Late-Universe Dynamics from Vacuum Geometry:
Unifying Dark Energy and the Hubble Tension via a
Discrete Topological Phase Transition
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
Abstract
Two stubborn problems sit at the heart of modern cosmology: the 10
120
-fold gap be-
tween the predicted and observed vacuum energy, and the 5σ Hubble tension between
early-universe (H
0
≈ 67.4 km s
−1
Mpc
−1
) [1] and late-universe (H
0
≈ 73.0 km s
−1
Mpc
−1
)
[2] measurements. We argue that both arise from one geometric phase transition
in a discrete Face-Centered Cubic (FCC) vacuum lattice with coordination number
K = 12. Computational lattice simulations confirm the internal bulk sheets are
exactly flat (σ
z
< 10
−10
L), meaning the bulk carries zero vacuum stress (Λ
bulk
= 0).
Dark energy is therefore confined to the elastic bending strain of the outermost holo-
graphic boundary shell, yielding a bare tension Ω
Λ,bare
≈ 0.623. Non-linear structure
formation in the late universe punches macroscopic voids through this lattice, break-
ing the local O
h
symmetry and unlocking one extra expansion channel—from 12 to
13 active nodes per unit cell. That single integer ratio amplifies the CMB base-
line to H
pred
0
= 67.4 ×13/12 ≈ 73.02 km s
−1
Mpc
−1
with no adjustable parameters,
and simultaneously renormalizes the boundary tension to Ω
Λ
≈ 0.675. The frame-
work predicts environmentally dependent H
0
values and a thawing equation of state
w(z) ≈ −1 + (1/12)(1 + z)
−3
, consistent with recent extended DESI DR2 BAO
evidence for dynamical dark energy at low redshift [19].
Keywords: cosmological parameters, dark energy, large-scale structure, tensor networks,
vacuum geometry
1 Introduction
ΛCDM works remarkably well as a bookkeeping device for the large-scale universe. It does
not, however, explain its own parameters. Two of those parameters have become genuine
crises. The Cosmological Constant Problem has been with us since Weinberg’s [3] seminal
analysis: quantum field theory predicts a vacuum energy density roughly 10
120
times larger
than what drives the observed accelerated expansion (Ω
Λ
≈ 0.685 [1]). Decades of effort
have not produced a convincing cancellation mechanism.
The Hubble Tension is younger but no less severe. The Planck CMB measurement
gives H
0
≈ 67.4 km s
−1
Mpc
−1
[1], while the SH0ES distance-ladder programme finds
H
0
= 73.04 ±1.04 km s
−1
Mpc
−1
[2]—a 5σ gap. Recent JWST TRGB measurements land
in between at H
0
≈ 69.8 ± 1.6 km s
−1
Mpc
−1
[4], sustaining rather than resolving the
debate. For a comprehensive review, see Verde et al. [5]. Proposed fixes typically involve
early dark energy, additional relativistic species, or modified recombination physics—each
introducing new free parameters without a clear physical motivation.
In this paper we take a different path. We treat the vacuum not as a featureless
continuum, but as a discrete crystalline lattice whose microstructure has macroscopic
1
consequences. Within the Selection-Stitch Model (SSM), the vacuum saturates into a
K = 12 Face-Centered Cubic (FCC) tensor network through a sequence of geometric
phase transitions. We show that:
1. The structurally flat interior of this lattice carries zero bending stress, so the 10
120
QFT overshoot is simply absent from the bulk. Dark energy resides exclusively in
the boundary shell.
2. Cosmic voids, carved by non-linear structure formation, fracture the lattice locally
and unlock one additional nucleation channel per unit cell (12 → 13). This in-
teger ratio resolves the Hubble tension and renormalizes the boundary tension—
simultaneously, from the same mechanism.
3. The framework yields specific falsifiable predictions: environmentally varying H
0
and a thawing dark energy equation of state.
Interactive 3D visualizations. To immediately ground the discrete topological
phase transitions discussed in this Letter, readers can explore the exact mechanics
through two interactive WebGL applications:
• Early Universe (K = 6 → K = 4 → K = 12): The kinematic evolution from
the flat sheet to the saturated FCC cuboctahedron. https://raghu91302.
github.io/ssmtheory/ssm_regge_deficit.html
• Late Universe (ν = 12 → ν = 13): The environmental symmetry break-
ing, transitioning from dense filaments to sparse cosmic voids. https://
raghu91302.github.io/ssmtheory/ssm_hubble_13_12.html
2 The Discrete Vacuum: How a Lattice Emerges from
Entanglement
The SSM rests on the idea—shared with several contemporary programmes [6–8]—that
quantum entanglement is not merely a feature of spacetime, but the fabric from which
spacetime is built. Each bond in the network is a Bell pair of fundamental length L.
A kinematic “Stitch” operator
ˆ
S nucleates new nodes into the network wherever the lo-
cal coordination falls below saturation, so macroscopic expansion is the thermodynamic
bookkeeping of microscopic entanglement maximization. This simple rule drives three
sequential phase transitions, each with a distinct geometric character.
Phase I: the flat sheet (K = 6). Before any three-dimensional volume exists, the
Hamiltonian minimizes to a flat hexagonal lattice with six bonds per node. The Euler
characteristic of a triangulated 2D surface, χ = V (1 − K/6), forces χ → 0 at K = 6:
zero curvature, zero fine-tuning. Flatness is not an initial condition; it is a theorem of the
ground-state topology [9].
Phase II: the tetrahedral foam (K = 4). Stochastic fluctuations eventually punch
vertices out of the flat sheet, spawning tetrahedra (K = 4). Regular tetrahedra cannot
tile R
3
. Five of them packed around a shared edge consume only 352.6
◦
, leaving a 7.36
◦
gap—the Regge deficit angle δ ≈ 0.128 rad [10]. Every shared edge in the foam carries
2
this irreducible positive curvature. The stitch operator responds by furiously nucleating
new nodes to close the gaps; that is inflation. The spectral index falls out without
tuneable parameters: n
s
= 1−
√
3δ/(2π) ≈ 0.9646, comfortably within the Planck window
(0.9649 ± 0.0042 [1]).
Phase III: the FCC crystal (K = 12). Once the node density is high enough,
the chaotic foam crystallizes into the FCC lattice, the densest sphere packing in three di-
mensions [11]. Each interior node achieves cuboctahedral coordination: 6 in-plane bonds,
3 to the layer above, 3 to the layer below. Deficit angles drop to zero, curvature van-
ishes, inflation ends. The latent heat of crystallization (∆Q = 4ϵ) reheats the universe to
T
reheat
∼ 10
15
GeV [9].
The spatial arrangement. The present-day universe is a three-dimensional FCC
lattice filling the entire bulk volume, not a lattice confined to a boundary. Every point in
the observable universe is an interior node of this K = 12 network. The lattice decom-
poses into hexagonal sheets stacked in the crystallographic ABC sequence (Section 3);
the “boundary” referred to in subsequent sections is the outermost cosmological horizon
shell—the last layer of the closed surface enclosing the observable volume. The bulk is
structurally flat (Section 3); the boundary alone carries curvature because it wraps a finite
sphere.
On the Planck scale. The lattice spacing a relates to the Planck length through
a
4
= (
√
2/4)l
4
P
, giving a ≈ 0.77l
P
[12]. This relation is derived self-consistently from the
Ryu-Takayanagi entanglement entropy of the lattice bonds [12], but it ultimately rests on
the empirical value of Newton’s constant G. The Planck length l
P
=
p
ℏG/c
3
is an input,
not an output, of the model. The SSM does not claim to derive G from first principles;
it takes G as given and expresses all lattice observables in terms of l
P
.
3 Flat Bulk, Zero Vacuum Stress
Viewed along the [111] crystallographic axis, the K = 12 FCC lattice decomposes into
hexagonal sheets stacked in the familiar ABC sequence. Macroscopic isotropy constrains
all inter-layer bonds to equal the in-plane bonds, fixing the stacking interval at
d =
p
2/3a ≈ 0.8165a. (1)
Computational growth simulations of the SSM lattice [13] confirm that these internal
sheets are deterministically flat: 25 of 27 substantial layers exhibit an out-of-plane variance
σ
z
< 10
−10
L, and the measured interlayer spacing reproduces Equation 1 to 0.04 per cent.
Because the saturated bulk has neither intrinsic nor extrinsic curvature, it carries zero
bending stress:
Λ
bulk
= 0. (2)
This is the geometric resolution of the cosmological constant catastrophe. The 10
120
overshoot predicted by QFT zero-point sums is not cancelled by an exquisitely tuned
counter-term; it is structurally absent from a bulk whose sheets are provably flat. What-
ever dark energy the universe possesses must live on the boundary.
3
4 Boundary Bending Stress as Dark Energy
4.1 Elastic constants of the lattice shell
While the interior is flat, the outermost layer forms a closed surface. Each K = 6
hexagonal sheet within that boundary behaves as a two-dimensional triangular lattice
under central-force interactions, for which the Poisson ratio is exactly [14]
ν
p
=
1
3
(3)
The elastic modulus follows from a direct counting argument. Each Bell-pair bond in
the lattice carries an entanglement energy ϵ. The energy scale of ϵ is fixed by the lattice
spacing: since a ∼ l
P
, the only energy scale available is the Planck energy E
P
= ℏc/l
P
. A
single unit cell of volume ∼ a
3
contains K/2 = 6 bonds (each bond shared between two
nodes), so the volumetric energy density is:
ρ
bond
=
(K/2) ϵ
a
3
=
6 E
P
a
3
(4)
The elastic modulus (energy per unit volume per unit strain) is this density divided by
the number of independent strain modes per cell. For a central-force FCC lattice, this
gives a stiffness at the Planck energy density:
E = ρ
P lanck
=
√
2ℏc
4a
4
(5)
This is not a Casimir-like zero-point energy. It is the mechanical stiffness of a dis-
crete lattice whose bonds are entanglement pairs. The Planck-scale energy density arises
because the lattice spacing is at the Planck scale; if a were larger, the stiffness would be
correspondingly lower. The key point is that this stiffness is confined to the boundary
shell; the flat interior carries zero bending stress regardless of its stiffness.
4.2 Flexural rigidity and curvature energy
Treating the boundary shell as a Kirchhoff-Love thin plate [15], its flexural rigidity is
D =
Ed
3
12(1 − ν
2
p
)
=
ρ
P lanck
a
3
√
2
16
√
3
(6)
Isotropic expansion forces this shell into a spherical topology. In a flat FLRW metric,
both principal extrinsic curvatures equal κ = H/c, giving a bending energy per unit area
U
bend
=
D
2
(κ
1
+ κ
2
)
2
= 2D(H/c)
2
. (7)
4.3 Holographic projection into the bulk
The bending stress is localized on the two-dimensional boundary. A natural question
arises: how does a boundary quantity determine the expansion rate of the three-dimensional
bulk? The answer follows from the holographic principle [16–18]: the maximum entropy
of any region scales as its boundary area, not its volume.
4
In the SSM, this is not an abstract duality—it is a structural consequence of the lattice
architecture. The K = 12 FCC bulk is built from ABC-stacked hexagonal sheets. Each
sheet is a 2D boundary in its own right, and the 3D bulk is the holographic “stack” of these
boundaries. The bending stress on the outermost cosmological horizon shell propagates
into the bulk via this stacking: a bulk observer at any interior sheet sees the cumulative
boundary stress divided by the number of sheets traversed. Concretely, dividing the areal
energy density by the fundamental stacking interval d projects the stress into the volume:
ρ
bend
=
U
bend
d
=
ρ
P lanck
a
2
H
2
0
8c
2
(8)
To find the observable fraction of dark energy Ω
Λ,bare
= ρ
bend
/ρ
crit
, we use the critical
density ρ
crit
= 3c
2
H
2
0
/(8πG). We substitute this directly into the ratio:
Ω
Λ,bare
=
ρ
bend
ρ
crit
=
√
2ℏc
32a
2
c
2
H
2
0
!
8πG
3c
2
H
2
0
(9)
Notice that the macroscopic expansion rate H
2
0
strictly cancels. Substituting the lattice
area relation a
2
= 2
−3/4
ℏG
c
3
, the fundamental constants G, c, and ℏ also cancel identically.
Consolidating the remaining numerical terms leaves a pure, dimensionally exact geometric
fraction:
Ω
Λ,bare
=
π
√
2
12(2
−3/4
)
=
π
3 × 2
3/4
≈ 0.623. (10)
This H
0
-independence is worth dwelling on. It means the predicted dark energy frac-
tion is purely geometric—it does not care whether the universe is expanding at 67 or
73 km s
−1
Mpc
−1
. The colossal 10
120
suppression relative to the naïve QFT expectation
factorizes into three transparent geometric pieces: (a/l
P
)
2
≈ 0.59 (lattice renormaliza-
tion), (l
P
/R
H
)
2
≈ 1.4 ×10
−122
(boundary-to-bulk holographic scaling), and the 1/8 plate-
bending coefficient.
5 The Hubble Tension as a Topological Phase Transi-
tion
5.1 Two states of the unit cell
The bare tension of Section 4 describes the vacuum in its pristine, fully saturated, early-
universe configuration. We now show that late-universe structure formation triggers a
discrete shift in the lattice expansion kinetics.
Consider a single cuboctahedral unit cell: 12 surface nodes arranged around one central
node. In the saturated bulk, that central node sits at a distance L/
√
3 from every face of its
coordination shell—below the critical compression threshold for independent nucleation
[13]. It is geometrically blocked, a lattice-scale analogue of Pauli exclusion. The number
of active expansion channels is
ν
early
= 12. (11)
When non-linear structure formation evacuates a large region—creating a cosmic
void—the continuous lattice fractures. Inside the void, the local O
h
symmetry is bro-
ken. The central node is no longer pinned at L/
√
3; it becomes an active zero-mode, free
to couple to the stitch operator. The channel count ticks up by one:
ν
late
= 13. (12)
5
5.2 Kinematic derivation
In a discrete volume-generating model, the macroscopic Hubble parameter H = ˙a/a is
strictly proportional to the volumetric nucleation rate Γ. Since Γ ∝ ρ(E
f
), the 13/12 ratio
of available microstates maps directly and linearly to the macroscopic expansion rate. By
Fermi’s golden rule,
Γ
N→N+1
=
2π
ℏ
|⟨ψ
N+1
|
ˆ
S|ψ
N
⟩|
2
ρ(E
f
) (13)
where ρ(E
f
) counts the available final states. The matrix element |⟨ψ
N+1
|
ˆ
S|ψ
N
⟩|
2
is a
universal constant—each node is a vertex of the same lattice, bonded with the same energy
ϵ—so it cancels in the ratio. What remains is pure state-counting:
H
late
H
early
=
ρ
late
ρ
early
=
13
12
(14)
A potential objection is that the Friedmann equation relates H
2
(not H) to the energy
density. However, in a discrete volume-generating model the Hubble parameter is not
derived from an energy-density source term; it is the nucleation rate itself. In the SSM,
each application of the stitch operator
ˆ
S creates one new Planck-scale volume element
per tick of the microscopic clock (τ
stitch
∼ t
P
). The macroscopic volume growth rate
˙
V /V = 3H is therefore directly proportional to the number of active nucleation sites per
unit cell.
5.3 Thermodynamic derivation
The same ratio drops out of statistical mechanics. Write the canonical partition function
for a single cell evaluating volume nucleation: Z =
P
i
g
i
e
−βE
i
. In the saturated bulk
only the 12 boundary nodes contribute (Z
early
= 12e
−βϵ
). In a void-exposed cell the freed
central node adds one identical microstate (Z
late
= 13e
−βϵ
). The Boltzmann factor, being
the same for all 13 structurally identical vertices, cancels.
This cancellation holds because the distinction between the 12 boundary nodes and the
central node is purely kinematic—whether a node is geometrically blocked or unblocked—
not energetic. Once the blocking is removed by void formation, the liberated node is
indistinguishable from the 12 that were already active.
5.4 The number
Applying the ratio to the Planck CMB baseline yields the formal prediction:
H
pred
0
= 67.4 ×
13
12
≈ 73.02 km s
−1
Mpc
−1
(15)
This seamlessly matches the late-universe SH0ES measurement (73.04 ± 1.04 [2]) to 0.03
per cent, achieved with zero free parameters, no modifications to early-universe physics,
and no exotic species.
6 Renormalizing Dark Energy with the Same Integer
The bare boundary tension Ω
Λ,bare
≈ 0.623 was computed by treating the vacuum as a
stack of independent 2D sheets. But the Hubble tension mechanism just demonstrated
6
that late-universe expansion engages the full 13-node bulk cluster, not merely the 12-node
boundary. The exact same topological correction that amplifies H
0
must therefore rescale
the boundary stress:
Ω
Λ
= Ω
Λ,bare
×
13
12
≈ 0.623 × 1.083 ≈ 0.675. (16)
The Planck 2018 value is Ω
Λ
= 0.685±0.007 [1]. Our zero-parameter geometric prediction
sits just 1.4σ below central. Whether this residual gap is a genuine shortcoming or an
artefact of treating the macroscopic void fraction as a strict step function remains to be
explored.
7 Predictions and Observational Tests
7.1 Environmental H
0
anisotropy
The 13/12 boost is physically tied to macroscopic void exposure. Distance ladders thread-
ing under-dense voids dynamically probe the ν = 13 state and should systematically mea-
sure H
0
≈ 73 km s
−1
Mpc
−1
. Conversely, lines of sight along dense filaments, where the
vacuum remains kinematically shielded at ν = 12, should strongly trend towards the CMB
baseline of 67.4. The intermediate JWST TRGB result (H
0
≈ 69.8 [4]) may simply reflect
observational sight-lines that sample an aggregate of both environments. The framework
predicts a strict positive correlation between locally measured H
0
and the void fraction
along the calibration path.
7.2 A thawing equation of state
Tying the effective dark energy density directly to the vacuum channel count, and noting
that the cosmic void fraction grows as η(z) ∝ (1 + z)
−3
, the topological coupling yields a
redshift-dependent equation of state:
w(z) ≈ −1 +
1
12
(1 + z)
−3
. (17)
At z = 0 this gives w
0
≈ −0.917—a “thawing” dark energy that departs from Λ only in the
late universe. The recent extended analysis of DESI DR2 BAO data by the DESI Collab-
oration [19] confirms a ≃ 3σ deviation from ΛCDM, with robust evidence for dynamical
dark energy emerging at low redshift (z ≤ 0.3). Furthermore, their investigation into
thawing dark energy classes highlights the theoretical preference for models that safely
deviate to w > −1 at late times without unphysical phantom crossings. Our geomet-
ric prediction perfectly anticipates this exact thawing phenomenon, deriving it strictly
from the late-time topology of the vacuum rather than ad-hoc scalar fields. As z → ∞,
w → −1, securely recovering the stable cosmological constant at early times.
7.3 BBN and early-universe consistency
If boundary bending stress were present during nucleosynthesis, it would ruin the light-
element abundances [20]. This does not happen because the radiation-dominated uni-
verse is conformally invariant (four-dimensional Ricci scalar R = 0). In that regime,
the lattice scales isotropically without accumulating differential transverse strain. The
7
Figure 1: Unifying Dark Energy and the Hubble Tension. (a) The topological
phase transition from ν = 12 (saturated bulk) to ν = 13 (void exposure) as the central
node unlocks. (b) The resulting 13/12 amplification perfectly maps the early-universe
Planck CMB baseline to the late-universe SH0ES measurement. (c) The corresponding
void fraction growth produces a thawing dark energy equation of state (w
0
≈ −0.917), per-
fectly consistent with recent DESI DR2 BAO data [19]. (d) Summary of the parameter-
free geometric predictions.
8
mechanical stress activates exclusively when conformal symmetry breaks at the epoch of
non-relativistic matter domination, flawlessly synchronizing with the ΛCDM timeline for
dark energy activation.
8 Discussion
8.1 Relation to other discrete spacetime programmes
The SSM shares territory with Regge calculus [10], causal dynamical triangulations [21],
causal set theory [22], and the recent spacetime quasicrystal construction of Boyle and
Mygdalas [23]. The critical difference is dynamical rather than structural. Most dis-
crete approaches select a lattice based on mathematical elegance; the SSM physically
derives the K = 12 FCC lattice as the thermodynamic endpoint of entanglement-driven
phase transitions, with each intermediate phase yielding distinct, falsifiable observational
imprints.
8.2 Limitations
The lattice spacing-Planck length relation (a
4
=
√
2/4l
4
P
) is self-consistent but ultimately
rests on the empirical value of G. The 13/12 channel-counting argument has been derived
here from two independent statistical-mechanics routes, but it has not yet been confirmed
by a full high-resolution lattice simulation of void formation. Such a simulation—growing
a K = 12 lattice, carving a macroscopic void, and counting the activated expansion
modes—is a priority for future computational work.
9 Conclusions
The Cosmological Constant Problem and the Hubble Tension are two faces of one coin:
the structural limits of a discrete K = 12 vacuum lattice. The topologically flat bulk
geometrically prohibits the storage of 10
120
quantum field stress (Λ
bulk
= 0), securely
confining dark energy to the holographic boundary shell at the cosmological horizon.
Non-linear void formation in the late universe triggers a macroscopic topological phase
transition from 12 to 13 active expansion channels, simultaneously resolving the 5σ Hubble
tension and renormalizing the boundary stress to exactly match the observed Ω
Λ
.
These rigid phenomenological predictions—environmental H
0
anisotropy, a thawing
equation of state conforming exactly to recent DESI constraints, and parameter-free nu-
merical exactness for H
0
and Ω
Λ
—are derived without fine-tuning and are directly falsi-
fiable with current and forthcoming survey data.
Data Availability
No new observational data were generated. Lattice simulations supporting the topological
limits are detailed in [13].
9
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