Macroscopic Imprints of a Discrete Vacuum:
Deriving the CMB Hemispherical Power Asymmetry
from K=12 Crystallization Kinematics
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
May 2026
Abstract
Standard continuous inflation predicts a vanishing dipole modulation amplitude (
A
= 0) for the
primordial scalar power spectrum. The Planck satellite measures a persistent hemispherical
power asymmetry at large angular scales (
ℓ ≤
64) with amplitude
A
= 0
.
066
±
0
.
021
[
7
,
1
]. We derive a zero-parameter geometric origin for this anomaly from the Selection-
Stitch Model (SSM), a
K=
12 face-centered cubic vacuum framework developed by Kulkarni
(arXiv:2603.20294); (Phys. Open 27, 100423). The vacuum emerges through a
K=
6
→
K=
4
→ K=
12 topological phase cascade, with the 3D metric generated behind a superluminal
2D causal phase front. The tunneling probability per lattice site for the 2D-to-3D lift is
p
=
e
−S
E
, where
S
E
= 3 is the Euclidean bounce action of the tetrahedral defect, equal to
the three skew-edge pairs of its bounding geometry. This gives
p
=
e
−3
≈
0
.
0498 [
3
]. The
tunneling probability enforces a fractional age gradient ∆
t/t
= 2
p
across the causal horizon.
Poisson defect-density scaling and the
√
n
amplitude statistics of additive sources translate
this gradient into a hemispherical modulation of the primordial scalar fluctuation amplitude,
A
=
p ≈
0
.
0498. The background temperature itself thermalizes through Thomson scattering
before recombination and carries no fossil of the asynchronous crystallization; the asymmetry
resides in the amplitude of the
δT/T ∼
10
−5
fluctuations, not in the 2.725 K monopole. The
prediction falls within 0
.
8
σ
of the Planck measurement. The derivation is analytically exact
and verified by direct simulation.
Keywords: Cosmic Microwave Background; hemispherical power asymmetry; discrete spacetime;
topological phase transitions; tensor networks.
1 Introduction
A central pillar of the standard ΛCDM cosmological model is the assumption of a continuous
spacetime manifold undergoing a period of exponential inflation. Standard inflation invokes a
spatially homogeneous scalar field to causally connect disparate regions of the primordial plasma,
and the time delay between the onset of cooling for any two spatially separated regions is forced
to zero (∆
t
(
d
) = 0). Standard inflation therefore predicts a perfectly isotropic CMB with a null
linear temperature gradient (∆T = 0).
Precision observations challenge this. The Planck satellite has confirmed a persistent, scale-
dependent hemispherical power asymmetry (a dipole modulation of the temperature fluctuation
amplitude) at large angular scales (
ℓ ≤
64), measured at
A
= 0
.
066
±
0
.
021 [
7
]. This signal is
inconsistent with
A
= 0 at greater than 3
σ
. No mechanism within standard continuous slow-roll
inflation produces this signal without invoking finely tuned, ad hoc secondary fields [1].
1
This paper demonstrates that the anomaly emerges natively from the discrete vacuum of the
Selection-Stitch Model (SSM). The framework models the emergence of space not as continuous
expansion but as a topological crystallization front propagating through a discrete network
[
6
,
4
]. The asymmetry resides in the amplitude of the primordial fluctuations (
δT/T ∼
10
−5
),
not in the 2.725 K background monopole. The background temperature thermalizes through
Thomson scattering with the ionized plasma well before recombination and retains no fossil of
the asynchronous crystallization. The defect-density distribution that sources the primordial
perturbations does not thermalize, and it carries the age-gradient signature into the observed
power spectrum.
2 The Crystallization Front: Physical Mechanism
In the SSM, the pre-geometric vacuum is a 2D hexagonal (
K=
6) entanglement network, the
topological ground state with zero curvature and maximum entropy [
6
]. The
K=
6 sheet is itself
reached from a
K=
1 Bell-pair seed via the stitch operator of the matter paper [
6
]; the present
paper focuses on the
K=
6
→ K=
12 transition that generates 3D spacetime. Three-dimensional
space does not pre-exist. It is generated by a topological phase transition.
Definition 1 (The
K=
6
→ K=
4
→ K=
12 Phase Transition). The generation of the 3D metric
proceeds via three stages:
1.
Nucleation. A fluctuation creates a tetrahedral (
K=
4) defect in the 2D sheet. This defect
has a Regge deficit angle
δ
= 2
π −
5
arccos
(1
/
3)
≈
0
.
128 rad, which stores elastic energy and
creates a local bulge into the third dimension.
2.
Propagation. A crystallization front propagates laterally across the 2D sheet. At each lattice
site reached by the front, two processes compete: lateral stitching (extending in-plane to the
next
K=
6 site without a tunneling barrier) and vertical lifting (generating a new layer via
ABC stacking, requiring tunneling through a potential barrier).
3.
Saturation. Tetrahedral clusters accumulate and pack into the densest 3D arrangement, the
FCC lattice with K=12, saturating the Kepler bound [2].
The SSM crystallization front replaces standard slow-roll inflation. The 2D causal front
propagates superluminally relative to the 3D metric it generates, thermalizing the primordial
network without requiring an inflaton field. Unlike standard inflation (where
v
=
∞
and ∆
t
= 0),
the SSM front has a finite superluminal velocity. This finite velocity produces a nonzero age
gradient across the observable universe.
3 The Tunneling Probability and Velocity Ratio
Theorem 1 (The Coleman Tunneling Amplitude and
S
E
= 3). The probability per lattice site
that a vertical 3D lift occurs is p = e
−3
≈ 0.0498.
Proof.
The probability per lattice site for a vertical (3D) lift follows the semiclassical tunneling
formula [3]:
p = e
−S
E
, (1)
where
S
E
is the Euclidean bounce action for the vacuum transition at the defect site. For the SSM
lattice, the bounce action follows directly from the geometry of the tetrahedral defect. The defect
possesses
c
skew
= 3 skew-edge pairs in the bounding tetrahedron (the same combinatorial invariant
that produces three color charges in the matter sector [
6
]). Each skew-edge pair contributes a
2
unit action to the tunneling barrier. The Euclidean action of the bounce solution is therefore the
topological invariant:
S
E
= c
skew
= 3. (2)
This equals the number of independent rotational degrees of freedom that must be overcome to
lift a 2D site into 3D. The tunneling probability is p = e
−3
.
Theorem 2 (The 3D Generation Velocity). The macroscopic generation velocity of the 3D spatial
bulk, v
3D
, is related to the 2D causal front velocity, v
front
, by v
3D
= p · v
front
.
Proof.
At each lattice site reached by the crystallization front, the site undergoes a vertical lift
with probability
p
. The lateral front advances at speed
v
front
, visiting sites at a rate
v
front
/L
per
unit time. Of these, a fraction
p
generate 3D volume. The effective 3D generation velocity is
therefore
v
3D
= p × v
front
. (3)
The exact velocity ratio is
v
3D
/v
front
=
p
p
2/3
due to the ABC stacking height
h
=
p
2/3 L
. The
geometric factor
p
2/3
cancels identically when computing the fractional age gradient ∆
t/t
, so it
can be absorbed into the definition v
3D
= p · v
front
.
Theorem 3 (Superluminal Phase Velocity and Causality). The 3D metric wall advances at
the speed of causation (
v
3D
=
c
), so the 2D causal phase front propagates superluminally at
v
front
≈ 20.1 c.
Proof.
By construction, the 3D metric is the lattice itself, and the boundary of the lattice is the
boundary of causally connected spacetime. The lattice-front propagation speed therefore sets the
cosmological speed limit:
v
3D
= c. (4)
This identification of the speed of light with the lattice propagation speed is a defining postulate
of the framework. The 2D causal front speed then follows geometrically from Theorem 2:
v
front
=
c
p
= c e
3
≈ 20.1 c. (5)
This is superluminal but no violation occurs: the 2D front propagates on the pre-geometric
boundary sheet, not through the 3D metric. The 3D spacetime in which
c
is the speed limit
exists only after the vertical lift. Every region of the observable universe was causally connected
on the 2D sheet (at 20
.
1
c
) before the 3D metric locked in (at
c
). This is how the SSM solves the
horizon problem without an inflaton field.
4 Derivation of the Macroscopic Age Gradient
Because
v
front
is finite, the 2D-to-3D topological transition enforces a distance-dependent time
delay.
Theorem 4 (The Horizon Age Gradient). The fractional age difference ∆
t/t
across the maximum
diameter of the observable 3D universe is exactly ∆t/t = 2p.
Proof.
Consider the 3D-crystallized patch at a cosmic time
t
after the onset of crystallization. Its
physical diameter follows from the expansion of the 3D metric wall:
d = 2 v
3D
t. (6)
3
The 2D crystallization front must traverse this exact diameter to reach the antipodal edge. The
time delay between Point A (where crystallization began) and Point B (the antipodal edge) is
∆t =
d
v
front
=
2 v
3D
t
v
front
. (7)
Substituting v
3D
= p · v
front
(equation 3):
∆t
t
= 2p = 2e
−3
≈ 0.0996. (8)
A kinematic cancellation:
v
front
cancels identically. The fractional age gradient depends only on
the discrete tunneling probability
p
. The tunneling probability
p
=
e
−3
is a topological invariant
of the tetrahedral defect. It is the same for every sheet at every epoch because every
K=
6
hexagonal layer presents the identical local geometry at the defect site. This universality makes
the age gradient a strict, permanent constant.
5 Translation to a CMB Power Asymmetry
The age gradient ∆
t/t
= 2
p
from Section 4 must couple to an observable in the CMB. Two routes
are conceivable. The gradient could modulate the mean background temperature
T
0
= 2
.
725 K,
producing an intrinsic temperature dipole of order
pT
0
∼
100 mK. This route is excluded by
observation: the measured CMB dipole is only 3
.
36 mK and is entirely accounted for by the
Solar System’s kinematic motion relative to the CMB rest frame, leaving no room for an intrinsic
primordial dipole at this level. The alternative route, which we adopt here, couples the age
gradient to the amplitude of the primordial fluctuations (
δT/T ∼
10
−5
) rather than to the
background monopole.
Why the background carries no fossil of the gradient
Before recombination, photons and the ionized plasma are tightly coupled by Thomson scattering
at large optical depths [
8
]. The mean temperature equilibrates locally on the Thomson time,
which at
z ≈
10
3
is many orders of magnitude shorter than the Hubble time. Any background
temperature gradient set up by the asynchronous crystallization erases through this equilibration
well before recombination. The observed
T
0
= 2
.
725 K monopole is the result of this thermalized
background and retains no age-gradient signature.
The primordial fluctuation amplitudes do not thermalize. They are statistical patterns im-
printed on the matter and photon distributions by the defect-creation process during crystallization.
Thomson scattering redistributes energy locally to maintain a common mean temperature, but it
does not erase the variance of perturbations around that mean. The defect-density distribution at
recombination retains the age-gradient signature, and the resulting fluctuation amplitudes carry
it into the observed power spectrum.
Defect density scales linearly with lattice age
In the SSM, primordial scalar perturbations are sourced by topological defects produced during
crystallization. Modeling the nucleation as a Poisson process with constant rate
λ
per unit volume
per unit time, the defect density at lattice age t is
n
def
(t) = λ t. (9)
4
This linear scaling is the Poisson default. It does not require any parameter choice beyond the
existence of a constant nucleation rate. The horizon-scale age gradient ∆
t/t
= 2
p
from Theorem
4 therefore imprints a defect-density gradient of identical fractional magnitude:
∆n
def
n
def
=
∆t
t
= 2p. (10)
From defect density to perturbation amplitude
A perturbation field
δ
(
x
) =
P
i
f
(
x − x
i
) built additively from
N
independent Poisson sources at
positions {x
i
} has variance proportional to the source density:
⟨|δ(x)|
2
⟩ = n
def
(x)
Z
|f|
2
d
3
x
′
=⇒ δ
rms
(x) ∝
p
n
def
(x). (11)
The fractional amplitude modulation is therefore half the fractional density modulation:
∆δ
rms
δ
rms
=
1
2
∆n
def
n
def
=
1
2
× 2p = p. (12)
Identification with the Planck modulation amplitude
The Planck hemispherical power asymmetry is defined by a dipole modulation of the primordial
fluctuation field [7]:
δT (ˆn) = (1 + A ˆn · ˆp ) δT
iso
(ˆn), (13)
where
ˆp
is the preferred axis and
A
is the amplitude variation across that axis. The SSM
mechanism produces exactly this kind of modulation through Equation (12):
Theorem 5 (The Hemispherical Power Asymmetry Amplitude). The hemispherical power
asymmetry amplitude predicted by SSM crystallization kinematics is
A
SSM
= p = e
−3
≈ 0.0498. (14)
The factor of 1
/
2 in Equation
(12)
comes from the
√
n
scaling of fluctuation amplitudes
for additive Poisson sources, not from any cosmological equation of state. The factor of 2 in
∆
n/n
= 2
p
comes from the diametric geometry of the horizon (Theorem 4). The two factors
cancel exactly, leaving A = p as a parameter-free structural prediction.
6 Comparison to Planck Observations
Table 1: Predicted hemispherical power asymmetry compared with observation. The SSM prediction lies
0.8σ from the Planck central value. Standard inflation predicts A = 0, excluded at greater than 3σ.
Model Dipole modulation A
Standard continuous inflation (v = ∞) 0.000 (exact)
SSM K=12 lattice (p = e
−3
) 0.0498
Planck 2015 measurement [7] 0.066 ± 0.021
The parameter-free SSM prediction of
A ≈
0
.
0498 resides within the 1
σ
bounds of the Planck
observation. Three features of this geometric derivation resolve secondary CMB anomalies.
5
Scale dependence. The age gradient is coherent on horizon scales, so it imprints a coherent
amplitude modulation only at low multipoles (
ℓ ≤
64, corresponding to angular scales above a few
degrees). For sub-horizon modes (high
ℓ
), a single mode-wavelength contains many independent
sub-regions of the lattice, each with its own local age, and the coherent gradient averages out.
The framework therefore naturally predicts that the asymmetry decays at high
ℓ
, in agreement
with the observed scale dependence [1, 7].
Preferred axis. The propagation direction of the initial 2D crystallization front spontaneously
breaks the
SO
(3) rotational symmetry of the primordial vacuum, and the first 3D lift explicitly
breaks spatial parity. This provides a natural, pre-geometric origin for the observed preferred
axis.
Horizon problem. As proven in Theorem 3, the 2D front propagates at
v
front
≈
20
.
1
c
,
acting as a superluminal causal conduit that thermalizes the primordial network before 3D lock-in.
The Thomson-coupled background continues to equilibrate after lock-in, erasing any residual
mean-temperature gradient.
7 Discussion
Sensitivity to
S
E
. The prediction
A
=
e
−S
E
depends strictly on the bounce action. If
S
E
were
2 instead of 3, the prediction would be
A
=
e
−2
≈
0
.
135, outside the Planck 2
σ
band. If
S
E
= 4,
then
A
=
e
−4
≈
0
.
018, only marginally consistent. The value
S
E
= 3 (the skew-edge count of the
tetrahedral bounding geometry, also responsible for the three color charges in the matter sector
[6, 5]) is the unique integer that falls within 1σ of the Planck measurement.
Independence from cosmological parameters. The derivation uses no continuous
cosmological parameters (
H
0
, Ω
m
, Ω
Λ
). The result
A
=
p
depends only on the discrete topology
of the lattice defect and on the linear Poisson defect scaling n
def
(t) ∝ t.
Poisson defect scaling. The linear time dependence
n
def
(
t
) =
λt
in Equation
(9)
is the
natural first-principles assumption for a constant-rate Poisson nucleation process. Alternative
scalings would alter the prediction. If
n
def
∝ t
1/2
, then ∆
n/n
=
p
and
A
=
p/
2; if
n
def
∝ t
2
, then
∆
n/n
= 4
p
and
A
= 2
p
. The linear scaling is the unique form consistent with a memoryless
nucleation process at constant rate, and it produces the observed
A ≈ p
agreement with no
further parameter choice.
Background versus fluctuation. A mean-temperature interpretation of the age gradient
would predict an intrinsic CMB dipole of order
pT
0
≈
100 mK, in conflict with the observed
kinematic (3
.
36 mK) dipole. The framework instead couples the gradient to the defect-density dis-
tribution that sources the primordial fluctuations. Thomson scattering thermalizes the background
but not the variance of imprinted perturbations, so the asymmetry survives in the fluctuation
power spectrum while leaving the monopole unmodulated.
Falsifiability. The SSM predicts
A
= 0
.
0498
±
0 (no scatter from the model). Future CMB
measurements (LiteBIRD, CMB-S4) with smaller error bars will either confirm or exclude this
value. If the true
A
is measured to be greater than 0
.
08 at 3
σ
, the SSM prediction is falsified. A
measurement of an intrinsic primordial CMB monopole dipole at the
∼
100 mK level would also
falsify the present framework, since the model predicts that the background does not retain the
age gradient.
6
8 Conclusion
The hemispherical power asymmetry of the CMB is the macroscopic kinematic fossil of a discrete
vacuum phase transition. The derivation proceeds in four exact steps:
1.
The vacuum crystallizes from 2D (
K=
6) to 3D (
K=
12) via a topological phase transition
with tunneling probability p = e
−3
per site.
2.
The 3D metric grows at speed
v
3D
=
p · v
front
, creating a fractional age gradient ∆
t/t
= 2
p
across the horizon.
3.
Linear Poisson defect scaling (
n
def
∝ t
) and the
√
n
amplitude statistics of additive sources
translate the age gradient into a fractional amplitude modulation ∆
δ/δ
=
p
across the
horizon, while the background temperature itself thermalizes through Thomson scattering
and carries no fossil of the gradient.
4.
The hemispherical power asymmetry amplitude equals the microscopic tunneling probability:
A = p = e
−3
≈ 0.0498.
The prediction, derived from lattice topology alone with zero free parameters, falls within
0
.
8
σ
of the Planck measurement (
A
= 0
.
066
±
0
.
021), while standard inflation predicts
A
= 0,
excluded at greater than 3σ.
References
[1]
Eriksen, H. K., et al. (2004). Asymmetries in the cosmic microwave background anisotropy
field. Astrophysical Journal 605, 14. doi:10.1086/382267.
[2] Hales, T. C. (2005). A proof of the Kepler conjecture. Annals of Mathematics 162, 1065.
[3]
Coleman, S. (1977). Fate of the false vacuum: Semiclassical theory. Physical Review D 15,
2929. doi:10.1103/PhysRevD.15.2929.
[4]
Kulkarni, R. (2026a). A 67%-rate CSS code on the FCC lattice: [[192
,
130
,
3]] from weight-12
stabilizers. arXiv:2603.20294.
[5]
Kulkarni, R. (2026b). The Mass-Energy-Information Equivalence: A bottom-up identifica-
tion of the particle spectrum via FCC lattice error correction. Physics Open 27, 100414.
doi:10.1016/j.physo.2026.100414.
[6]
Kulkarni, R. (2026c). Matter as incomplete crystallization: Quark charges, color confinement,
and the proton mass from a single extra node in the vacuum lattice. Physics Open 27, 100423.
doi:10.1016/j.physo.2026.100423.
[7]
Planck Collaboration (2016). Planck 2015 results. XVI. Isotropy and statistics of the CMB.
Astronomy & Astrophysics 594, A16. doi:10.1051/0004-6361/201525830.
[8] Weinberg, S. (1972). Gravitation and Cosmology. Wiley.
7
A Computational Verification
The analytic result
A
=
p
=
e
−3
is verified by direct numerical simulation of the crystallization
kinematics.
Setup. A one-dimensional lattice of
N
= 900 sites models the lateral axis of the crystallization
front. The front propagates at normalized speed
v
front
= 1, crystallizing site
x
at time
t
c
(
x
) =
x/v
front
. At each crystallized site, vertical (3D) growth proceeds at rate
v
3D
=
p ×v
front
, giving a
local 3D diameter
D
3D
(
x
) = 2
v
3D
(
t
obs
−t
c
(
x
)). Under the Poisson defect-accumulation model of
Section 5, the perturbation amplitude sourced by lattice defects at observation time
t
obs
scales as
δ
rms
(
x
)
∝
p
n
def
(x) ∝
(
t
obs
− t
c
(
x
))
1/2
at sites where the local lattice age has been accumulating
defects at the constant Poisson rate. The hemispherical asymmetry at any observation point
x
0
is
A(x
0
) =
|δ
rms
(x
0
− D
3D
/2) − δ
rms
(x
0
+ D
3D
/2)|
⟨δ
rms
⟩
. (15)
Deterministic result. Scanning across 70 independent observation points yields a mean
asymmetry
A
= 0
.
0498
±
0
.
0005, matching the analytic prediction
p
=
e
−3
= 0
.
0498 to within
0.1%.
Calibration. Varying the tunneling probability over two orders of magnitude (
p
= 0
.
005 to
p = 0.30) confirms that the simulation reproduces A = p to four significant figures at low p and
to better than 2
.
5% throughout the scanned range (Figure 1b). The linear relationship is not
an input. It is an emergent output of the diametric horizon geometry combined with the
√
n
defect-amplitude scaling.
Stochastic verification. Replacing the deterministic vertical growth rate with a stochastic
Binomial
(
n, p
) tunneling process over 200 Monte Carlo runs gives
A
= 0
.
0496
±
0
.
0090, confirming
that shot noise from individual tunneling events does not alter the systematic gradient.
0 50 100 150 200 250 300
Lateral position
x
0
10
20
30
40
50
60
Vertical position
z
D
3D
=2
pt
front
position
(a) Crystallization age map
0.00 0.05 0.10 0.15 0.20 0.25 0.30
Tunneling probability
p
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Asymmetry
A
(b) Calibration:
A
=
p
=
e
−
3
A
=
p
Planck:
0
.
066
±
0
.
021
Simulation
p
=
e
−
3
Stochastic (
N
= 200
)
0
50
100
150
200
250
Age (since crystallization)
Figure 1: (a) Crystallization age map in the
xz
-plane. The front propagates laterally (rightward) at
speed
v
front
; vertical 3D growth occurs at rate
v
3D
=
p v
front
. Darker shading indicates older regions,
which have accumulated more lattice defects and therefore source larger primordial fluctuation amplitudes.
The white arrow marks the 3D diameter
D
3D
= 2
pt
. (b) Calibration curve: simulated asymmetry
A
versus tunneling probability
p
. Filled circles, deterministic simulation at 10 values of
p
. Marker with error
bar, stochastic Monte Carlo (
N
= 200) at
p
=
e
−3
. Star, the SSM prediction
A
=
p
=
e
−3
≈
0
.
0498. The
red band shows the Planck measurement
A
= 0
.
066
±
0
.
021, within which the SSM prediction falls at
0.8σ. Standard ΛCDM predicts A = 0, excluded by Planck at greater than 3σ.
Source code. The Python code reproducing the results above is available at
https://github.com/raghu91302/ssmtheory/blob/main/cmb_asymmetry_simulation.py
8
The script uses NumPy, builds the 1D lattice, applies the crystallization front and tunneling
kinematics described in Section 3, and computes the amplitude asymmetry under the Poisson
defect-scaling hypothesis of Section 5. Output reproduces the deterministic and stochastic results
above in seconds on a standard laptop.
9
