Resolving the Hubble Tension via a Topological Phase
Transition
in a Discrete Vacuum Tensor Network
Raghu Kulkarni
Independent Researcher, Calabasas, CA 91302, USA
raghu@idrive.com
February 13, 2026
Abstract
The persistent discrepancy between early-universe (H
0
≈ 67.4 km/s/Mpc) and late-universe
(H
0
≈ 73.0 km/s/Mpc) measurements of the Hubble constant has grown into a 5σ crisis for the
standard ΛCDM model. While proposed solutions often invoke new physics such as Early Dark
Energy or interacting radiation, these models typically require fine-tuning to fit the data. In
this work, we explore a geometric resolution based on the Selection-Stitch Model (SSM),
which treats the vacuum not as a continuum, but as a discrete Face-Centered Cubic (FCC)
lattice saturated at the Kepler packing limit (K = 12). We demonstrate that the formation
of cosmic voids at late times triggers a topological phase transition in the vacuum’s unit cell.
As local density drops, the lattice relaxes from a “Shielded” state (ν = 12) to an “Exposed”
state (ν = 13), activating a previously inert nucleation channel. This discrete topological shift
predicts an intrinsic expansion boost of ξ = 13/12 ≈ 8.3%. When applied to the Planck CMB
baseline, this geometric mechanism yields a local Hubble constant of H
0
≈ 73.02 km/s/Mpc,
aligning with SHOES measurements to within 0.03% without the need for arbitrary parameters.
1 Introduction
For over a decade, precision cosmology has been haunted by a statistically significant disagreement
in the measurement of the universe’s expansion rate, the Hubble constant (H
0
). Measurements an-
chored in the physics of the early universe—specifically the Cosmic Microwave Background (CMB)
observed by Planck—yield a value of 67.4 ± 0.5 km/s/Mpc [1], assuming the standard ΛCDM
cosmological model. Conversely, direct local measurements using the Cepheid-Supernova distance
ladder (SHOES) consistently report a higher value of 73.04 ± 1.04 km/s/Mpc [2].
This discrepancy, now exceeding 5σ significance [3], suggests a crack in our understanding of the
cosmos. Independent techniques such as strong gravitational lensing (H0LiCOW) [4] and Mega-
maser observations [5] support the higher local value, while Tip of the Red Giant Branch (TRGB)
measurements [6, 7] sit between the two camps, painting a complex observational landscape.
1
The theoretical response has been prolific. Proposals to resolve the tension generally fall into
two categories: modifying the early universe (e.g., Early Dark Energy [8, 9]) or modifying late-
time physics (e.g., interacting dark energy [10], void models [11]). However, these solutions often
introduce new scalar fields or arbitrary parameters that must be finely tuned to reconcile the CMB
damping tail with local expansion [12].
In this paper, we take a different approach. Rather than adding new fields to a continuous space-
time, we ask whether the discrepancy might arise from the discrete geometry of spacetime itself.
Building on the Selection-Stitch Model (SSM) [13], we model the vacuum as a physical lattice
that undergoes a phase transition analogous to crystallization. We show that the “Hubble Ten-
sion” naturally emerges as the ratio of degrees of freedom between the dense early universe and the
void-dominated late universe.
2 The Discrete Vacuum Framework
Standard cosmology assumes spacetime is a smooth manifold. However, approaches ranging from
Causal Set Theory [18] to Cellular Automaton interpretations of Quantum Mechanics [19] suggest
that at the Planck scale, spacetime may be discrete. The SSM formalizes this by positing that the
vacuum ground state is a tensor network that saturates the packing limit of information.
2.1 The FCC Ansatz
We assume the vacuum relaxes into a Face-Centered Cubic (FCC) lattice. This choice is phys-
ically motivated by the sphere packing problem: the FCC lattice achieves the maximum possible
density (ρ = π/
√
18) for packing spheres in 3D space, a result proven as the Kepler Conjecture
[20]. The fundamental unit of this lattice is the Cuboctahedron, a polyhedron with a central
node surrounded by 12 identical nearest neighbors. This fixes the lattice coordination number at
K = 12.
2.2 The Cosmological Dictionary
In a discrete geometry, cosmological expansion is not the stretching of a rubber sheet, but the
continuous nucleation of new lattice sites—a form of volumetric growth [21]. To relate this to the
Hubble parameter H(t), we follow a three-step derivation:
1. Metric Definition: The cosmological scale factor a(t) is proportional to the linear dimension
of the lattice volume: a(t) ∝ V (t)
1/3
.
2. Nucleation Rate: The rate of volume creation
˙
V is determined by the number of active
nucleation channels (ν) available on the surface of the unit cells:
˙
V ∝ ν · V (t)
2/3
.
3. Hubble Parameter: Substituting these into the definition H(t) ≡ ˙a/a:
H(t) =
˙a
a
=
1
3
˙
V
V
∝
νV
2/3
V
= νV
−1/3
(1)
2
For any fixed epoch (fixed volume V ), the expansion rate is linearly proportional to the topological
channel count ν. Thus, H ∝ ν.
3 The Topological Phase Transition
We propose that the vacuum is not static. Just as water behaves differently than ice, the vacuum
lattice exhibits different connectivity phases depending on the local matter density.
3.1 Phase I: The Shielded Lattice (Early Universe)
In the early universe (z ≫ 2), matter density is high and homogeneous. The vacuum lattice is
compressed and structurally supported by this density. In this “Solid Phase,” the Cuboctahedral
unit cell is intact. The central node is completely surrounded by its 12 neighbors. Due to steric
hindrance—or “caging” effects common in dense granular media [22]—the central node cannot
participate in surface nucleation. It is topologically shielded.
• Active Channels: Only the 12 surface nodes are active.
• Effective DoF: ν
early
= 12.
This phase characterizes the recombination epoch, meaning the Planck CMB measurement is ef-
fectively calibrating the expansion rate of a ν = 12 lattice.
3.2 Phase II: The Exposed Lattice (Late Universe)
As the universe expands, gravitational collapse draws matter into filaments, leaving behind vast,
empty regions known as Cosmic Voids [23, 24]. Today, these voids occupy over 77% of the uni-
verse’s volume [25]. In these low-density regions, the lattice undergoes a relaxation process we
term “Thawing.” Vacancies appear in the coordination shell of the unit cells. Crucially, removing
a neighbor breaks the cage symmetry. At the void interface, the mean-field vacancy density corre-
sponds to roughly one missing neighbor per cell. This opens a geometric path for the central node
to couple directly to the vacuum boundary.
• Activation: The central node, previously inert, activates as a nucleation site.
• Active Channels: 12 surface vectors + 1 newly exposed bulk vector.
• Effective DoF: ν
late
= 12 + 1 = 13.
This transition from a shielded solid to an exposed mesh represents a fundamental phase change
in the vacuum’s growth efficiency.
3
4 Derivation and Results
4.1 The Geometric Boost
If the expansion rate is proportional to the active topological channels (H ∝ ν), then the ratio
between the late-time expansion rate and the early-time rate is simply the ratio of the integers 13
and 12:
ξ =
H
late
H
early
=
ν
late
ν
early
=
13
12
≈ 1.0833... (2)
This “Geometric Boost” is an intrinsic property of the FCC lattice thawing into a void-dominated
state.
4.2 Comparing with Observation
We test this prediction by applying the boost factor to the early-universe baseline established by
Planck. Using the Planck 2018 result (H
CMB
= 67.4 ± 0.5 km/s/Mpc) [1]:
H
pred
= 67.4 ×
13
12
= 73.02 ± 0.54 km/s/Mpc (3)
This prediction is remarkably consistent with the latest local measurements. The SHOES collab-
oration reports 73.04 ± 1.04 km/s/Mpc [2], differing from our geometric prediction by less than
0.03%.
Table 1: Comparison of the SSM prediction with major observational constraints.
Dataset / Method Value (H
0
) Tension (σ) Reference
Planck 2018 (CMB) 67.4 ± 0.5 – [1]
SHOES (Cepheids/SNIa) 73.04 ± 1.04 5.0σ [2]
H0LiCOW (Lensing) 73.3
+1.7
−1.8
∼ 3σ [4]
Megamasers (MCP) 73.9 ± 3.0 ∼ 2σ [5]
TRGB (CCHP) 69.8 ± 1.9 ∼ 1σ [6]
SSM Prediction (13/12) 73.02 < 0.1σ This Work
4.3 Internal Consistency: The “+1” Principle
It is worth noting that this “K → K + 1” counting principle appears elsewhere in the SSM frame-
work, suggesting a deep structural unity.
• Fine Structure Constant: α
−1
= 136+1 = 137. (136 vacuum channels + 1 self-interaction
node) [14].
• Weinberg Angle: sin
2
θ
W
= 3/13. (3 spatial degrees out of 12 + 1 total bulk degrees) [13].
• Hubble Tension: ξ = 13/12. (12 shielded channels + 1 exposed center).
4
In each case, the physical observable is determined by the activation of the central node which is
geometrically present but normally inert in the bulk phase.
5 Discussion
5.1 Summary of Predictions
The Hubble Tension derivation joins a family of zero-parameter predictions derived from the same
K = 12 lattice geometry.
Table 2: The Grand Summary of SSM Predictions.
Observable SSM Ratio Predicted Observed Reference
H
0
Boost 13/12 1.0833 1.0836 This Work
α
−1
137 137 137.036 [14]
Ω
DM
/Ω
b
5.40 5.40 5.36 [16]
n
s
26/27 0.9630 0.9649 [17]
m
p
/m
e
1836 1836 1836.15 [15]
5.2 Falsifiable Predictions
The geometric nature of the SSM leads to distinct, testable falsification criteria:
1. Void Anisotropy: The local expansion rate should be systematically higher inside large
cosmic voids and lower along dense supercluster filaments. Future surveys like Euclid and
DESI will provide the resolution to test this density-dependence explicitly.
2. Transition Redshift: The onset of the Hubble tension should correlate with the epoch of
void percolation (z ∼ 1 − 2), appearing as a “knee” in the expansion history rather than a
smooth drift.
3. Thawing Equation of State: The transition is not instantaneous; it tracks the percolation
of the cosmic void network. This implies a dynamic Dark Energy equation of state with
w(z) > −1 at late times (thawing). A measurement of phantom energy (w < −1) would
falsify the mechanism.
6 Conclusion
The Hubble Tension may not require new particles or modified gravity, but rather a better un-
derstanding of the vacuum’s microstructure. By modeling spacetime as a discrete FCC lattice, we
find that the tension corresponds to a simple topological phase transition: the uncaging of the unit
cell’s central node in cosmic voids. The resulting ratio of 13/12 predicts a local Hubble constant of
73.02 km/s/Mpc, essentially resolving the 5σ crisis. This result invites us to reconsider the vacuum
not as a passive stage, but as a dynamic, crystallizing medium.
5
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