Vacuum Fluctuations as the
Stiffness of Spacetime
Induced Regge Gravity on the Selection–Stitch Lattice
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
Abstract
The gravitational sector of the Selection–Stitch Model (SSM) is Regge calculus on the
D
4
lattice, with its action assumed. We derive the action from the vacuum fluctuations of quantum
fields, now measured directly, which we postulate predate spacetime and persist in it as the
substrate of the lattice’s bonds and defects; a saturation rule keeps them from forming new
bonds inside existing space. If the fluctuations are regulated covariantly and see the lattice’s
edge lengths as a piecewise-flat geometry, integrating them out induces an Einstein–Hilbert
term (Sakharov’s mechanism), which on such a geometry is exactly the Regge action. We verify
this on the hinge itself: from the exact spectrum of a cone, the conical heat-kernel coefficient
is
1
12
(
α
−1
− α
) to six digits, giving the hinge action
−
(8
πG
ind
)
−1
Aδ
[1 +
δ/
4
π
+
O
(
δ
2
)]. The
fluctuations cannot instead be the lattice’s own: lattice-regulated fields, including those priced
by error-correction costs, induce a shape-dependent vacuum energy that gives the graviton a
mass. A defect’s verification cost, counted in proper-time ticks, couples as a rest mass, so it
is the source in Einstein’s equation. The induced Newton constant is 1
/G
ind
=
C
Λ
2
/
12
π
with
C
=
N
s
(1
−
6
ξ
) +
N
Weyl
+ 2
N
Dirac
−
4
N
V
. With the cutoff fixed by mode counting, the SSM’s
area-law
G
requires
C
= 4
.
2–5
.
9; three generations of the Spin(10) spinor with the Standard
Model gauge group and a minimal Higgs give
C
= 4, a match within a factor 1
.
5 with no
adjustable parameter, and minimal Pati–Salam content is excluded. The covariance postulate,
the identification of ticks with proper time, and the cutoff scheme are the main limitations.
1 Introduction
In the Selection–Stitch Model (SSM) the structure of spacetime is a lattice of Planck-scale bonds,
face-centered cubic in space and the
D
4
root lattice in four dimensions, on which a quantum
error-correcting code runs [
1
,
2
]. Matter is a defect of that lattice, and mass is the cost of verifying
it [
3
]. Gravity is the Regge calculus of the lattice’s edge lengths [
4
]: curvature sits on the triangular
hinges, and the action is
S
=
−
(8
πG
)
−1
P
σ
A
σ
δ
σ
. On the closed
D
4
complex this action reproduces
the Fierz–Pauli and Einstein–Hilbert terms exactly at two derivatives, and its field equations give
the Newtonian limit and post-Newtonian
β
=
γ
= 1 [
5
]. In that work, as in almost all of Regge
calculus, the action itself is assumed.
This paper derives it. The proposal is physical before it is technical: the lattice and its defects
float on the vacuum’s fluctuations, and gravity is the stiffness those fluctuations give to space, in
the sense of Sakharov’s induced gravity [
6
,
7
]. Section 2 reviews the measured fluctuations and
states the SSM’s premise about them, the saturation rule that keeps them from creating new space,
and the covariance postulate the derivation of the action uses. Section 3 explains why they cannot
be the lattice’s own fields. Section 4 derives the Regge action and its source, Sec. 5 computes
1
Newton’s constant and compares it with the SSM’s independent area-law value, and Secs. 6–7 collect
consistency checks and limitations.
What is new. Sakharov’s mechanism itself is standard. This paper adds four things. It shows
that the fluctuations inducing the lattice’s dynamics cannot be the lattice’s own fields: every
lattice-regulated sector we test, including those priced by the SSM’s error-correction costs, gives the
graviton a mass (Sec. 3). It verifies the Regge hinge term directly on the conical singularity, from
an exact spectrum, rather than through the smooth-curvature identity alone (Sec. 4.3). It derives
the source: a defect’s verification cost, counted in proper-time ticks, couples to the geometry as a
rest mass, which gives the equivalence principle and makes verification cost the source of curvature
(Sec. 4.5). And it tests the induced Newton constant against the SSM’s independent area-law value,
with a cutoff fixed by mode counting rather than chosen (Sec. 5).
Scope of the claims. The derivation of the Regge action is conditional on Postulate 2, the
covariance of the vacuum fluctuations, and that of the source on Postulate 3; everything after them
is an exact identity, a standard heat-kernel result, a counting argument, or a computation reported
with its numerics. Postulate 1, that the fluctuations predate spacetime, frames the model but is not
used in either derivation. The comparison of
G
is an order-one test under two stated cutoff schemes,
not a precision prediction. The induced-action computations are Euclidean and one-loop.
2 The ocean and the lattice
The ocean is observed. The ground state of a quantum field is not empty: its field values
fluctuate at every wavelength. The fluctuations were first inferred from their effects, the Lamb shift
of hydrogen [
8
,
9
], read by Welton as the jitter of an electron in the vacuum field [
10
], and the
Casimir force between conductors [
11
]. They are now measured directly: the electric field of the
vacuum has been sampled in time [
12
], its correlation spectrum measured [
13
], and its correlations
detected between points outside each other’s light cone [14], and moving boundaries convert them
into real photons [
15
]. Whether the fluctuations should be called virtual particles is an interpretive
question [16]; their existence and spectrum are not. We call them the ocean.
Postulate 1 (The ocean). The vacuum fluctuations predate the lattice, and with it spacetime, and
persist in it. They are the substrate from which the lattice’s bonds and its defects formed.
Before the lattice. The SSM’s account of what precedes the lattice is the following. Before
there is a lattice there are only these fluctuations: degrees of freedom defined by their algebra
alone, with no position, since location is a lattice property. Among all possible rulebooks, the
one realized is one in which some structure can persist; quantum theory is the natural candidate,
because it permits relation without prior geometry (entanglement), combines variation with heredity,
and supports self-verification. The first persistent structure is the smallest closed cycle of verified
relation, three degrees of freedom pairwise entangled: the triangle. Adjacency is verified relation,
so the triangle is the first geometry and its verification cycle is the first clock. From it the SSM’s
crystallization proceeds as published: the triangle seed grows into a two-dimensional sheet, lifts into
three dimensions, and completes the transition to the close-packed lattice, trapping the remnant
nodes that are matter [
2
]. The lattice that emerges is polycrystalline even from a single seed: its
bulk is FCC, and the interior nodes short of full coordination sit at
K
= 10–11, on the boundaries
between crystallites of different orientations [
2
]. The lattice and its defects are both made of stitched
fluctuations.
2
After the lattice: the saturation rule. Stitching converts fluctuations into bonds but does
not exhaust them; the ocean persists around the finished lattice and its defects. It does not keep
creating space, and the reason is the rule that created space in the first place. A bond forms only
where a node can accept one, and a node already surrounded by the maximal number of neighbors
at bond distance cannot: that maximum is the kissing number, 12 in three dimensions [
17
] and
24 in four [
18
], which the FCC slice and the
D
4
lattice attain. The SSM’s growth kinematics
applies exactly this rule [
2
]. Away from defects and grain boundaries every node is saturated, so the
ocean cannot stitch there: the existing lattice excludes new bonds at the points it already occupies.
Stitching remains possible only where coordination is incomplete: at growth fronts, at the grain
boundaries between crystallites, and at the boundaries of vacancies, where it would compete with a
vacancy’s evaporation [
19
]; we do not treat that competition here. The ocean therefore coexists
with space rather than adding to it. Once space exists, the persisting fluctuations are the field
modes on its geometry, and so acquire location from it; the measured vacuum fluctuations above
are this present-day remainder.
The covariance postulate. Two further properties of the ocean matter for gravity: it has no
rest frame, and it sees the lattice as geometry. The measured correlations of the vacuum field
outside the light cone [
14
] are those of the Lorentz-invariant vacuum of quantum field theory, and
are consistent with the first. We state both together.
Postulate 2 (Covariant ocean). The persisting vacuum fluctuations are those of quantum fields
coupled to the piecewise-flat geometry defined by the lattice’s edge lengths, minimally unless stated
otherwise, and regulated covariantly, by a proper-time cutoff Λ that respects the symmetry of the
geometry rather than that of the lattice. The scale of Λ is set by the lattice’s mode density (Sec. 5).
Covariance is what makes the ocean frictionless and non-dissipative for matter in uniform motion:
a Lorentz-invariant vacuum is a pure state with no preferred frame, so a uniformly moving body
cannot detect its motion through it, and there is no bath to decohere into. Accelerated motion
does couple to the vacuum (the Unruh effect), as it must. Postulate 1 is the model’s premise;
Postulate 2 is the assumption the derivation of the Regge action uses, and Sec. 3 shows that it
cannot be weakened to fields regulated by the lattice itself. A third postulate, for the source, is
stated in Sec. 4.5.
3 Why the fluctuations cannot be the lattice’s own
Two results close the alternative in which the lattice’s own sectors induce its dynamics.
The stabilizer sector induces nothing. The code’s check Hamiltonian is
H
=
P
i
J
i
(1
−S
i
)
/
2,
where the checks
S
i
are commuting Pauli products with eigenvalues
±
1. Each term is nonnegative,
and all terms vanish on the code space, where every
S
i
= +1; the code space is nonempty because
the checks form a valid CSS stabilizer group [
1
]. The ground energy is therefore exactly zero for
every choice of positive couplings
J
i
, and the ground space does not depend on them. However the
couplings are made to depend on the edge lengths, neither changes, so the pure code contributes
no induced action, curvature term or otherwise. With the other sign convention,
H
′
=
−
P
i
J
i
S
i
,
the ground energy is
−
P
i
J
i
: an ultralocal sum of one term per check, which depends on the
edge lengths only through each check’s own couplings and contains no deficit angles, so it is not a
curvature term either. Vacuum fluctuations, and with them any induced gravity, come only from
non-commuting dynamics.
3
bond couplings real scalar U(1) gauge Wilson–Dirac
QEC pricing per bond (J ∝ 1/ℓ) 0.492 0.475 0.500
QEC pricing per check support 0.167 0.253 0.167
exact continuum kinetic tensor 0.143 – –
Table 1: Shape response of the induced vacuum energy of lattice-regulated fields under uniform
strain, relative to the covariant value (covariance requires 1). QEC: error-correction. The gauge
field lives on the root links with root-triangle plaquettes. For the scalar all nine traceless directions
give the same value, and for the other fields every direction tested does, as the rank-four isotropy of
D
4
requires. Momentum grids 16
4
and 24
4
(scalar, Dirac, control) and 8
4
and 12
4
(gauge) agree to
four digits. The Wilson–Dirac operator is
D
(
k
) =
P
j
t
j
(
γ · ˆn
j
)
e
ik·d
j
+
r
(1
− cos k · d
j
)
, summed
over the 24 root vectors
d
j
with
ˆn
j
=
d
j
/|d
j
|
and bond amplitudes
t
j
, including a Wilson term [
20
]
that gaps the doublers; the ratio shown is at
r
= 1 and lies between 0
.
49 and 0
.
56 for 0
.
25
≤ r ≤
4.
Lattice-regulated fields give the graviton a mass. For the dynamical sectors we computed
the one-loop vacuum energy
W
(
ϵ
) of fields living on the
D
4
lattice under a uniform strain
g
= 1 +
ϵ
,
with bond couplings depending on bond lengths. A uniform strain is a flat geometry in other
coordinates, so the Regge action gives it zero cost for any shape; an induced action compatible
with Regge must depend on it only through the volume. If
W
depends on
ϵ
only through
√
det g
,
its gradient at
ϵ
= 0 is
1
2
w δ
µν
, which defines the volume response
w
, and its second derivative
along any traceless unit direction is
−
1
2
w
. The test is therefore the shape (traceless) response of
W
against
−
1
2
w
, and it suffices at
k
= 0: no term at
O
(
k
2
) can repair a shape dependence of the
uniform vacuum energy. Couplings are priced in two ways suggested by the SSM’s error-correction
costs, with each bond’s quantum costing
ℏc/
4
ℓ
: per bond,
J
e
∝
1
/ℓ
e
(for a gauge plaquette, the
mean of 1
/ℓ
over its three bonds); and per check support,
J ∝
1
/
¯
ℓ
with
¯
ℓ
the mean bond length of
the check (a vertex star, or a plaquette). Table 1 lists the ratio of the measured shape response to
the covariant value.
Every coupling rule fails, including one built to reproduce the continuum kinetic tensor exactly.
A shape-dependent vacuum energy is an induced shear stiffness at the lattice scale, which gives the
graviton a mass of order the cutoff. The same holds for any energy that depends on the shape of a
uniform strain, including an elastic energy intrinsic to the lattice, which is why the lattice can have
no appreciable stiffness of its own (Sec. 4.4); for the bosonic fields the induced stiffness is negative,
so it would also destabilize the lattice against shear. For a power-law coupling J ∝ ℓ
−p
covariance
is restored only at a field-dependent exponent,
p
∗
= 4
.
1996 for the scalar and 4
.
6899 for the gauge
field, so no single rule works. Because the shape excess is close to
1
4
w
for every species, bosonic
and fermionic fields cancel it approximately when their lattice mode counts balance, but only to a
few percent. The obstruction is generic: the modes near the lattice cutoff see the crystal’s shape.
Postulate 2 removes it by construction, since a covariantly regulated ocean has no preferred shape.
4 Derivation of the Regge action and its source
4.1 Sakharov’s mechanism
For a field with Laplace-type fluctuation operator
D
=
−
(
∇
2
+
E
), the one-loop Euclidean effective
action with proper-time cutoff Λ is
Γ = −
s
2
Z
∞
Λ
−2
dt
t
Tr e
−tD
, Tr e
−tD
=
1
(4πt)
2
Z
√
g
a
0
+ t a
1
+ t
2
a
2
+ . . .
, (1)
4
with
s
= +1 for bosons and
s
=
−
1 for fermions (for which
D
=
D/
2
), and
a
1
=
tr
(
E
+
R/
6) [
21
].
The a
0
term is a cosmological term of order Λ
4
; the a
1
term is
Γ ⊃ −
C Λ
2
192π
2
Z
√
g R ≡ −
1
16πG
ind
Z
√
g R,
1
G
ind
=
C Λ
2
12π
, (2)
where
C
counts the fields in units of a minimally coupled real scalar. From the coefficients
a
1
of
each species (Appendix A),
C = N
s
(1 − 6ξ) + N
Weyl
+ 2N
Dirac
− 4N
V
, (3)
with
ξ
the scalar curvature coupling and
N
V
the number of gauge fields. Minimally coupled scalars
and fermions induce a positive Newton constant, a conformally coupled scalar (
ξ
=
1
6
) none, and
gauge fields a negative one. This is Sakharov’s “metrical elasticity of space” [
6
], here supplied by
the vacuum fluctuations; the curvature-squared terms
a
2
are logarithmic in Λ and suppressed at
long wavelength.
4.2 The lattice as piecewise-flat geometry
The SSM’s edge lengths define a piecewise-flat geometry: flat inside every four-simplex, with
curvature concentrated on the triangular hinges. For such a geometry the Einstein–Hilbert integral
is exactly [4, 22]
Z
√
g R = 2
X
σ
A
σ
δ
σ
, (4)
where
A
σ
is the area of hinge
σ
and
δ
σ
its deficit angle. Substituting Eq.
(4)
into Eq.
(2)
gives the
Regge action with Newton’s constant
G
ind
. Because a piecewise-flat geometry is singular at the
hinges, we verify this step directly on the singularity.
4.3 The hinge term from the conical heat kernel
Near a hinge the geometry is a flat plane times a two-dimensional cone of angle 2
πα
, with
δ
=
2
π
(1
−α
). The heat-kernel trace on a cone carries a vertex term beyond the smooth expansion. For
a scalar it is [23, 24]
Tr e
−tD
vertex
=
1
12
1
α
− α
, (5)
independent of
t
, and in four dimensions the hinge contributes
A
σ
(4
πt
)
−1
times this. We verify
Eq.
(5)
independently of the literature. For a cone-disk of radius 1 and angle 2
πα
with a Dirichlet
boundary, the spectrum is
j
2
ν,k
with
ν
=
|m|/α
, the zeros of the Bessel functions
J
ν
. We compute
it exactly, form the heat trace, remove the area and perimeter terms of the Weyl expansion, and
fit the constant, which must equal the boundary term
α/
6 plus the vertex term. Figure 1(a) and
Table 2 show the result: the vertex coefficient agrees with Eq.
(5)
to six digits at every cone angle
tested, from a deficit of 7.2
◦
to 240
◦
, and the flat disk returns the boundary value 1/6 exactly.
Inserting the vertex term into Eq. (1) gives, per hinge,
Γ
σ
= −
Λ
2
96π
A
σ
1
α
− α
= −
1
8πG
ind
A
σ
δ
σ
h
1 +
δ
σ
4π
+ O(δ
2
σ
)
i
, (6)
written for one minimally coupled scalar and normalized with its contribution to
G
ind
. The linear
term is the Regge action, with the same
G
ind
as Eq.
(2)
; this is the content of Eq.
(4)
on the
singularity itself. For every species the linear term is fixed by its smooth coefficient
a
1
, because a
5
α δ vertex, numerical
1
12
(α
−1
− α) linear (Regge) ratio
0.98 7.2
◦
0.003367 0.003367 0.003333 1.010
0.95 18
◦
0.008553 0.008553 0.008333 1.026
0.90 36
◦
0.017593 0.017593 0.016667 1.056
0.80 72
◦
0.037500 0.037500 0.033333 1.125
0.50 180
◦
0.125000 0.125000 0.083333 1.500
Table 2: Conical vertex coefficient from the exact Bessel spectrum of a cone-disk, against the closed
form, Eq. (5), and its linear (Regge) part (1 − α)/6. The ratio is (1 + α)/2α = 1 + δ/4π + O(δ
2
).
0 50 100 150 200 250
deficit angle
δ
(deg)
0.00
0.05
0.10
0.15
0.20
0.25
conical heat-kernel constant
(a) hinge term of the ocean
exact
1
12
(
α
−
1
−
α
)
Regge (linear),
1
−
α
6
from exact spectrum
SM SM
+
ν
R
SM+
ν
R
conf. H
left
right
Pati
Salam
(16 scalars)
20
10
0
10
C
in
1
/G
ind
=
C
Λ
2
/
12
π
area law
(C = 4.2 5.9)
(b) induced Newton constant
Figure 1: (a) The hinge term of the ocean: the conical heat-kernel coefficient from the exact
spectrum (circles), the closed form, Eq.
(5)
, and its linear part, which is the Regge term. (b) The
induced Newton constant, Eqs.
(2)
–
(3)
, for candidate field contents, against the range the SSM’s
area law requires under the two mode-count cutoffs (shaded).
cone with a small deficit can be smoothed into a regular geometry with
R
√
gR
= 2
Aδ
[
25
]. This
includes the curvature couplings in
E
, which on the regularized cone are supported at the tip; for
a gauge field this tip term is the contact term found by Kabat [
26
], and with it the conical and
smooth coefficients agree. The O(δ
2
) corrections are species-dependent curvature-squared terms.
Where hinges meet. The cone computation treats an isolated hinge. On the lattice, hinges
meet along edges and at vertices, and the cutoff length Λ
−1
≃ L
f
/
5–
L
f
/
4
.
2 (Sec. 5) is not far below
the lattice spacing. This does not affect the Regge term: the coefficient of
t
in the heat trace is
1
6
R
√
gR
for any metric, and the conical term is exact at linear order in each deficit, so contributions
of different hinges add. It does affect the
O
(
δ
2
) terms, which depend on how hinges are arranged.
These are curvature-squared terms, suppressed at long wavelength since
δ ∼
(
ka
)
2
h
for a metric
perturbation
h
of wavenumber
k
, but at a cutoff this close to the lattice scale their coefficients carry
order-one scheme dependence. Summing over hinges and species,
Proposition 1. Under Postulate 2, the one-loop action the vacuum fluctuations induce on the SSM
lattice is, at leading order in the deficits and the long-wavelength expansion,
S = −
1
8πG
ind
X
σ
A
σ
δ
σ
,
1
G
ind
=
CΛ
2
12π
, (7)
6
plus a cosmological term, Sec. 6, and curvature-squared corrections.
4.4 Two layers: discrete geometry and continuous fluctuations
The derivation joins two layers of different character.
The first is the lattice, which supplies spacetime’s structure and location. It is discrete: its
connectivity is fixed, with every node away from defects and grain boundaries saturated at the
kissing number, and its geometry is carried by its edge lengths. Each four-simplex is flat, and
curvature is concentrated in deficit angles on the triangular hinges, so curvature is a property of the
lengths, not of the connectivity. In the vacuum every deficit vanishes and the lattice is exactly flat.
Near a mass the edge lengths depart from their vacuum value by an amount of order
GM/
(
rc
2
),
while no bond is added, removed, or rerouted. In the SSM a defect’s elastic strain energy is negligible
compared with its verification cost, so the lattice has no appreciable stiffness of its own, and it must
not have one: a stiffness of its own would give the graviton a mass (Sec. 3).
The second layer is the vacuum fluctuations. They are continuous and carry no structure: they
are covariant field modes living on the lattice’s piecewise-flat geometry (Postulate 2). They do
not bend. Their energy depends on the geometry they live on, and that dependence supplies the
stiffness the lattice lacks. The two layers meet at the hinges. Inside each simplex the modes are
smooth, but every hinge is a conical singularity, whose vertex term, Eq.
(5)
, converts the discrete
deficit into its share of the Regge action.
The division of labor is then complete. A defect’s mass, its verification cost (Sec. 4.5), pushes
the edge lengths away from flat, producing small deficits on every hinge around it. The fluctuations’
energy rises with those deficits, with stiffness 1
/G
set by their content and cutoff. The geometry
settles where the two balance, which is Einstein’s equation at long wavelength. The response is
nonlinear, because the Regge action is nonlinear in the edge lengths, and it is long-ranged, because
mass is a conserved source and the covariant fluctuations admit no graviton mass. Structure, and
with it error correction, belongs entirely to the lattice; the fluctuations contribute only resistance to
its deformation.
Spacetime, in the sense that includes its dynamics, is the two layers together. The lattice’s
verification cycles supply the ticks of its clocks, but the relative rates of those clocks are set by its
geometry, and the geometry is fixed only through the fluctuations’ stiffness. The fluctuations also
exist where there is no lattice, before crystallization and inside vacancies, so spacetime is made from
them rather than they from it.
4.5 The source: verification cost as rest mass
The derivation so far fixes the geometry’s side of Einstein’s equation. Its other side, the source, is
a defect’s mass, which in the SSM is its verification cost [
3
]: a defect of type
x
performs a fixed
number
C
x
of verification operations per tick. We now show that this cost couples to the geometry
as a rest mass, with no further assumption about the defect.
In the SSM, time is counted in verification ticks: the triangle’s verification cycle is the first clock
(Sec. 2), and every clock since is a count of ticks. In Regge geometry, proper time along a worldline
is fixed by the edge lengths it traverses. The one physical input is to identify the two:
Postulate 3 (Ticks are proper time). The number of verification ticks along a defect’s worldline is
its proper time in units of a fixed tick duration
τ
0
, the proper time being the one the lattice’s edge
lengths assign.
7
Proposition 2. Under Postulate 3, a defect performing
C
x
verification operations per tick, each
accumulating a fixed phase φ
0
, has the worldline action
S
x
= −ℏφ
0
C
x
Z
dτ
τ
0
≡ −M
x
c
2
Z
dτ, M
x
c
2
= C
x
ε, ε =
ℏφ
0
τ
0
, (8)
where
φ
0
is fixed by the SSM’s calibration of the bond energy. The coupling to the geometry depends
on the defect only through M
x
, so gravitational and inertial mass are both the verification cost.
The proof is a count. Each verification operation is a step of a quantum process and accumulates
a fixed phase
φ
0
, an action
ℏφ
0
. The number of operations per tick,
C
x
, is a count and does not
depend on the geometry, and by Postulate 3 the number of ticks along a worldline is
R
dτ/τ
0
. The
action is
ℏφ
0
times the number of operations, Eq.
(8)
. It is the point-particle action of general
relativity, and nothing about the defect’s internal structure enters it except the count
C
x
; that is
the weak equivalence principle, obtained here from the definition of mass as verification cost rather
than assumed. The defect’s phase turns at the angular frequency
M
x
c
2
/ℏ
=
C
x
φ
0
/τ
0
: its rate of
verification sets its Compton frequency.
In the static weak-field limit
dτ
=
√
g
44
dt
, so
S
x
≃ −M
x
c
2
R
(1 +
1
2
h
44
)
dt
: the defect couples to
the geometry through
1
2
M
x
h
44
, which is exactly the source coupling used in Ref. [
5
]. Varying the
induced action, Eq.
(7)
, together with Eq.
(8)
summed over defects gives the Regge equations with
a source, and at long wavelength Einstein’s equation,
G
µν
= 8
πG
ind
T
µν
, with
T
µν
built from the
verification-cost density. Two statements follow, and they are the precise sense in which verification
cost is proportional to curvature. Locally, for slowly moving defects of cost density
ρ
, the time-time
Ricci curvature is
R
tt
= 4
πG
ind
ρ
, the local form of Poisson’s equation. Outside the defects, where
the cost density vanishes, the tidal curvature is fixed by the total enclosed cost, and on the
D
4
lattice the static field it produces has the Newtonian normalization exactly, with
β
=
γ
= 1 at
second order [5].
5 Newton’s constant
The SSM fixes
G
independently, through its area law: one bit of entanglement per bond cell,
matched to the Bekenstein–Hawking entropy, gives
G
=
L
2
f
/
(4
ln
2) [
19
]. The bonds of the spatial
FCC slice are the
D
4
roots with no time component, so
L
f
is also the
D
4
nearest-neighbor distance.
Equation (7) must agree with this value. Two inputs are needed: the cutoff and the field content.
The cutoff. We fix Λ by mode counting rather than by hand, requiring one mode per lattice
site per field component. For
D
4
with nearest-neighbor distance
L
f
the volume per site is
L
4
f
/
2.
Two identifications are natural. If the heat-kernel mode density at the cutoff, (4
π
Λ
−2
)
−2
, equals
the site density, then Λ
L
f
= (32
π
2
)
1/4
= 4
.
216. If instead the momentum ball
|p| <
Λ holds one
mode per site, as in a Debye cutoff, then Λ
4
/
32
π
2
= 2
/L
4
f
and Λ
L
f
= (64
π
2
)
1/4
= 5
.
013. The
second is compatible with our proper-time coefficient, because the quadratic divergence has the
same coefficient, Λ
2
/
16
π
2
, in proper-time and sharp-momentum regularization. The area law then
requires
C
required
=
48π ln 2
(ΛL
f
)
2
= 5.88 (heat-kernel density), 4.16 (Debye count). (9)
The spread between the two measures the scheme dependence of the comparison.
8
G
ind
/G
area
field content C heat-kernel Debye
Standard Model (45 Weyl, 12 gauge, minimal Higgs) 1 5.88 4.16
three 16 (48 Weyl), 12 gauge, minimal Higgs 4 1.47 1.04
three 16, 12 gauge, conformal Higgs (ξ =
1
6
) 0 no induced G
left–right: 15 gauge, 48 Weyl, 16 real scalars 4 1.47 1.04
Pati–Salam: 21 gauge, 48 Weyl, 16 real scalars −20 negative G
Table 3: The coefficient
C
of Eq.
(3)
and the ratio of the induced to the area-law Newton constant
under the two mode-count cutoffs, which require
C
= 5
.
88 and
C
= 4
.
16. The 16 real scalars of the
left–right model are a bidoublet and two doublets; the Pati–Salam row uses the same scalar sector.
The content. We take the SSM’s matter content to be three generations, each a Spin(10) spinor
16
, which contains one Standard Model generation and a right-handed neutrino [
27
,
28
]. The SSM
program proposes that the three generations come from the triality of the
D
4
cell, but this is not
yet derived, and we treat the content as an input. Table 3 evaluates Eq.
(3)
for this content and for
alternatives. The field masses do not enter: they shift C by amounts of order m
2
/Λ
2
≲ 10
−34
.
For the SSM’s content the induced Newton constant has the right sign and agrees with the
area-law value to within a factor 1
.
5 under either cutoff, 1
.
47 and 1
.
04, with no adjustable parameter
(Fig. 1b). The comparison also constrains content. Pati–Salam with this scalar sector induces a
negative Newton constant; it would need about 21 more real scalars for a positive one, which larger
Pati–Salam Higgs sectors can supply. A conformally coupled Higgs induces no gravity, and the
Standard Model without right-handed neutrinos underproduces it by a factor 4–6.
Two cautions keep this in proportion. First,
C
= 4 is a near-cancellation, 48 + 4
−
48, between
terms of order fifty; changing the cutoff of one species relative to another by a modest factor changes
C
at order one. The sign and the order-one agreement are the robust content; the precise ratio is
scheme-dependent, as the two cutoffs show. Second, either mode count is a principled identification,
not a derivation, of how a continuum regulator relates to the lattice spacing.
6 Consistency
With the
D
4
Regge sector. On the closed
D
4
complex the Regge action’s quadratic and cubic
terms equal Fierz–Pauli and Einstein–Hilbert exactly, and its static field gives the Newtonian
normalization and
β
=
γ
= 1 [
5
]. Those results took the Regge action as input; here it is an output,
and they carry over. The coefficient of
t
in the heat trace is
1
6
R
√
gR
for any metric, so the induced
action equals the Regge action at two derivatives to all orders in the metric perturbation, and
differs from it only at four derivatives. Every result of Ref. [
5
] is a two-derivative statement, so
each holds for the induced action, with
G
=
G
ind
. The chain is closed: the vacuum fluctuations
induce Einstein–Hilbert, which on the lattice is the Regge action, which at long wavelength is
Einstein–Hilbert again.
With the cosmological constant. The
a
0
term of Eq.
(1)
is a cosmological term of order Λ
4
times the difference of bosonic and fermionic degrees of freedom, 28 against 96 for the SSM’s content
(three copies of the
16
, the Standard Model gauge group, and a minimal Higgs). It requires the
self-tuning of a self-bound vacuum [
29
], in which the gravitating combination is
−P
and vanishes in
equilibrium; the ocean picture is natural for this mechanism, since a self-bound medium adjusts
9
to zero pressure. The mechanism acts through a density, so it can cancel a vacuum energy that
depends on volume but not one that depends on shape. That is why the shape terms of Sec. 3 are
fatal for lattice-regulated fields, and why Postulate 2 matters here too: under it no shape term
arises, and the vacuum energy that remains is exactly the kind self-tuning can remove.
With the matter coupling. Defects source the geometry through their center-of-mass worldlines
with the action of Proposition 2, whose static limit is the coupling assumed in Ref. [
5
]; the vacuum
fluctuations respond to the geometry. The far field depends only on the total mass, by the Newtonian
normalization of Ref. [
5
], so the internal structure of a defect does not enter. Because the stabilizer
vacuum has exactly zero energy for any couplings (Sec. 3), a defect’s verification cost is its whole
energy, with no vacuum offset to subtract, so the mass that sources the geometry is unambiguous.
With Lorentz invariance. The induced action is covariant, but the lattice geometry it acts on
is not: the Regge action on
D
4
carries anisotropic corrections at relative order (
ka
)
2
, tabulated
for the static field in Ref. [
5
]. Lorentz violation therefore enters only through the lattice geometry,
suppressed by (E/E
P
)
2
at energy E, and not through the vacuum fluctuations themselves.
With the lattice no-go. Section 3 showed that lattice-regulated fields induce a graviton mass.
Under Postulate 2 the shape terms vanish by symmetry, and no balance between bosons and fermions
is needed for the graviton to stay massless. Balance still enters, through
C
and the cosmological
term, but as a condition on magnitudes, not on covariance.
7 Limitations and status
Table 4 grades each claim. The main limitations are these.
The postulates. Postulate 1 is the model’s premise: that the fluctuations persist is observed, but
that they predate spacetime is not directly testable. Postulate 2 is assumed. The derivation of the
action moves the assumption from the Regge action to the covariance of the vacuum fluctuations,
which is better grounded, since the observed vacuum is Lorentz invariant, but it is not derived
from the lattice. Its weakest joint is that it asks for a regulator covariant in form but set at a
scale the lattice defines; how a Lorentz-invariant cutoff comes to sit at the lattice spacing is not
explained here. That the lattice has no appreciable stiffness of its own (Sec. 4.4) is taken from
the SSM, not rederived here, though Sec. 3 shows it is required. Postulate 3, which identifies the
count of verification ticks with the proper time of the Regge geometry, is the one input of the source
derivation; it is natural in a model that defines time by those ticks, but it is assumed.
Newton’s constant. The value of
G
ind
is reliable to a factor of a few, so the agreement to within
a factor 1
.
5 in Sec. 5 is partly fortuitous; its robust content is the sign and the order of magnitude.
The two mode counts of Sec. 5 differ by a factor 1
.
41 in
G
, and
C
is a near-cancellation, sensitive to
the relative cutoffs of different species. The field content is an input: neither the three generations
nor the scalar sector is yet derived in the SSM, and each real scalar changes C by one.
What is omitted. Graviton loops and the fluctuations of the lattice’s own geometry are omitted;
in induced gravity the graviton is not a fundamental field of the ocean. The
O
(
δ
2
) hinge corrections
are computed for a scalar at an isolated hinge only, and the phase per operation
φ
0
in Proposition 2
is fixed by the SSM’s calibration rather than derived. The cosmological term is removed by imported
10
claim status
vacuum fluctuations exist now measured [12, 13, 14]
fluctuations predate spacetime (Postulate 1) premise, not directly testable
saturation rule: no new bonds inside space kissing-number theorems [17, 18]
first-triangle account of crystallization interpretation
lattice has no appreciable stiffness of its own (Sec. 4.4) SSM result; required by Sec. 3
stabilizer sector induces no action exact (commuting checks)
lattice-regulated fields induce a graviton mass (Table 1) computed
Sakharov coefficients, Eq. (3) standard heat kernel, rederived
R
√
gR = 2
P
Aδ on piecewise-flat geometry exact identity
conical vertex coefficient, Eq. (5) computed from the exact spectrum
Postulate 2 (covariance) assumed
Regge action induced, Proposition 1 derived, given Postulate 2
Postulate 3 (ticks are proper time) assumed
verification cost couples as rest mass, Proposition 2 derived, given Postulate 3
field content: three 16, scalar sector input, not derived in the SSM
G
ind
against the area law
sign and order of magnitude robust; fac-
tor 1.5 scheme-dependent
two-derivative D
4
results hold for the induced action follows (Sec. 6)
Lorentz violation only through the lattice geometry follows (Sec. 6)
cosmological term cancelled imported self-tuning [29]
Table 4: Status of the paper’s claims, in the order of the paper: premises, the no-go, the derivations,
and their consistency.
self-tuning [
29
], not derived. Whether the ocean restitches at the boundary of a vacancy, in
competition with its evaporation, is left open. The induced-action computations are Euclidean
and one-loop, and the account of the ocean before the lattice is interpretation, not used in either
derivation.
8 Conclusion
The vacuum fluctuations are measured, and in the Selection–Stitch Model we take them to be older
than space: the lattice and its defects are made of them, and the existing lattice excludes new
stitching by the rule that built it. Gravity then need not be postulated as Regge calculus. If the
fluctuations around the lattice are covariant, integrating them out induces the Einstein–Hilbert
action, which on the lattice’s piecewise-flat geometry is exactly the Regge action; the conical heat
kernel confirms this on the hinge itself, with a curvature-squared correction of relative size
δ/
4
π
.
A defect’s verification cost, counted in the ticks that define time, couples as a rest mass, so it is
the source of that curvature and the equivalence principle follows. The fluctuations cannot instead
be the lattice’s own fields, whose induced vacuum energy depends on the shape of the lattice and
gives the graviton a mass. The induced Newton constant, with the cutoff set by the lattice’s mode
density and three generations of the Spin(10) spinor taken as the SSM’s content, has the right sign
and agrees with the SSM’s area-law value to within a factor of 1
.
5 under either mode count, and
Pati–Salam content with a minimal scalar sector is excluded. The lattice is what bends, the defects
are what bend it, and the vacuum’s fluctuations are what resist. The most consequential open
question is where that picture meets its own boundary: whether the fluctuations restitch the edge
of a vacancy faster than it evaporates.
11
A Heat-kernel coefficients
For
D
=
−
(
∇
2
+
E
) acting on a bundle of rank
n
,
a
1
=
tr
(
E
) +
nR/
6. A real scalar with coupling
ξR
has
E
=
−ξR
, so
a
1
= (
1
6
−ξ
)
R
. A Dirac fermion has
D/
2
=
−∇
2
+
R/
4 on four components, so
a
1
= 4(
1
6
−
1
4
)
R
=
−
1
3
R
, and its effective action
−
1
2
log det D/
2
carries
s
=
−
1; a Weyl fermion is half
of this. A gauge field in Feynman gauge has
D
=
−∇
2
+
Ric
on vectors, so
a
1
=
4
6
R − R
=
−
1
3
R
,
and its two Faddeev–Popov ghosts subtract 2
×
1
6
R
, giving
−
2
3
R
with
s
= +1. Relative to a minimal
real scalar (
a
1
=
1
6
R
,
s
= +1), the contributions to
C
are 1
−
6
ξ
, 2, 1, and
−
4 for a scalar, Dirac
fermion, Weyl fermion, and gauge field. The cosmological coefficient
a
0
=
tr
1 counts degrees of
freedom with the sign
s
: a real scalar counts 1, a gauge field 4
−
2 = 2 after its ghosts, and a Weyl
fermion 2 with the opposite sign. For the SSM’s content (three copies of the
16
, the Standard
Model gauge group, and a minimal Higgs) this gives 4 + 12
×
2 = 28 bosonic against 48
×
2 = 96
fermionic degrees of freedom.
B Reproducibility
The archive
ocean regge.zip
contains the scripts for every number in this paper.
cone heat.py
computes the Bessel spectra of cone-disks and fits the heat-kernel constant (Table 2);
newton.py
evaluates Eq.
(3)
under both mode-count cutoffs (Table 3);
make figs.py
draws Fig. 1. The
lattice results of Table 1 are reproduced by
induced k0.py
(scalar),
rotor k0.py
(gauge field),
fermion k0.py
(Wilson–Dirac),
geometric control.py
(continuum kinetic tensor), and
root p.py
and
rotor p2.py
(the exponents
p
∗
). All scripts require only Python 3 with numpy, scipy, and
matplotlib, and the archive’s README.md gives the command for each.
Declarations
Funding. No funding was received for this study.
Competing interests. The author declares no competing interests.
Use of AI tools. During the preparation of this work, the author used Claude Opus 5.5 (Anthropic)
to write and run the verification and analysis scripts, generate the figures, analyze the numerical
results, and draft and edit the manuscript text and L
A
T
E
X source. The author has reviewed and
edited the output and takes full responsibility for the content of this publication.
Data availability. All code is openly available in the archive
ocean regge.zip
at
https://
github.com/raghu91302/ssmtheory (Appendix B); no experimental data were used.
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