Macroscopic Imprints of a Discrete Vacuum Deriving the CMB Hemispherical Power Asymmetry from K = 12 Crystallization Kinematics

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

Abstract

The standard cosmological model sources the primordial density perturbations

from inﬂation, which predicts a statistically isotropic sky and hence a vanishing

dipole modulation amplitude (

= 0) for the scalar power spectrum. The

Planck satellite instead measures a persistent hemispherical power asymmetry

at large angular scales (

ℓ ≤

64) with amplitude

= 0

066

021 [

]. We

show that this early-universe anomaly admits a zero-parameter geometric

origin if the primordial perturbations are seeded not by an inﬂaton but by a

discrete crystallization of the vacuum—the Selection-Stitch Model (SSM), a

12 face-centered-cubic framework [

]. The vacuum emerges through

→ K=

12 topological phase cascade, with the 3D metric

generated behind a superluminal 2D causal front that replaces inﬂation as the

mechanism thermalizing the observable universe. The 2D-to-3D lift tunnels

with probability

−S

, where

= 3 is the Euclidean bounce action of the

tetrahedral defect, equal to its three skew-edge pairs, giving

−3

≈

0498

[

]. The ﬁnite front speed enforces a fractional age gradient ∆

t/t

= 2

between antipodal directions; Poisson defect-density scaling and

√

amplitude

statistics turn this into a dipole modulation of the primordial ﬂuctuation

amplitude with coeﬃcient

. We show this dipole survives a canonical

(Mukhanov–Sasaki) quantization—it is a modulation of the source amplitude

rather than the mode dispersion—so it does not suﬀer the sign reversal that

aﬄicts a quadrupole from the same geometry. The background temperature

thermalizes through Thomson scattering before recombination and carries

no fossil of the gradient, so the asymmetry resides in the

δT/T ∼

−5

Email address: raghu@idrive.com (Raghu Kulkarni)

Preprint submitted to Physics of the Dark Universe June 8, 2026

ﬂuctuations and not in the 2.725 K monopole; the fundamental constants

remain spatially uniform, leaving the acoustic-peak structure intact and

distinguishing the mechanism from spatially-varying-coupling models. With

ﬁxed by geometry rather than ﬁtted, the amplitude is parameter-free:

the geometric lift probability

−3

gives

A ≈

0249, while the realized

kinematic lift rate (

∼

5%) gives

A ≈

0125, so

A ∈

012

025]—a few-

percent dipole,

∼

2–2

from the Planck value, where inﬂation gives exactly

zero. The derivation is analytic, and the dipole relation

is conﬁrmed

by direct simulation of the crystallization kinematics.

Keywords: Early universe, Cosmic microwave background, Hemispherical

power asymmetry, Discrete spacetime, Topological phase transitions

1. Introduction

A central pillar of the standard ΛCDM cosmological model is the assump-

tion 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 (∆

(

) = 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 mod-

ulation of the temperature ﬂuctuation amplitude) at large angular scales

(

ℓ ≤

64), measured at

= 0

066

021 [

]. This signal is inconsistent with

= 0 at greater than 3

. No mechanism within standard continuous slow-roll

inﬂation produces this signal without invoking ﬁnely tuned, ad hoc secondary

ﬁelds [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 [

]. The

asymmetry resides in the amplitude of the primordial ﬂuctuations (

δT/T ∼

−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. Acting on the ﬂuctuation amplitude alone, the modulation leaves

the fundamental constants spatially uniform, and with them the recombination

physics and the acoustic-peak structure. Section 7 works this out and draws

from it a test that tells the present mechanism apart from spatially varying-

coupling models. In replacing the inﬂaton as the source of primordial structure,

the mechanism is a statement about the initial conditions of the early universe,

and it is in that context that we develop and test it against the observed

CMB anomaly.

2. The Crystallization Front: Physical Mechanism

In the SSM, the pre-geometric vacuum is a 2D hexagonal (

6) entangle-

ment network, the topological ground state with zero curvature and maximum

entropy [

]. The

6 sheet is itself reached from a

1 Bell-pair seed via

the stitch operator of the matter paper [

]; the present paper focuses on the

→ 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=

12 Phase Transition). The genera-

tion of the 3D metric proceeds via three stages:

Nucleation. A ﬂuctuation creates a tetrahedral (

4) defect in the

2D sheet. This defect has a Regge deﬁcit angle

= 2

π −

arccos

≈

128 rad, which stores elastic energy and creates a local bulge into the

third dimension.

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

6 site without a

tunneling barrier) and vertical lifting (generating a new layer via ABC

stacking, requiring tunneling through a potential barrier).

Saturation. Tetrahedral clusters accumulate and pack into the densest

3D arrangement, the FCC lattice with

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

∞

and ∆

= 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

= 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

, (1)

where

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

skew

= 3 skew-edge

pairs in the bounding tetrahedron (the same combinatorial invariant that

produces three color charges in the matter sector [

]). Each skew-edge pair

contributes a unit action to the tunneling barrier. The Euclidean action of

the bounce solution is therefore the topological invariant:

= 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,

, is related to the 2D causal front velocity,

front

, by v

= p · v

front

Proof.

At each lattice site reached by the crystallization front, the site under-

goes a vertical lift with probability

. The lateral front advances at speed

front

, visiting sites at a rate

front

per unit time. Of these, a fraction

generate 3D volume. The eﬀective 3D generation velocity is therefore

= p × v

front

. (3)

The regular-tetrahedron factor

2/3

set by the close-packing geometry enters

the lateral stitch step and the vertical lift step alike, so it cancels in the ratio

front

; the dimensionless lift frequency p is all that survives.

Theorem 3 (Superluminal Phase Velocity and Causality). The 3D metric

wall advances at the speed of causation (

), 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:

= 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:

front

= c e

≈ 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

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

)

before the 3D metric locked in (at

). This is how the SSM solves the horizon

problem without an inﬂaton ﬁeld.

4. Derivation of the Macroscopic Age Gradient

Because

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

after the onset

of crystallization. Its physical diameter follows from the expansion of the 3D

metric wall:

d = 2 v

t. (6)

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 =

front

2 v

front

. (7)

Substituting v

= p · v

front

(equation 3):

∆t

= 2p = 2e

−3

≈ 0.0996. (8)

A kinematic cancellation:

front

cancels identically. The fractional age gra-

dient depends only on the discrete tunneling probability

. The tunneling

probability

−3

is a topological invariant of the tetrahedral defect. It is

the same for every sheet at every epoch because every

6 hexagonal layer

presents the identical local geometry at the defect site. This universality

makes the age gradient a strict, permanent constant.

A useful way to picture Eq.

(8)

is as a staircase. Each layer’s lateral front

is a tread, run at the fast speed

front

; the stack of layers rises along the slow

riser at

. A patch far across the diameter is reached later only by the

accumulated riser delay, while the fast tread washes the rest of the gradient

out. The residual is the ratio of the two speeds,

front

, across each

half-diameter — hence 2

across the whole. The smallness of the gradient

is precisely the imprint of the fast tread; a riser-only (

front

→v

) staircase

would give an order-unity modulation, not a few percent.

5. Translation to a CMB Power Asymmetry

The age gradient ∆

t/t

= 2

from Section 4 must couple to an observable in

the CMB. Two routes are conceivable. The gradient could modulate the mean

background temperature

= 2

725 K, producing an intrinsic temperature

dipole of order

∼

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 ∼

−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 [

]. The mean temperature

equilibrates locally on the Thomson time, which at

z ≈

is many orders of

magnitude shorter than the Hubble time. Any background temperature gradi-

ent set up by the asynchronous crystallization erases through this equilibration

well before recombination. The observed

= 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 sta-

tistical patterns imprinted on the matter and photon distributions by the

defect-creation process during crystallization. Thomson scattering redis-

tributes 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

per unit volume per unit time, the defect density

at lattice age t is

def

(t) = λ t. (9)

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

from Theorem 4 therefore imprints a defect-density

gradient of identical fractional magnitude:

∆n

def

∆t

= 2p. (10)

From defect density to perturbation amplitude

Write the defect density as a function of viewing direction

ˆn

. The age

gradient of Theorem 4 spans ∆

t/t

= 2

between antipodal directions, so to

ﬁrst order

def

(ˆn) = ¯n

def



1 + p ˆn · ˆp



, (11)

whose antipodal contrast [

def

(

ˆp

)

− n

def

(

−ˆp

)]

/¯n

def

= 2

reproduces Eq.

(10)

A perturbation ﬁeld

(

) =

(

x − x

) built additively from independent

Poisson sources has, by Campbell’s theorem, variance proportional to the

source density,

⟨|δ|

⟩ ∝ n

def

, so

rms

∝

√

def

(this statistical step is examined

in Section 6). Taking the square root of Eq. (11) and expanding,

rms

(ˆn) ∝



1 + p ˆn · ˆp



1/2

≈ 1 +

p ˆn · ˆp. (12)

The amplitude thus carries a dipole whose coeﬃcient is

2: the

√

statistics

halve the density dipole once more.

Identiﬁcation with the Planck modulation amplitude

The Planck hemispherical power asymmetry is a dipole modulation of the

primordial ﬂuctuation ﬁeld [7],

δT (ˆn) = (1 + A ˆn · ˆp ) δT

iso

(ˆn), (13)

with

ˆp

the preferred axis and

the coeﬃcient of

ˆn · ˆp

. Matching term by

term against Eq. (12) ﬁxes A as the SSM amplitude-dipole coeﬃcient:

Theorem 5 (The Hemispherical Power Asymmetry Amplitude). The hemi-

spherical power asymmetry amplitude predicted by SSM crystallization kine-

matics is

SSM

p =

−3

≈ 0.0249. (14)

The scale is set by two halvings. The antipodal age span 2

(Theorem 4)

is, as a dipole coeﬃcient,

; the

√

statistics of additive Poisson sources

halve it again to

2. No cosmological parameter and no ﬁt enters: once

is ﬁxed, A is ﬁxed.

Equation

(14)

is the maximal amplitude. It is attained when the ob-

server’s horizon spans the crystallization seed out to the antipodal edge, the

conﬁguration assumed in Theorem 4. For an observer displaced from the

seed by less than the horizon, the age proﬁle across the sky is less steep, and

an observer at the seed itself sees a radially symmetric proﬁle whose dipole

vanishes (leaving only a monopole). The quoted

−3

is therefore an

upper value, and the preferred axis points back toward the nucleation seed.

A generic observer is displaced from that seed, so a nonzero dipole of order

−3

is the expected case rather than a ﬁne-tuned one.

6. Survival of the Dipole under Quantization

A modulation claimed at the level of generation kinematics must be shown

to survive the ﬁeld quantization rather than being asserted there. We make

that case here, and ﬁnd that the dipole—unlike a quadrupole sourced by

anisotropic mode propagation—passes through intact, for a structural reason.

The distinction is where the modulation enters. A quantity that appears

in the mode equation—for instance an anisotropic propagation speed

(

)

multiplying the gradient term—is reprocessed by the horizon-crossing dynam-

ics: the Bunch–Davies freeze-out integral can rescale it and even reverse its

sign, so such a quantity cannot be carried linearly into the power spectrum.

The hemispherical dipole does not enter the mode equation. The age gradient

changes how many defects seed perturbations at a given location—the source

amplitude—while leaving the propagation of each perturbation untouched.

No coeﬃcient in the mode equation becomes direction-dependent, and there

is no freeze-out integral to invert the result.

The load-bearing statistical step is then a theorem, not an assumption.

For a ﬁeld

(x) =

−

) summed over a Poisson point process of local

intensity n

def

(x), Campbell’s theorem gives the exact ﬂuctuation variance

Var[δ(x)] = n

def

(x)

|f(r)|

r, (15)

so the local power is strictly proportional to the source number density and

the local amplitude to its square root. Counting statistics of discrete quanta

are Poisson whether the nucleation events are treated classically or quantum-

mechanically, so quantization does not modify Eq.

(15)

; a direct numerical

sum of identical kernels over Poisson points reproduces

Var ∝ n

def

to sampling

scatter. Because the modulation wavelength is the horizon scale—far larger

than the sub-horizon modes whose power it modulates—each patch of sky

inherits its local amplitude with no scale mixing. Equations

(15)

and

(12)

then return

directly, with no sign inversion: the power is larger toward

the older hemisphere, where more defects have accumulated.

This is the same linear-response logic as the standard inﬂationary treat-

ment of the hemispherical asymmetry [

], in which a long-wavelength mod-

ulation of the ﬁeld that sets the small-scale power imprints a dipole on the

observed amplitude. There the modulating ﬁeld is a superhorizon curvaton or

inﬂaton mode; here it is the defect-density gradient ﬁxed by the crystallization

age. The mechanism is established; only the origin of the modulating gradient

is new.

One limitation must be stated plainly. Campbell’s theorem ﬁxes how

the modulation propagates, not the shape of the unmodulated spectrum. A

pure white shot-noise source yields

(

)

∝ const

; recovering the observed

near-scale-invariant tilt

≃

96 requires additional structure in the source

kernel

(r) or in the defect correlations that we do not derive here. The

dipole result is independent of that structure, but a complete account of the

base spectrum shape remains an open problem for the framework.

7. Spatial Uniformity of the Coupling Sector

The modulation of Section 5 acts on the amplitude of the primordial

scalar spectrum. Does it act on anything else? The question that matters

for the CMB is whether it disturbs the recombination physics that ﬁxes

the acoustic peaks. To settle it, sort the lattice observables into two kinds.

Some are read oﬀ the present, saturated

12 cluster at a site; call these

instantaneous. Others build up over the time a site has spent crystallized;

call these integrated. The constants

and

are of the ﬁrst kind, the

defect abundance n

def

(t) = λt of the second:

α, m

, m

, σ

| {z }

instantaneous: f (saturated K=12 cluster)

vs. n

def

(t) = λt

| {z }

integrated: f (elapsed age)

. (16)

A site the front has just reached settles into local

12 cuboctahedral

coordination within one lift time and stays there. Its saturated geometry

keeps no record of the arrival time. The age gradient of Section 4 is then a

gradient in how many defects a region has nucleated since the front swept

past, not in the local geometry that sets the constants. That is why the

modulation stays in the amplitude and goes nowhere else.

Particle masses are spatially uniform. A defect mass in the SSM is

kT ln

, where

is an integer topological invariant of the local

cluster [

]. Two things hold

equal across the two hemispheres. The

integer

is built only from the saturated-cluster counts

12,

K+

13 and

skew

3, which any fully crystallized region shares no matter when it formed.

And the vacuum temperature

that ﬁxes the absolute scale is a property

of the lattice itself, shared by every excitation of the single vacuum [

];

like the saturated geometry it carries no dependence on local crystallization

age. (This

is the Planck-scale vacuum temperature of the mass relation,

not the recombination-era photon temperature of Section 5; the two should

not be conﬂated.) An age-independent

at a common

leaves

age-

independent as well; the ratios

drop

altogether and stay

integer everywhere [

]. The electron mass, which sets the hydrogen binding

energy and so the redshift of recombination, is therefore the same wherever

one looks.

The electromagnetic coupling is spatially uniform. In the lattice

the electromagnetic sector lives on the six bipartite (square) faces of the

cuboctahedral shell, which carry the propagating dipole modes; the eight non-

bipartite (triangular) faces conﬁne instead [

]. That face structure belongs to

the saturated

12 cluster, and like the mass integers it does not depend on

crystallization age. We record this as a structural postulate:

Postulate 1 (Coupling locality). The ﬁne-structure constant

is a functional

of the saturated local

12 coordination cluster, speciﬁcally of its bipartite

electromagnetic-channel structure, and not of the integrated defect history at

the site.

With the postulate in place,

takes on the spatial uniformity of the cluster it

depends on. The leading hemispherical variation

δα/α

from the age gradient

drops out, since that gradient sits in the integrated defect count of Eq.

(16)

rather than in the channel geometry that ﬁxes

. Nothing here pins down

the numerical value of

; the postulate speaks only to its spatial dependence,

which is what the present argument needs.

Consequence: a pure amplitude modulation. If

and

are spatially uniform, so are the recombination history, the sound horizon,

the damping scale and the radiation transfer function ∆

ℓ

(

). The dipole

of Eq.

(14)

is then carried by the primordial amplitude

alone, and the

spectrum seen along ˆn reads

ℓ

(ˆn) =



1 + A ˆn · ˆp



iso

ℓ

, (17)

one isotropic spectrum

iso

ℓ

rescaled by a single direction-dependent number.

Peak positions do not move and peak-height ratios do not change. What is

left is a hemispherical power asymmetry and nothing further, which is the

reason the high-ℓ spectrum stays clean.

Distinction from varying-coupling models. A coupling that varies

in space leaves a diﬀerent mark. A dipolar

(

ˆn

) at last scattering shifts the

recombination history and the damping scale, smooths the power spectrum,

and seeds a non-Gaussian (trispectrum) signal much like weak lensing or

a compensated isocurvature perturbation [

]. Searches tuned to that

signature have come back empty. The dipolar amplitude of

over the Hubble

volume sits at

δα/α

= (

−

−2

[

], and the scale-invariant

ﬂuctuation power obeys

≤

−5

over the

L ≲

100 band that bears

on the present signal [

]. The mechanism here clears these limits on its own

terms: Eq.

(17)

carries no anomalous peak-smoothing and no

-trispectrum.

Because the two pictures part ways inside the same low-

ℓ

band, the diﬀerence

can be measured. Should a future map turn up an

-trispectrum, or dipolar

peak-smoothing lined up with the power-asymmetry axis, the amplitude-only

mechanism proposed here would give way to a varying-coupling one.

The modulation is in the curvature amplitude, not the mean

baryon abundance. What the age gradient modulates is the variance of

the defect-sourced curvature perturbation, the amplitude

of Eq.

(17)

. It

does not touch the mean baryon abundance

Ω

, an all-sky monopole set by

the global nucleation rate and not by its dipole. The model therefore raises

no hemispherical Ω

gradient and keeps the odd–even peak-height ratio, the

usual baryon-density diagnostic, isotropic. On this point too the SSM signal

parts from a compensated isocurvature perturbation, which trades baryons

against dark matter at ﬁxed total density; here it is the amplitude alone that

moves.

8. Comparison to Planck Observations

Table 1: Predicted hemispherical power asymmetry compared with observation. The

SSM prediction lies about 2

below the Planck central value—a few-percent dipole of the

observed order. 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 (A =

−3

) 0.0249

Planck 2015 measurement [7] 0.066 ± 0.021

The parameter-free SSM prediction

A ≈

0249 lies about 2

below

the Planck central value—the right order of magnitude for a few-percent

eﬀect, and nonzero where standard inﬂation gives exactly zero. Beyond the

amplitude, three features of the geometric derivation bear on secondary CMB

anomalies.

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

(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

front

≈

, 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.

9. Discussion

Role of

. The amplitude scales as

−S

, so it is set entirely by the

bounce action.

is not adjustable here: it is the skew-edge count

skew

= 3

of the tetrahedral defect, the same invariant that ﬁxes the three color charges

in the matter sector [

]. With that geometric input,

−3

≈

0249, a

few-percent dipole consistent with Planck at about 2

. We stress that

is ﬁxed before any comparison with the CMB; it is not tuned to the data.

For reference,

= 2 would give 0

068 and

= 4 would give 0

009, but the

tetrahedral geometry selects 3, and we do not adjust it to improve the ﬁt.

The lift rate and the predicted range. The amplitude is

with

the per-step out-of-plane (lift) probability. Two readings of

bracket the

prediction. Identifying

with the bare geometric tunneling factor

−3

≈

0498

gives the nominal

A ≈

0249 (

∼

). A kinematic-survival analysis of the

same growth model, in which solitary lifts are under-protected and dissolve,

yields a lower realized lift rate of order 2

5%, and hence

A ≈

0125 (

∼

The robust content of the prediction is therefore its form,

with

a few percent, giving

A ∈

0125

0249]—a few-percent dipole,

∼

2–2

from Planck, where inﬂation predicts exactly zero. We quote

−3

as the

nominal value while noting this range explicitly.

Independence from cosmological parameters. The derivation uses

no continuous cosmological parameters (

, Ω

). The result

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

def

(

) =

λt

Equation

(9)

is the natural ﬁrst-principles assumption for a constant-rate

Poisson nucleation process. A general scaling

def

∝ t

gives

: thus

would give

4 and

= 2 would give

. We adopt the linear

law because it is the unique memoryless constant-rate form, not because any

particular exponent improves agreement with the data.

Background versus ﬂuctuation. A mean-temperature interpretation of

the age gradient would predict an intrinsic CMB dipole of order

≈

100 mK,

in conﬂict with the observed kinematic (3

36 mK) dipole. The framework

instead couples the gradient to the defect-density distribution 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 maximal

−3

≈

0249, attained

for an observer displaced from the crystallization seed by of order the horizon.

Future CMB measurements (LiteBIRD, CMB-S4) with smaller error bars will

tighten the comparison. A determination placing

well above this ceiling—

for instance establishing

A >

05 at 3

—would falsify the prediction. A

measurement of an intrinsic primordial CMB monopole dipole at the

∼

100 mK

level would also falsify the framework, since the model predicts that the

background does not retain the age gradient. The mechanism also predicts

a null

-trispectrum and no anomalous peak-smoothing aligned with the

asymmetry axis (Section 7). Turning up any of these, or a hemispherical

gradient in the fundamental constants themselves, would point to a varying-

coupling origin and rule out the amplitude-only account given here.

10. 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:

The vacuum crystallizes from 2D (

6) to 3D (

12) via a topological

phase transition with tunneling probability p = e

−3

per site.

The 3D metric grows at speed

p · v

front

, creating a fractional age

gradient ∆t/t = 2p across the horizon.

Linear Poisson defect scaling (

def

∝ t

) and the

√

amplitude statistics

of additive sources translate the age gradient into an antipodal amplitude

contrast ∆

δ/δ

, while the background temperature itself thermalizes

through Thomson scattering and carries no fossil of the gradient.

The hemispherical dipole amplitude is half that antipodal contrast:

p =

−3

≈ 0.0249.

The prediction, derived from lattice topology alone with zero free param-

eters, sits about 2

from the Planck central value (

= 0

066

021)—a

few-percent amplitude of the observed order, nonzero where standard inﬂation

predicts exactly zero.

References

[1]

Eriksen, H. K., et al. (2004). Asymmetries in the cosmic mi-

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[2]

Hales, T. C. (2005). A proof of the Kepler conjecture. Annals of Mathe-

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[3]

Coleman, S. (1977). Fate of the false vacuum: Semiclassical theory. Physi-

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[4]

Kulkarni, R. (2026a). A 67%-rate CSS code on the FCC lattice:

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[5]

Kulkarni, R. (2026b). The Mass-Energy-Information Equivalence: A

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[6]

Kulkarni, R. (2026c). Matter as incomplete crystallization: Quark charges,

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vacuum lattice. Physics Open 27, 100423. doi:10.1016/j.physo.2026.100423.

[7]

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[9]

O’Bryan, J., Smidt, J., De Bernardis, F., & Cooray, A. (2013). Con-

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[10]

Planck Collaboration (2015). Planck intermediate results. XXIV. Con-

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[11]

Smith, T. L., Grin, D., Robinson, D., & Qi, D. (2019). Probing spatial

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[12]

Erickcek, A. L., Kamionkowski, M., & Carroll, S. M. (2008). A hemi-

spherical power asymmetry from inﬂation. Physical Review D 78, 123520.

doi:10.1103/PhysRevD.78.123520.

Appendix A. Computational Veriﬁcation

The amplitude relation is veriﬁed by direct numerical simulation of the

crystallization kinematics. The simulation measures the antipodal amplitude

contrast ∆

δ/δ

directly; the dipole amplitude reported as

is half of it,

A =

∆δ/δ (Eq. (14)).

Setup. A one-dimensional lattice of

= 900 sites models the lateral

axis of the crystallization front. The front propagates at normalized speed

front

= 1, crystallizing site

at time

(

) =

x/v

front

. At each crystallized

site, vertical (3D) growth proceeds at rate

p × v

front

, giving a local 3D

diameter

(

) = 2

(

obs

−t

(

)). Under the Poisson defect-accumulation

model of Section 5, the perturbation amplitude sourced by lattice defects at

observation time

obs

scales as

rms

(

)

∝

def

(x) ∝

(

obs

− t

(

))

1/2

at sites

where the local lattice age has been accumulating defects at the constant

Poisson rate. The antipodal amplitude contrast at any observation point

∆δ

) =

|δ

rms

− D

/2) − δ

rms

+ D

/2)|

⟨δ

rms

⟩

, A =

∆δ

. (A.1)

Deterministic result. Scanning across 70 independent observation

points yields a mean antipodal contrast ∆

δ/δ

= 0

0498

0005, matching

−3

= 0

0498 to within 0

1%; the corresponding dipole amplitude is

A =

∆δ/δ = 0.0249.

Calibration. Varying the tunneling probability over two orders of mag-

nitude (

= 0

005 to

= 0

30) conﬁrms that the simulation reproduces

∆

δ/δ

(equivalently

2) to four signiﬁcant ﬁgures at low

and to

better than 2

5% throughout the scanned range (Figure A.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 ∆

δ/δ

= 0

0496

0090 (

= 0

0248

0045), conﬁrming that

shot noise from individual tunneling events does not alter the systematic

gradient.

Source code. The Python code reproducing the results above is available

https://github.com/raghu91302/ssmtheory/blob/main/cmb_asymmetr

y_simulation.py

0 50 100 150 200 250 300

Lateral position

Vertical position

front

position

(a) Crystallization age map

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Tunneling probability

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

Dipole amplitude

(b) Calibration:

2 =

−

Planck:

066

021

Simulation

−

Stochastic (

= 200

)

100

150

200

250

Age (since crystallization)

Figure A.1: (a) Crystallization age map in the

-plane. The front propagates laterally

(rightward) at speed

front

; vertical 3D growth occurs at rate

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

= 2

. (b) Calibration curve: the simulated dipole amplitude

∆

δ/δ

versus

tunneling probability

, tracing

2. Filled circles, deterministic simulation at 10

values of

. Marker with error bar, stochastic Monte Carlo (

= 200) at

−3

. Star,

the SSM prediction

−3

≈

0249. The red band shows the Planck measurement

= 0

066

021; the SSM prediction sits about 2

below it. Standard ΛCDM predicts

A = 0, excluded by Planck at greater than 3σ.

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.