln10\) of a single decade, the Gran Sasso row alone requires \(C<3\times10^{-7}\). The \(E^2\) spectrum is largest at the top of each band, so a fit to the full spectrum is stronger.

The bound \(\eqref{eq:emission}\) keeps the near-cone part of the noise, \(k=E/\hbar c\). There the electron–nucleus and electron–electron terms of the emission rate carry \(j_0(Er/\hbar c)\) in place of the Gaussian overlap \(e^{-r^2/4R_0^2}\). Screening reduces the rate by \(11\%\) at \(50\) keV and by about \(1\%\) at \(100\)–\(140\) keV, where the \(E^2\) spectrum drives the fit; it is large only below \(10\) keV, where the spectrum is small. The fit with and without screening gives the same bound. This closure is an experimental exclusion within the semiclassical emission model  used by the searches.

6 Laws that read the time from the state

Hypothesis (L) was used tacitly by Lemma 4.1, as it is by Myrvold’s theorem . Bedingham’s covariant model  shows what dropping it buys. A pointer field \(a(x)\) with one quantized degree of freedom per spacetime point, \([a(x),a^\dagger(x')]=\delta^4(x-x')\), is written by matter through \(A(x)=\int g(x,y)\,(a+a^\dagger)(y)\,d^4y\) with \(g\) supported in the future cone of \(x\).

A pointer channel is thus inside case (a): it has no back-action and realizes arm (i). Its one-particle spectrum is all of \(\mathbb{R}^4\), which supplies off-shell, zero-frequency weight in a Poincaré-invariant vacuum; it does not shield the matter vacuum. A Lorentz-invariant \(g\) supported in the future cone depends only on the proper time \(\tau\) of \(y-x\); its integral over \(\{1<\tau<2,\ |\mathbf{y}|<L\}\) grows as \(L^2\), so it cannot be normalized. A state-independent invariant pointer kernel is therefore a distribution. For the retarded Riesz kernels, with Fourier transform \((q^2)_{\rm ret}^{-\alpha/2}\), \(|\tilde g|^2=|q^2|^{-\alpha}\) is smooth and positive on the timelike shell above the pair threshold, and Lemma 4.1 applies: at \(\alpha=1\), the Diósi–Penrose case, the vacuum excitation is infinite. For a general invariant tempered distribution supported in the future cone we conjecture the same; the regularity of \(|\tilde g|^2\) on the timelike shell is the open step.

Bedingham’s model avoids this by reading the state: his smearing is \(g\propto\exp\{-k\,\bar T^{\mu\nu}(x)(y-x)_\mu(y-x)_\nu\}\), normalized on the future cone, with \(\bar T^{\mu\nu}=\langle T^{\mu\nu}\rangle\) on the past light cone of \(x\) . That this state dependence is how the model evades Myrvold’s theorem was stated by Muciño, Okon, Sudarsky and Wiedemann . In our reading, normalizing on the non-compact cone forces \(\tilde g(q\neq0)\to0\) as \(\bar T\to0\), so the coupling switches off exactly at \(\bar T=0\) and is on for every \(\bar T>0\).

The cosmic microwave background alone gives \(\bar T^{00}=4.2\times10^{-14}\) J m\(^{-3}\) in voids, before neutrinos and dark matter, so in voids \(u\) is the rest frame of the cosmic medium and the vacuum is excited at a finite, frame-cut, nonzero rate. Physically this is an aether carried by the matter content: a time direction selected spontaneously by the state, as the microwave background selects its rest frame, rather than written into the law. Matter’s energy flow is one possible carrier of \(u\); the argument does not show that it is the only one.

The nonlinearity is concrete. Send a body by a quantum coin either to rest or to speed \(v\). Its branch-averaged energy flow, and with it \(u\), moves at \(v/2\), so each branch moves at \(v/2\) relative to \(u\). If the decoherence of a body moving at \(w\) through the frame scales as \(\gamma_w^3\) relative to its frame-free value, as it does for an energy-density kernel defined in the frame (rate \(\gamma_w^2\) times the rest rate, against \(1/\gamma_w\) for a frame-free rate), each branch’s rate shifts by \(\gamma_{v/2}^3-1\simeq3v^2/8c^2\), while either branch prepared alone shifts by zero. At \(v=1\) m s\(^{-1}\) the shift is \(4\times10^{-18}\). A dynamics that is nonlinear over mixtures can signal , though nonlinear dynamics need not when the nonlinearity acts only through locally accessible data ; here \(u\) reads \(\langle T\rangle\) inside the past light cone, which blocks the obvious steering protocols, but we have not shown that it does not signal. A matter density read from the state is also how relativistic collapse models define their local beables .

6.1 The chain, assembled

Read as a contrapositive: if an isolated, inertial superposition decoheres at \(\Gamma=C\EG/\hbar\) with \(C\) of order one, then, outside the exceptions of Remark 2.1, the law of the noise is not Lorentz invariant: the distinction between states of motion is either written into the law, which has a compact stabilizer when its spectrum reaches timelike momenta, or read by the law from the state, as in Proposition 6.2.

7 The fingerprint of a frame

Theorem 2.2 says what an observed rate requires. A frame theory that meets the requirement has been outlined, but it is not complete: it signals at a level suppressed by \(G\), and its definition on curved backgrounds is open. In its minimal form the classical sector of geometry is the instantaneous constraint field on constant-mean-curvature slices, sourced by a continuous record, at Gaussian resolution \(R_0\) and with least decoherence, of the spacelike part of the full constraint source; everything else, transverse-traceless exchange and constraint exchange at timelike momentum, is quantum. The spacelike part of the stress tensor of the quantum sector with exact four-momentum is \[\begin{equation} T^{\mathrm{SL}}_{\mu\nu}(x)=\int\!\frac{d^4q}{(2\pi)^4}\,\theta\!\left(k^2-\frac{\omega^2}{c^2}\right)e^{iq\cdot x}\,\tilde T_{\mu\nu}(q); \label{eq:TSL} \end{equation}\] it annihilates the vacuum to all orders, since the spectrum condition puts every \(\langle n|\tilde T(q)|\Omega\rangle\) in the closed forward cone. Its free-field core, the \(a^\dagger a\) collapse operator with a dark vacuum and a preferred ordering, is Pearle’s ; its floor \(C\geq1\), with \(C=1\) at least decoherence, is Tilloy and Diósi’s , with Kafri, Taylor and Milburn’s classical channel  as the two-body precursor. The model is non-local, so it evades Theorem 4.4 by bringing its own ordering, the CMC slices, whose unit normal is \(u\); it signals at a level suppressed by \(G\), through the non-local response of \(T^{\mathrm{SL}}\), in the frame of the slices. Its gravitational-wave sector, cosmology and costs are left to a separate paper; here we use only its emission spectrum, which is where a frame becomes observable.

The cut spectrum.

A charge kicked by the record can emit a photon of energy \(E\) only if the transferred four-momentum is spacelike, \(c|\mathbf{k}|>E/\hbar\). The white integrand \(\int d^3k\,k^2\,e^{-k^2R_0^2}/k^2\) acquires the factor \(\theta(c|\mathbf{k}|-E/\hbar)\), and \[\begin{equation} \frac{d\Gamma}{dE}=\frac{d\Gamma}{dE}\Big|_{\rm DP}\;S\!\left(\frac{ER_0}{\hbar c}\right), \qquad S(x)=\frac{4}{\sqrt\pi}\int_x^\infty\!u^2e^{-u^2}du=\operatorname{erfc}x+\frac{2x}{\sqrt\pi}\,e^{-x^2} \label{eq:cutoff} \end{equation}\] for the self term; the cut overlaps of the atomic pair terms are given in Appendix 10. Below \(\hbar c/R_0\) the spectrum is Diósi–Penrose’s, above it it falls as \(e^{-(ER_0/\hbar c)^2}\) (Fig. 1a). The cut \(q^2>0\) is Lorentz invariant; the frame enters through the smearing, which is in \(\mathbf{k}\) and not in \(q^2\). By Proposition 5.3 no covariant smearing produces a falling spectrum: a Gaussian cutoff in the spontaneous X-ray spectrum, at the energy \(\hbar c/R_0\) that also sets the heating, is the observable fingerprint of a frame. The direction of the laboratory’s motion through the frame leaves no usable trace: the cosmic-microwave-background velocity \(\beta=1.23\times10^{-3}\) changes the cut emission only at order \(\beta^2x^4\), below \(10^{-4}\) for \(x\leq3\).

The window.

A digitized-figure refit of the XENONnT SR0 spectrum with the cut emission (Appendix 10) bounds the model at \(C=1\) by \(R_0>1.93\) Å, the 95% limit constrained to the median sensitivity. The unconstrained limit, \(2.11\) Å, is stronger only because the threshold bin fluctuates low, \(4\) events against \(6.9\) expected. Least decoherence makes the decoherence smearing equal to the smearing of Newton’s force, so the \(52~\mu\)m inverse-square-law test  caps \(R_0<4.0\times10^{-5}\) m in the infinite-slab approximation. The window \[\begin{equation} 1.93\times10^{-10}~\mathrm{m}<R_0<4.0\times10^{-5}~\mathrm{m}\qquad(C=1) \label{eq:window} \end{equation}\] spans 5.32 decades. MAJORANA and Gran Sasso no longer bind once the cut is applied: their edges fall to \(0.18\) Å and \(0.006\) Å. At the lower edge \(\hbar c/R_0=1.02\) keV.

Detection.

A cut signal near the edge is an excess concentrated in the first keV above threshold, over \(80\%\) in \(1\)–\(2\) keV. At the exposure that detects it at \(5\sigma\), about \(1.3\times10^3\) t y for \(R_0\) at \(1.1\) times the observed edge, the best fits of the Diósi–Penrose \(1/E\) shape, the covariant \(E^2\) shape and a coloured (Lorentzian) shape are each rejected at \(\geq4.9\sigma\), and the spectrum measures \(R_0\) to \((2.03,2.84)\) Å at \(1\sigma\) with \(C\) free. That is a statistical floor: at the edge the signal sits where the detection efficiency is least certain, \(\pm18\%\) at \(1.3\) keV. The edge moves with \(C\) only as \(C^{0.13}\) and with exposure hardly at all, to a median \(2.93\) Å at \(10^4\) t y; above about \(3\) Å the window is dark to xenon detectors with a \(1\) keV threshold, and the lever is the threshold.

8 Discussion

8.1 What an observation would mean

Theorem 2.2 turns a decoherence measurement into a statement about symmetry. If an isolated, inertial, equal-mass superposition decoheres at \(\Gamma=C\EG/\hbar\) with \(C\) of order one, then a reciprocal bath behind it that meets the conditions of the drag theorem at kilometre scales has \(T_{\mathrm{eff}}>3.3\times10^{16}C\) K, and the law of the noise is not Lorentz invariant. With matter the local quantum field theory we know, a vacuum that the noise does not excite without bound, and no further degrees of freedom, a noise law is not Lorentz invariant, and if its spectrum reaches timelike momenta its Lorentz stabilizer is compact and it has a rest frame; a record with feedback has non-covariant kernels. If instead the law reads its time direction from the state, it is a dynamical aether, nonlinear over mixtures. The two readings are not separated by present data. The escapes are named: an infrared modification of Newton’s law, non-Gaussian or jump records, non-local couplings, which bring their own foliation, dynamics without time ordering, which escape the fourth-order argument but not the X-ray bound, and new degrees of freedom whose law depends on the state.

The statement is independent of any model. Gravity enters only through the normalization of the noise, \(4\pi C\hbar G/k^2\), and through the Newtonian infrared limit of hypothesis (G). The theorem says nothing about why the noise should be gravitational.

8.2 Relation to earlier work

The obstruction to covariant collapse is due to Ghirardi, Grassi and Pearle , Pearle , Myrvold , Jones, Gasbarri and Bassi  and Diósi ; Lemma 4.1 is its form for a decoherence noise and claims no new mechanism. The tachyonic spacelike noise, and the observation that it re-excites the vacuum beyond lowest order, are Pearle’s . The pointer field and its state-dependent smearing are Bedingham’s ; the reading of that state dependence as the evasion of Myrvold’s theorem is in Ref. , and the role of a state-defined matter density in relativistic collapse is discussed in Ref. . The classical-channel floor \(C\geq1\) and least decoherence are Tilloy and Diósi’s , after Kafri, Taylor and Milburn , with the decoherence–diffusion trade-off of Oppenheim et al.  as its general form; the covariant classical–quantum theory and its Newtonian limit are theirs . The signalling risk of nonlinearity over mixtures is Gisin’s , and Polchinski’s construction shows it can be avoided . Flash-sourced gravity  and Poissonian hybrids  are the jump-record alternatives. The free-field core of the model of Section 7 is Pearle’s quasi-relativistic collapse operator .

New here, within the sources we have read: the link of Theorem 4.4 from a causal, exactly Newtonian classical response with complete positivity to infinite vacuum excitation; the closure of the spacelike-support door for the Diósi–Penrose normalization, by fourth-order excitation and by X- and \(\gamma\)-ray data (Lemma 5.2, Propositions 5.3 and 5.4); and the refit of the XENONnT spectrum with the cut emission of a frame theory (Appendix 10).

8.3 Open problems

• Case (b) for kernels invariant under a non-compact subgroup, such as \(SO(2,1)\), that are Newtonian only at \(\omega=0\) (Remark 4.2).

• The amplitude condition of (Q) in a general interacting theory, beyond the tree-level bilinears for which Lemma 5.2 verifies it.

• The general form of the invariant-pointer step: whether every Lorentz-invariant tempered distribution supported in the future cone has \(|\tilde g|^2\) positive and locally integrable on the timelike shell (Section 6).

• A theorem for non-local couplings: whether every ordering-dependent generator needs a foliation, or an ordering-independent prescription exists.

• The curved-spacetime statement: frame or foliation for a coloured noise in Friedmann–Robertson–Walker spacetime.

• Whether a state-borne time direction signals.

• Jump records that keep the Diósi–Penrose scaling over the laboratory band.

9 Conclusions

A Diósi–Penrose decoherence rate breaks Lorentz invariance. For state-independent laws acting on local quantum field theory through a classical noise, the law is not Lorentz invariant when the vacuum is excited at a finite rate, and its stabilizer is compact if its spectrum reaches timelike momenta; for a Gaussian record with an exactly Newtonian response the kernels are not covariant. The one covariant noise that is dark at lowest order is excited at fourth order and is limited by X-ray data to \(C<2\times10^{-8}\). A law that reads its time from the state is a dynamical aether. A frame shows as a Gaussian cutoff of the spontaneous X-ray spectrum near \(\hbar c/R_0\), which a xenon detector can see if \(R_0\) lies near \(2\) Å. A single-body interferometer in the saturated regime measures \(C\); a cutoff in the spontaneous X-ray spectrum would show the frame.

10 The XENONnT refit

Data.

The XENONnT collapse search  uses the SR0 electronic-recoil data and background model of Ref. , \(1.16\) t y over \(1\)–\(140\) keV, with nine background components. No numerical data release exists. The figures of Ref.  are vector graphics, from which we read the data points (\(1\) keV bins over \(1\)–\(30\) keV, \(2\) keV bins over \(1\)–\(140\) keV), the background model and its eight plotted components, the total efficiency with its \(1\sigma\) band, and the \(^{37}\)Ar response. Every bin count times \(1.16\) t y is an integer to \(0.02\), \(3658\) events in all. The \(^{37}\)Ar line fits a skew-normal response of width \(0.279\sqrt{E/\text{keV}}\) keV and shape \(1.61\).

Emission model.

The signal is the semiclassical spontaneous-emission rate of xenon of Piscicchia et al. , which XENONnT uses: the self terms \((N_p^2+N_e)/E\) and the electron–electron and electron–nucleus pair terms, each with the Gaussian overlap \(e^{-r^2/4R_0^2}\) and the photon phase \(\mathrm{sinc}(Er/\hbar c)\), with the orbital radii of Ref. . At \(R_0=4.5\) Å it reproduces XENONnT’s own Diósi–Penrose curve to \(5\%\) at \(17\), \(47\) and \(140\) keV, so it is \(C=1\) in Diósi units. The spacelike record restricts every overlap to \(|\mathbf{k}|>E/\hbar c\). Since \(e^{-r^2/4R_0^2}=(4/\sqrt\pi)\int_0^\infty u^2e^{-u^2}j_0(ur/R_0)\,du\), the cut overlap is \[\begin{equation} F(s,x)=\frac{4}{\sqrt\pi}\int_x^\infty\!u^2e^{-u^2}j_0(us)\,du =\frac1s\Bigl[\frac{2}{\sqrt\pi}e^{-x^2}\sin sx+s\,\operatorname{Re}\bigl(e^{-x^2+isx}\,w(\tfrac s2+ix)\bigr)\Bigr], \label{eq:cut-overlap} \end{equation}\] with \(s=r/R_0\), \(x=ER_0/\hbar c\) and \(w\) the Faddeeva function; \(F(0,x)=S(x)\) of Eq. \(\eqref{eq:cutoff}\).

Likelihood.

Poisson over \(84\) bins; eight background components profiled with the published constraints (\(^{214}\)Pb free in its range, \(^{85}\)Kr at \(90\pm22\%\), materials, \(^{136}\)Xe and solar neutrinos constrained, \(^{124}\)Xe, \(^{133}\)Xe and \(^{83m}\)Kr free); one efficiency nuisance scaling every bin by its \(1\sigma\) band; the signal folded with the response and the efficiency; true energies below \(1\) keV, where the emission model is not valid, dropped. The one-sided \(95\%\) limit uses \(q=2.706\), which reproduces XENONnT’s \(90\%\)-to-\(95\%\) ratio. Applied to the uncut spectrum the fit gives \(4.48\) Å against the published \(4.5\) Å; the residual factor \(1.012\) on the signal is kept throughout.

Results.

At \(C=1\) the cut spectrum gives \(R_0>2.11\) Å (\(95\%\) CL) observed and a median expected limit of \(1.93\) Å. The observed limit lies above the median because the threshold bin fluctuates low (\(4\) events against \(6.9\) expected), so we quote the power-constrained \(1.93\) Å. Variants of the response, of the true-energy floor (\(0.5\) or \(1.5\) keV) and of the efficiency treatment move the observed limit within \(1.90\)–\(2.16\) Å. At the observed edge the signal is \(3.4\) events, \(2.8\) of them in the \(1\)–\(2\) keV bin. The edge moves as \(C^{0.13}\) rather than \(C^{1/3}\) because the signal \(\propto CR_0^{-3}S(x)\) falls steeply with \(R_0\). At the threshold, \(x=E_{\rm th}R_0/\hbar c\approx1.07\), the slope \(d\ln[R_0^{-3}S]/d\ln R_0\) would give \(C^{0.21}\); the fitted \(0.13\) corresponds to an effective \(x\approx1.6\), about \(1.5\) keV, where the folded signal carries its weight after resolution and efficiency. Hence a factor-2 error in the threshold efficiency moves the edge by about \(10\%\), and \(10^4\) times the SR0 exposure moves the median limit only to \(2.93\) Å. The same likelihood with the covariant spectrum of Proposition 5.4, screened or not, gives \(C<1.8\times10^{-8}\); its counts are driven by the top of the band.

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