Distinguishing a Two-Step Quantum–Classical Threshold from
Collapse Models
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
October 2026
Abstract
A model of the vacuum as a face-centered-cubic lattice predicts a two-step quantum–classical
threshold, at 13.1 and 22.6 µg, from the hypothesis that a center-of-mass excitation stays co-
herent only while its reduced Compton wavelength is resolvable by the lattice. Collapse models
also limit macroscopic coherence, so the two must be told apart by more than a scale. We derive
three discriminators from the model’s postulates and compute the counterpart in Di´osi–Penrose
collapse and in continuous spontaneous localization (CSL). The thresholds depend on total mass
alone, while both collapse models depend on density. Below 13.1 µg the model predicts no intrin-
sic decoherence, where collapse takes 1 s for Di´osi–Penrose and about a microsecond for CSL at
its experimental bound, for 10
−15
kg over 1 µm. The model predicts no measurable heating; read
as position localization, its own rate scale would exceed the LISA Pathfinder bound by 66 orders
of magnitude, so coherence loss above the threshold must be dephasing in the energy basis. If
its effective mode mass is the relevant mass, a 16.2 µg mechanical cat state then bounds the
irreversible coupling at 38 orders below its natural scale. A worked levitated-particle example
and a public script complete the analysis.
Keywords: quantum–classical threshold; Di´osi–Penrose collapse; continuous spontaneous local-
ization; macroscopic superposition; decoherence; lattice vacuum; Compton wavelength; mechanical
cat states
1 Introduction
Where quantum mechanics stops applying to massive objects, if it stops at all, is an open experi-
mental question. Collapse models answer it with a new physical process. The best known, due to
Di´osi and Penrose [1, 2, 3] (DP), ties the collapse of a spatial superposition to the gravitational
self-energy of the difference between its branches, and predicts a collapse time τ
DP
= ℏ/E
G
. Con-
tinuous spontaneous localization [4, 5] (CSL) instead adds a universal stochastic localization with a
rate λ and a length r
C
, both constrained by experiment; at the standard r
C
, its originally proposed
strength is now excluded [6].
1
A recent paper [7] proposed a different answer from a discrete model of the vacuum, a face-
centered-cubic (FCC) lattice with Bell-pair bonds. It predicts two thresholds: a reversible defor-
mation of the center-of-mass mode at m
soft
= 13.1 µg, and loss of coherence at m
hard
= 22.6 µg,
in the exact ratio
√
3. That paper noted that the scale itself is not new, since DP also predicts a
scale of order the Planck mass. It distinguished the two models by the two-step structure and by
the absence of any dependence on the separation ∆x between branches.
Both distinctions require reaching the 13–23 µg window, which is beyond present superposition
experiments. Here we derive three further discriminators from the same postulates, compute the
DP and CSL counterparts of each, and identify where they can be tested. Two of them make
definite predictions far below the window, much closer to present experiments, and one of these
is already tested by existing null results. Deriving the third exposes a consistency requirement on
the original model and refines what its hard threshold means. That refinement turns an existing
experiment, the 16.2 µg mechanical cat states of Bild et al. [8], into a constraint on the model, and
a worked levitated-particle example shows how a single experiment could separate all three models.
Section 2 summarizes the framework for readers new to it. Each claim is labeled by status:
derived from the stated postulates, computed from a stated formula, or open. A public script
reproduces every number (Data Availability Statement).
2 The Selection–Stitch Model in Brief
2.1 The Vacuum Lattice
The Selection–Stitch Model (SSM) treats the vacuum as a discrete network rather than a continuum.
Its nodes form a face-centered-cubic (FCC) lattice, the densest packing of equal spheres, in which
every node has twelve nearest neighbors arranged as a cuboctahedron. Each bond is a maximally
entangled pair of qubits, a Bell pair. In the series, this lattice arises from a growth process of in-
plane “stitches” and rare out-of-plane “lifts” that crystallizes into FCC packing [9], and it carries
a quantum error-correcting code, so the vacuum behaves as a protected quantum memory [10].
These results motivate the framework, but the derivations below use only the postulates stated in
Section 3.
2.2 Matter as Trapped Nodes
Matter, in this picture, is a defect of the vacuum: an extra node trapped in one of the lattice’s
tetrahedral voids, which cannot relax into the twelve-coordinated bulk and so persists as a localized
object [9]. A defect moves by hopping between adjacent voids [11]. A macroscopic body is a large
collection of such defects. The hypothesis P3 of Section 3 treats the center-of-mass motion of such
a body, when it is in a superposition, as one collective excitation of the lattice, characterized by its
total rest energy.
2
2.3 One Length Scale
The lattice has one length scale, the bond length L. It is fixed by a single calibration, matching the
entanglement entropy of the code to the Bekenstein–Hawking area law, which gives L =
√
4 ln 2 ℓ
P
≈
1.665 ℓ
P
, about 2.7 × 10
−35
m [7]. At wavelengths much longer than L the lattice is invisible.
Provided all low-energy modes share one propagation speed, low-energy physics is isotropic and
Lorentz-invariant, with violations suppressed by (E/E
P
)
2
, about 10
−56
at optical energies [7].
Ordinary quantum mechanics holds to extraordinary precision, except where the hypothesis below
applies.
2.4 How a Threshold Arises
The lattice becomes relevant only when some scale associated with a body’s center-of-mass motion
approaches L. The SSM’s hypothesis is that the relevant scale is the reduced Compton wavelength,
λ
c
= ℏ/(mc). Unlike the size of a body, λ
c
shrinks as the mass grows, and it equals L at m =
ℏ/(Lc) ≈ 13 µg. The lattice scale is Planckian, but the mass at which λ
c
reaches it is macroscopic.
The two thresholds come from two lengths of the cuboctahedral geometry, the edge L and the
circumradius L/
√
3 of a triangular face (Figure 1).
L/
√
3
L
λ
c
= L: m
soft
= ℏ/(Lc)
λ
c
= L/
√
3: m
hard
=
√
3 m
soft
Figure 1: A triangular face of the cuboctahedron formed by a node’s twelve neighbors. The soft
threshold is the mass whose reduced Compton wavelength equals the edge L; the hard threshold,
the mass whose reduced Compton wavelength equals the circumradius L/
√
3. Their ratio,
√
3, is
fixed by the geometry alone.
3 The Threshold and Its Postulates
We restate only what the derivations use; Ref. [7] gives the full construction. The published model
rests on three postulates: P1, the FCC network with Bell-pair bonds; P2, the calibration of L
described in Section 2.3; and P3, Compton representability.
P3 (Compton representability). A center-of-mass excitation, characterized by its total
rest energy mc
2
, remains coherently representable on the lattice only while its reduced Compton
wavelength λ
c
= ℏ/(mc) is resolvable by the lattice geometry.
Representability first degrades when λ
c
= L and fails when λ
c
equals the circumradius L/
√
3
3
of a triangular face:
m
soft
=
ℏ
Lc
= 13.07 µg, m
hard
=
√
3 m
soft
= 22.64 µg. (1)
The lattice correction to the kinetic operator is Hermitian. Between the thresholds it deforms
the center-of-mass mode deterministically, by δ(m) =
1
6
(m/m
soft
)
2
, rather than collapsing it. Any
irreversible component was modeled in Ref. [7] by a phenomenological rate law,
Γ(m) =
0, m ≤ m
soft
,
Γ
0
(m/m
soft
)
2
− 1
2
, m
soft
< m < m
hard
,
Γ
0
=
ℏ
mL
2
, (2)
with coherence not sustained above m
hard
. That paper treated the absolute scale Γ
0
as undeter-
mined and called the vanishing below m
soft
a modeling choice. Section 6 shows that the latter
follows from P3, and Section 7 constrains the form of any irreversible component. Throughout, we
read the model as Ref. [7] constructs it, in which the lattice correction is Hermitian and loss of
representability is the only intrinsic source of non-unitarity.
4 The Collapse-Model Comparison
4.1 Di´osi–Penrose
For a uniform sphere of mass m and radius R in a superposition of two positions a distance d apart,
the DP energy is [1, 3]
E
G
=
Gm
2
R
×
2λ
2
−
3
2
λ
3
+
1
5
λ
5
, λ ≤ 1,
6
5
−
1
2λ
, λ ≥ 1,
λ =
d
2R
, (3)
with τ
DP
= ℏ/E
G
. We checked Eq. (3) against a direct Fourier-space evaluation of the grav-
itational self-energy, which agrees to 0.01% for d/R from 10
−3
to 8. For small separations,
E
G
→ Gm
2
d
2
/(2R
3
). [computed]
Two limits matter below. For d ≪ R, E
G
≈
2π
3
G m ρ d
2
with ρ the density. For d ≫ R,
E
G
→
6
5
Gm
2
/R. For d below the atomic spacing, the uniform sphere underestimates E
G
unless
the DP smearing length exceeds that spacing, because real matter concentrates its mass in nuclei.
The XENONnT search bounds the DP smearing length to R
0
> 4.9 × 10
−10
m [6], larger than the
atomic spacing, so the uniform sphere is close to the true value there as well. The DP times quoted
below are upper bounds, or close to the true values, and the separations used for the material
comparison of Section 5 lie above the atomic spacing.
4
4.2 Continuous Spontaneous Localization
In CSL the decoherence rate between branches of a body separated by d is [4, 5]
Γ
CSL
(d) =
λ
m
2
0
(4πr
2
C
)
3/2
Z
d
3
k
(2π)
3
e
−k
2
r
2
C
|˜ρ(k)|
2
[1 − cos(k · d)] , (4)
where m
0
is the nucleon mass and ˜ρ the Fourier transform of the mass density. For a single nucleon
it reduces to λ[1−e
−d
2
/(4r
2
C
)
], which the script reproduces exactly. [computed ] We take the standard
r
C
= 10
−7
m. Searches for spontaneous X-ray emission bound the rate from above [12], and the
XENONnT detector has recently tightened the bound to λ/r
2
C
< 3.0 × 10
−3
s
−1
m
−2
for white
noise [6]. At r
C
= 10
−7
m this gives λ < 3 × 10
−17
s
−1
, which excludes the originally proposed
λ = 10
−16
s
−1
[13]; free test masses give weaker bounds [14, 15]. We quote CSL results at the
largest allowed rate, λ = 3 × 10
−17
s
−1
. Colored or dissipative extensions of CSL, and larger r
C
,
remain less constrained [6]. CSL heats every body at [5]
dE
dt
=
3λℏ
2
m
4m
2
0
r
2
C
, (5)
which for 1 kg is about 9 × 10
−18
W at that rate. [computed ]
5 First Discriminator: The Thresholds Depend on Mass Alone
Derivation. P3 makes representability a property of the center-of-mass excitation characterized
by its total rest energy. The thresholds of Eq. (1), the deformation δ(m), and the rate law of
Eq. (2) are therefore functions of m alone. Density, radius, shape, composition, and the separation
∆x do not enter. Internal energy contributes to the rest mass, but for a room-temperature solid the
shift is ∆m/m ≈ 3 ×10
−12
. Molecule interferometry beyond 25 kDa shows that the center-of-mass
quantum mechanics of composite objects is governed by their total mass [16]. Extending this to
microgram bodies is part of the content of P3, not a consequence of that evidence. [derived]
Contrast. Both collapse models depend on how the mass is distributed. At equal mass, the
DP rate scales as the density for d ≪ R and as its cube root for d ≫ R. The CSL rate scales as the
density for d ≫ r
C
and more steeply for d ≪ r
C
, where the surface of the body dominates. Table 1
gives the factor by which collapse is faster than for silica, for spheres of mass m
soft
.
Table 1: Factor by which collapse is faster than for silica, for spheres of equal mass m = m
soft
, in
Di´osi–Penrose (DP) and in CSL. The CSL factors do not depend on λ. The lattice model predicts
identical thresholds and identical deformation for every row.
Material ρ (kg/m
3
) DP, 1 nm DP, 1 mm CSL, 1 nm CSL, 1 mm
silica 2200 1.00 1.00 1.00 1.00
sapphire 3980 1.81 1.24 2.20 1.81
gold 19,300 8.77 2.17 18.1 8.76
osmium 22,590 10.27 2.29 22.3 10.25
5
The test is a ratio. Two bodies of equal mass and different density, in the same apparatus, either
share their coherence limit or do not. It needs no absolute calibration, and it does not depend on
the value of L.
6 Second Discriminator: No Intrinsic Decoherence Below m
soft
Derivation. The rate law of Eq. (2) models the irreversible component of a failure of repre-
sentability. Under P3, representability holds for every m < m
soft
, so there is no failure. Since loss
of representability is the model’s only intrinsic source of non-unitarity (Section 3), there is nothing
irreversible to model. The only lattice effect there is the Hermitian correction to the kinetic op-
erator, which generates unitary evolution and conserves purity. The lattice produces no intrinsic
decoherence below m
soft
, for any separation and at any time. The first line of Eq. (2) states the
same result. Within the representability-breakdown mechanism it follows from P3 and needs no
separate modeling choice. [derived]
Contrast. Both collapse models predict a rate rather than a threshold, so every superposition
eventually collapses. Figure 2 and Table 2 give the collapse times for silica spheres far below m
soft
.
10
−18
10
−17
10
−16
10
−15
10
−14
10
−13
10
−12
10
−11
10
−10
10
−9
10
−8
10
−7
10
−13
10
−9
10
−5
10
−1
10
3
10
7
lattice model: no intrinsic
decoherence for m < m
soft
m
soft
m
hard
1 s
mass m (kg)
collapse time (s)
DP d = 1 nm DP d = 1 µm
DP d = 1 mm
CSL (bound) d = 1 µm
Figure 2: Collapse time for silica spheres: Di´osi–Penrose, from Eq. (3), at three separations, and
CSL at the XENONnT bound, from Eq. (4), at 1 µm. The shaded region, every mass below m
soft
,
is where the lattice model predicts no intrinsic decoherence at any time. The gray lines mark m
soft
(solid) and m
hard
(dashed).
6
Table 2: Collapse times for silica spheres below m
soft
, in Di´osi–Penrose (DP) and in CSL at the
XENONnT bound, λ = 3 × 10
−17
s
−1
. Weaker CSL gives proportionally longer times. The lattice
model predicts no intrinsic decoherence for every entry.
Mass Model d = 1 nm d = 1
µ
m d = 1 mm
10
−15
kg DP 3 × 10
5
s 1.0 s 0.6 s
CSL 0.4 s 1 × 10
−6
s 1 × 10
−6
s
1 ng DP 3 ×10
2
s 4 × 10
−4
s 6 × 10
−6
s
CSL 3 × 10
−3
s 8 × 10
−9
s 1 × 10
−9
s
1
µ
g DP 0.3 s 3 × 10
−7
s 7 × 10
−11
s
CSL 3 × 10
−5
s 8 × 10
−11
s 9 × 10
−13
s
These masses lie between about 10 and 10
7
times below m
soft
, and Figure 2 extends the compar-
ison to 10
−18
kg. All of them lie above the present record for spatial superpositions, 2.9 ×10
−22
kg
(Section 10.1). The lightest are in the range targeted by space-based proposals [17], and all are far
closer to present experiments than the threshold window. With environmental decoherence sup-
pressed, coherence maintained beyond a collapse model’s predicted time would falsify that model,
and intrinsic decoherence observed there would falsify the lattice model. This is a test of the model
that does not require reaching the threshold window.
Below m
soft
the lattice model agrees with standard quantum mechanics, which also predicts no
intrinsic decoherence. The test can falsify the lattice model and can exclude collapse models, but
a null result does not confirm the lattice model over standard quantum mechanics. The model’s
distinctive positive content remains the thresholds.
7 Third Discriminator: No Anomalous Heating or Radiation
7.1 Microscopic Matter
Every atom, nucleon, and electron lies far below m
soft
. By Section 6, no irreversible channel
acts on them, so the lattice model predicts no spontaneous radiation from ordinary matter. This is
consistent with the underground X-ray searches that excluded the parameter-free version of DP [18]
and bound CSL from above [12, 6], since both models produce such radiation. CSL also heats bulk
matter, by Eq. (5), at up to about 9 × 10
−18
W per kilogram at its present bound. [derived ]
7.2 A General Condition for Heating
Consider a Markovian irreversible channel in Lindblad form [19], with operator
ˆ
L and rate γ. An
observable
ˆ
A evolves as
d⟨
ˆ
A⟩
dt
⊃ γ
D
ˆ
L
†
ˆ
A
ˆ
L −
1
2
{
ˆ
L
†
ˆ
L,
ˆ
A}
E
, (6)
which vanishes when
ˆ
L commutes with
ˆ
A. A channel built from functions of momentum and mass,
the quantities that P3 involves, conserves the energy and the entire momentum distribution of
a free body. It produces no heating, no momentum diffusion, and no radiation. In a trap the
potential breaks this commutation for a function of momentum alone. For
ˆ
L = (Lˆp/ℏ)
2
at rate Γ
0
7
the heating is 2ω
2
L
2
⟨p
2
⟩/ℏ. It is suppressed by L
2
, to about 2 ×10
−54
W for a 1 kg body on a 1 Hz
spring at room temperature. A channel diagonal in the excitation’s own Hamiltonian,
ˆ
L = f (
ˆ
H),
commutes with it in a trap as well, and produces no heating at all. A position-localizing channel,
such as those of collapse models [5], does not commute with the kinetic energy; for
ˆ
L = ˆx it heats
at γℏ
2
/(2m). We confirmed these statements numerically, on a discretized free particle and on a
truncated oscillator. [derived ]
7.3 A Consistency Requirement on the Model
Above m
hard
the model states that coherence is not sustained, with rate scale Γ
0
= ℏ/(mL
2
). Every
macroscopic body lies above m
hard
, so the form of this channel is constrained by the absence of
anomalous heating in ordinary matter. Suppose the channel localized position on the lattice scale
L. It would then drive momentum diffusion at Γ
0
ℏ
2
/L
2
, and a 1.93 kg free test mass would heat at
about 3 × 10
35
W. The LISA Pathfinder force-noise level, of order 10
−15
N Hz
−1/2
[20], limits the
rate of any such channel to below about 3 × 10
−32
s
−1
. That is 66 orders of magnitude below Γ
0
.
[computed]
The irreversible component above m
hard
must therefore either commute with the Hamiltonian
of the collective excitation, or have a position-localizing part at least 66 orders of magnitude weaker
than Γ
0
. The first is the natural reading of P3, which concerns the rest energy of the excitation
and its kinetic dispersion; for a free center-of-mass mode it reduces to a function of p
2
/2m. So
P3 cannot consistently mean position localization at the lattice scale. Its breakdown concerns the
collective quantum representation, not random kicks to constituent matter. On that reading the
model predicts no measurable anomalous heating or radiation for bodies of any mass. [derived
consistency requirement]
7.4 What Then Loses Coherence
A channel diagonal in the Hamiltonian leaves the energy distribution unchanged. For a free particle
it also leaves the momentum distribution unchanged, including the far-field interference fringes of a
spatial superposition; this too was confirmed numerically. Above m
hard
, the lattice model predicts
dephasing between different energy components of the collective excitation, in the spirit of energy-
basis intrinsic decoherence [21], not loss of spatial fringes. The appropriate test is a state with
coherence between distinct energy components, such as the mechanical cat states of a resonator [8],
not matter-wave interferometry. This leaves the thresholds and the
√
3 ratio unchanged and changes
only the observable that would reveal them.
8 A Constraint from an Existing Experiment
Section 7 implies that any irreversible lattice effect in the window acts as dephasing between
energy components of the excitation. Mechanical cat states containing coherence between distinct
oscillator-energy components have been made. Bild et al. [8] prepared cat states of a bulk acoustic
mode with an effective mass of 16.2 µg, superpositions of two opposite-phase coherent states. The
8
two components have the same mean energy, but the superposition carries coherence among the
oscillator’s energy eigenstates, which an energy-dephasing channel would destroy. A companion
analysis [22] followed the decay of their Wigner negativity over tens of microseconds, observing it
for up to 40 µs. The published threshold paper already placed this experiment inside the window [7].
The bound. At m = 16.2 µg, m/m
soft
= 1.239, and the rate law of Eq. (2) at its full scale
gives Γ
0
[(m/m
soft
)
2
−1]
2
≈ 2.6 ×10
42
s
−1
. The observed coherence requires an intrinsic rate below
about 1/(40 µs) = 2.5 ×10
4
s
−1
. The irreversible coupling in the window must therefore lie at least
38 orders of magnitude below its natural scale. The intermediate regime cannot be governed by the
original full-strength irreversible rate; in practice the window is reversible. This agrees with the
threshold paper’s own view that the reversible deformation is the prediction that does not depend
on the rate model. That deformation is large here, δ = 0.256, but it is a deterministic shift and is
invisible to a measurement of decoherence. [computed ]
The caveat. The mode is an acoustic vibration of a crystal, not the center of mass of a
free body, and its effective mass is a property of the mode shape. Whether P3 applies to it with
m = 16.2 µg is an assumption, shared with Ref. [7]. If the relevant mass is below m
soft
instead,
the experiment lies in the null regime of Section 6, and it is again consistent with the lattice
model. In both cases the experiment is compatible with the threshold framework, and on the
stated assumption it rules out the full-strength irreversible rate in the window. [open]
9 A Fourth Signature: Reversibility
The effect in the window that does not depend on the rate model is the Hermitian, time-independent
lattice correction. On a superposition of two energy levels it produces a static, deterministic relative
phase. For a two-level encoding in which a π pulse reverses that relative phase, a Hahn-echo
sequence cancels it, whereas irreversible dephasing is not restored. The window effect is a pure
phase, set by the deformation δ(m) ∝ (m/m
soft
)
2
, that preserves contrast and vanishes under an
echo; collapse models produce a loss of contrast that no echo restores, accompanied by heating.
[derived ] The size of this phase for a given experiment is not yet predicted, because the model
supplies no map from δ(m) to the energy shift of a specific mode. [open] Deriving that map, an
energy shift ∆E(m, ω, . . .) for a given mode, is the next calculation the model needs; it would turn
the reversible window into a direct spectroscopic prediction.
10 Experimental Outlook
10.1 Where Experiments Stand
Table 3 places the predictions against present experiments. The largest masses brought into spa-
tial superposition are sodium nanoclusters of about 172 kDa, or 2.9 × 10
−22
kg, in matter-wave
interferometry [23], after molecules beyond 25 kDa in 2019 [16]. Levitated silica nanoparticles of
about 10
−18
kg have been cooled to their motional ground state, from room temperature [24] and in
cryogenic free space [25], but have not yet been delocalized over distances comparable to their size.
The two spheres of the worked example below are 10
4.5
and 10
6.5
times heavier than the record
9
superposition. The acoustic-mode cat states of Section 8 lie inside the threshold window. At the
other extreme, the center-of-mass motion of a 10 kg mirror has been cooled to an average occu-
pation of 10.8 [26], at a mass far above m
hard
. That state is thermal, with no coherence between
energy components, so it does not test the window; it shows that such masses can be brought close
to the quantum regime. [computed ]
Table 3: Present experiments, by mass, and the parts of this paper they bear on.
Platform Mass Achieved Bears on
sodium
nanoclusters [23]
2.9 × 10
−22
kg spatial superposition; mass record collapse-model
bounds
levitated silica
sphere [24, 25]
∼10
−18
kg motional ground state; no spatial
superposition yet
second
discriminator;
worked example
acoustic-mode
cat [8, 22]
16.2 µg
(effective)
cat states, coherence to 40 µs the window
(Section 8)
gravitational-wave
mirror [26]
10 kg thermal occupation 10.8 above m
hard
; no
coherence test
10.2 The Discriminators at a Glance
Table 4 summarizes the discriminators and the experiments that address them.
Table 4: The discriminators, with the predictions of each model and the experiments that address
them.
Discriminator Lattice
model
Di´osi–
Penrose
CSL Test
equal mass,
different
density
identical
thresholds
rate ∝ ρ
(d ≪ R), ρ
1/3
(d ≫ R)
rate ∝ ρ
(d ≫ r
C
),
steeper below
same apparatus, two
materials
below m
soft
no intrinsic
decoherence
1 s at
10
−15
kg,
1 µm
≈1 µs at
10
−15
kg,
1 µm
levitated
particles [24, 25, 17]
heating and
radiation
none
measurable
radiation;
parameter-
free form
excluded [18]
heating and
radiation;
original
strength
excluded [6]
radiation searches; free test
masses [20]
reversibility pure phase,
removed by
echo
contrast loss contrast loss echo on states with energy
coherence [8]
The second discriminator is the most practical, because its prediction applies at masses far
closer to present experiments than the threshold window. The first needs no absolute calibration,
because it is a ratio. The third is already consistent with existing null results.
10
10.3 A Worked Example
Table 5 gives the predictions for two levitated silica spheres. The first, of radius 100 nm, is within
an order of magnitude of the mass of the ground-state-cooled particles of Refs. [24, 25]; holding its
superposition for 10 ms would test white-noise CSL at its largest allowed rate, while DP would need
hours. The second, of 10
−15
kg, is a goal of space-based proposals [17]; holding it for several seconds
would exclude both collapse models. For both spheres the three models give distinct outcomes:
decoherence at the CSL time, decoherence at the DP time, or none. [computed ]
Table 5: Predicted intrinsic collapse times for two levitated silica spheres. The lattice model
predicts no intrinsic decoherence in either case.
Sphere Separation DP CSL, XENONnT
bound
R = 100 nm,
9.2 × 10
−18
kg
100 nm 6 × 10
3
s 8 × 10
−3
s
R = 480 nm, 10
−15
kg 1 µm 1.0 s 1 × 10
−6
s
11 Discussion
Consequences for the threshold model. Three points refine Ref. [7] without changing its
thresholds or its
√
3 ratio. The vanishing of the irreversible rate below m
soft
follows from P3, given
the model’s reading in Section 3; it is not a separate modeling choice. The irreversible component
cannot be position localization at the stated rate scale, and on the natural reading it is dephasing
in the energy basis. On that reading, and if P3 applies to the mode of Ref. [8], the window is
effectively reversible. All follow from P3 and that reading; the second also uses the Markovian
treatment of the channel, and the third the effective-mass assumption of Section 8.
Dependence on the bond length. The absolute window depends on L, which is tied to
one calibration. The first and third discriminators and the fourth signature do not depend on L
in form; the 66-order margin of Section 7 scales as L
−4
and moves by less than one order for any
L up to 2.8 ℓ
P
. The second depends on L through the location of m
soft
, which stays above every
mass in Table 2 for any L up to 2.8 ℓ
P
, where m
soft
= 7.8 µg.
Limitations. P3 is a hypothesis, not a theorem, and every discriminator here inherits that
status. The DP estimates use uniform spheres; below the atomic scale they are upper bounds on
the collapse time, or close to it. The CSL figures assume white noise and r
C
= 10
−7
m, at the
present XENONnT bound; colored or dissipative extensions weaken the X-ray bounds [5, 6]. The
analysis of heating assumes a Markovian channel. A microscopic, operator-level derivation of the
representability breakdown would fix the size of the reversible phase and the absolute rate, and
remains open.
11
12 Conclusions
The lattice model and collapse models predict limits on macroscopic coherence, but they differ in
kind. The lattice threshold depends on mass alone; it permits no intrinsic decoherence below 13.1 µg
at any time; and it produces no measurable heating or radiation. The last is also a consistency
requirement, which shows that P3 cannot mean position localization at the lattice scale; the model’s
loss of coherence above its threshold must be, to any measurable extent, dephasing between energy
components, not loss of spatial fringes. An existing 16.2 µg mechanical cat state does not conflict
with the threshold framework and, if P3 applies to its quoted effective mode mass, constrains
the irreversible coupling in the intermediate window to at least 38 orders of magnitude below the
nominal rate scale. The most accessible test lies far below the threshold window, where collapse
models predict decoherence within a second or much less and the lattice model, like standard
quantum mechanics, predicts none.
Author contributions. Conceptualization, methodology, formal analysis, investigation, writing—
original draft preparation, writing—review and editing, and visualization: R.K. The author has read
and agreed to the published version of the manuscript.
Funding. This research received no external funding.
Data availability. No new datasets were generated. A Python script that reproduces every
number in this paper (31 labeled checks, NumPy only) is available at https://github.com/rag
hu91302/ssmtheory/raw/main/discriminator_scripts.zip.
Conflicts of interest. The author declares no conflicts of interest.
References
[1] L. Di´osi, “A universal master equation for the gravitational violation of quantum mechanics,”
Phys. Lett. A 120, 377 (1987). doi:10.1016/0375-9601(87)90681-5.
[2] L. Di´osi, “Models for universal reduction of macroscopic quantum fluctuations,” Phys. Rev. A
40, 1165 (1989). doi:10.1103/PhysRevA.40.1165.
[3] R. Penrose, “On gravity’s role in quantum state reduction,” Gen. Relativ. Gravit. 28, 581
(1996). doi:10.1007/BF02105068.
[4] G. C. Ghirardi, P. Pearle, and A. Rimini, “Markov processes in Hilbert space and continu-
ous spontaneous localization of systems of identical particles,” Phys. Rev. A 42, 78 (1990).
doi:10.1103/PhysRevA.42.78.
[5] A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, “Models of wave-function
collapse, underlying theories, and experimental tests,” Rev. Mod. Phys. 85, 471 (2013).
doi:10.1103/RevModPhys.85.471.
[6] E. Aprile et al. (XENON Collaboration), “Challenging spontaneous quantum collapse with
XENONnT,” Phys. Rev. Lett. (2026). doi:10.1103/2jm3-4976; arXiv:2506.05507.
12
[7] R. Kulkarni, “A two-step quantum–classical threshold at 13.1–22.6
µ
g with exact ratio
√
3 from
a close-packed vacuum lattice,” Quantum Rep. 8, 78 (2026). doi:10.3390/quantum8030078.
[8] M. Bild, M. Fadel, Y. Yang, U. von L¨upke, P. Martin, A. Bruno, and Y. Chu,
“Schr¨odinger cat states of a 16-microgram mechanical oscillator,” Science 380, 274 (2023).
doi:10.1126/science.adf7553.
[9] R. Kulkarni, “Matter as incomplete crystallization: Quark charges, color confinement, and the
proton mass from a single extra node in the vacuum lattice,” Phys. Open 27, 100423 (2026).
doi:10.1016/j.physo.2026.100423.
[10] R. Kulkarni, “A 67%-rate CSS code on the FCC lattice: [[192, 130, 3]] from weight-12 stabiliz-
ers,” arXiv:2603.20294 (2026). arXiv:2603.20294.
[11] R. Kulkarni, “Defect migration in the D
4
vacuum lattice: Hop parity, site symmetry, and
soliton kinematics,” Symmetry 18, 1571 (2026). doi:10.3390/sym18091571.
[12] K. Piscicchia et al., “CSL collapse model mapped with the spontaneous radiation,” Entropy
19, 319 (2017). doi:10.3390/e19070319.
[13] G. C. Ghirardi, A. Rimini, and T. Weber, “Unified dynamics for microscopic and macroscopic
systems,” Phys. Rev. D 34, 470 (1986). doi:10.1103/PhysRevD.34.470.
[14] M. Carlesso, A. Bassi, P. Falferi, and A. Vinante, “Experimental bounds on col-
lapse models from gravitational wave detectors,” Phys. Rev. D 94, 124036 (2016).
doi:10.1103/PhysRevD.94.124036.
[15] B. Helou, B. J. J. Slagmolen, D. E. McClelland, and Y. Chen, “LISA Pathfinder
appreciably constrains collapse models,” Phys. Rev. D 95, 084054 (2017).
doi:10.1103/PhysRevD.95.084054.
[16] Y. Y. Fein, P. Geyer, P. Zwick, F. Kia lka, S. Pedalino, M. Mayor, S. Gerlich, and M.
Arndt, “Quantum superposition of molecules beyond 25 kDa,” Nat. Phys. 15, 1242 (2019).
doi:10.1038/s41567-019-0663-9.
[17] R. Kaltenbaek et al., “Macroscopic quantum resonators (MAQRO),” Exp. Astron. 34, 123
(2012). doi:10.1007/s10686-012-9292-3.
[18] S. Donadi, K. Piscicchia, C. Curceanu, L. Di´osi, M. Laubenstein, and A. Bassi, “Underground
test of gravity-related wave function collapse,” Nat. Phys. 17, 74 (2021). doi:10.1038/s41567-
020-1008-4.
[19] G. Lindblad, “On the generators of quantum dynamical semigroups,” Commun. Math. Phys.
48, 119 (1976). doi:10.1007/BF01608499.
[20] M. Armano et al., “Beyond the required LISA free-fall performance: new
LISA Pathfinder results down to 20
µ
Hz,” Phys. Rev. Lett. 120, 061101 (2018).
doi:10.1103/PhysRevLett.120.061101.
13
[21] G. J. Milburn, “Intrinsic decoherence in quantum mechanics,” Phys. Rev. A 44, 5401 (1991).
doi:10.1103/PhysRevA.44.5401.
[22] B. Schrinski, Y. Yang, U. von L¨upke, M. Bild, Y. Chu, K. Hornberger, S. Nimmrichter, and
M. Fadel, “Macroscopic quantum test with bulk acoustic wave resonators,” Phys. Rev. Lett.
130, 133604 (2023). doi:10.1103/PhysRevLett.130.133604.
[23] S. Pedalino et al., “Probing quantum mechanics with nanoparticle matter-wave interferome-
try,” Nature 649, 866 (2026). doi:10.1038/s41586-025-09917-9.
[24] U. Deli´c, M. Reisenbauer, K. Dare, D. Grass, V. Vuleti´c, N. Kiesel, and M. Aspelmeyer,
“Cooling of a levitated nanoparticle to the motional quantum ground state,” Science 367, 892
(2020). doi:10.1126/science.aba3993.
[25] F. Tebbenjohanns et al., “Quantum control of a nanoparticle optically levitated in cryogenic
free space,” Nature 595, 378 (2021). doi:10.1038/s41586-021-03617-w.
[26] C. Whittle et al., “Approaching the motional ground state of a 10-kg object,” Science 372,
1333 (2021). doi:10.1126/science.abh2634.
14
