Geometric Evaporation: Solving the Primordial Black
Hole Constraint via Lattice Tension in an Isometric
Holographic Polycrystalline Vacuum
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
Abstract
Standard semiclassical gravity predicts that black holes evaporate via Hawking radiation
with a lifetime scaling of τ ∝ M
3
[1]. This slow decay rate imposes strict constraints on
the abundance of Primordial Black Holes (PBHs), as those formed in the early universe
(M ∼ 10
15
g) would persist today, conflicting with gamma-ray background observations [2].
We propose an alternative decay mechanism based on the Selection-Stitch Model (SSM) [3],
where the vacuum is modeled as a discrete Face-Centered Cubic (FCC) tensor network. By
recognizing the 3D polycrystalline bulk as an isometric holographic projection of a continuous
2D boundary, the vacuum functions mathematically as an Isometric Tensor Network (isoTNS)
[4]. This structure retains genuine mechanical properties (polycrystalline domain walls and
discrete tension) while simultaneously preserving exact Lorentz invariance. Treating the event
horizon as a topological vacancy, we apply the Allen-Cahn equation [5] alongside quantum
rate theory to derive a "Geometric Evaporation" mode driven by physical lattice tension.
We analytically derive a recession velocity of
˙
R = −(c/2)(l
P
/R
H
), yielding a much faster
decay law of τ ∝ M
2
. Quantitatively, a 10
15
g PBH evaporates in 0.45 milliseconds via this
geometric channel, compared to ∼ 14 billion years via Hawking radiation. A discrete Peierls
Locking mechanism with a correlation length of L
corr
∼ 1 fm ensures that this rapid channel
cleanly eradicates microscopic PBHs while exponentially shutting off to leave macroscopic
astrophysical black holes perfectly stable.
Keywords: Primordial Black Holes, Geometric Evaporation, Isometric Tensor Networks,
Polycrystalline Vacuum, Allen-Cahn Dynamics
1. Introduction
The theoretical lifespan of a black hole is currently governed by the celebrated Hawking
radiation mechanism, which treats the surrounding vacuum as a continuous quantum field
[1]. This thermal emission yields a mass loss rate of
˙
M
Hawk
∝ M
−2
, leading to an overall
lifetime that scales as τ ∝ M
3
.
While mathematically elegant in the macroscopic limit, the standard Hawking derivation
relies on Quantum Field Theory in Curved Spacetime (QFTCS). This continuous approxima-
Email address: raghu@idrive.com (Raghu Kulkarni)
tion notoriously suffers from the Trans-Planckian problem: the outgoing thermal modes must
originate from field fluctuations near the horizon with wavelengths infinitely smaller than the
Planck length. If spacetime possesses a fundamental discrete cutoff, this continuous thermal
approximation must eventually break down at microscopic scales. Furthermore, the standard
τ ∝ M
3
scaling introduces a severe fine-tuning constraint when considering Primordial Black
Holes (PBHs).
If PBHs formed during the extreme density fluctuations of the early universe with masses
below 10
15
g, the Hawking scaling dictates they should be in the final, explosive stages of
evaporation today. The resulting gamma-ray background should be easily detectable, yet
constraints from Fermi-LAT and Planck strictly limit the allowed abundance of such relics
[2]. This forces cosmology into an uncomfortable corner: either PBHs did not form in this
mass range, or our understanding of black hole decay at the discrete trans-Planckian limit is
incomplete.
In this Letter, we derive a fundamentally faster decay mode arising naturally from the
mechanics of discrete quantum gravity. Utilizing the Selection-Stitch Model (SSM), we model
the vacuum not as an empty continuum, but as an emergent Face-Centered Cubic (FCC) ten-
sor network. To resolve the apparent conflict between a structured lattice and macroscopic
continuous symmetries, we define the 3D bulk as an isometric holographic projection of an
exactly symmetric 2D boundary [4]. Functioning mathematically as an Isometric Tensor
Network (isoTNS), this quasilocal holography allows the 3D bulk to act as a genuine phys-
ical substrate—retaining real mechanical properties such as localized polycrystalline grain
boundaries, topological defects, and discrete surface tension—while inheriting exact contin-
uous Lorentz invariance from the underlying isometry of the projection [3].
In this framework, a black hole is not merely a region of infinite metric curvature, but
a literal topological vacancy in the vacuum lattice. We demonstrate that the elastically
stretched 3D polycrystalline isoTNS lattice exerts a physical Young-Laplace pressure at-
tempting to mechanically "heal" this defect, driving a geometric decay scaling of τ ∝ M
2
.
Crucially, this geometric evaporation acts as an additive channel to standard Hawking ra-
diation (
˙
M
total
=
˙
M
Hawk
+
˙
M
Geo
), dominating the decay of microscopic primordial black
holes while becoming exponentially suppressed by lattice locking at macroscopic astrophysi-
cal scales.
2. The Physical Model: A Hole in the Polycrystalline Isometric Fabric
2.1. Isometric Tensor Network Topology
We postulate that the vacuum ground state is governed by a saturated FCC lattice with
a coordination number of K = 12. A black hole represents a localized region of broken
connectivity (K ≪ 12).
Because the 3D bulk operates as an isometric tensor network [3], it mathematically sup-
ports the structural mechanics of a physical polycrystalline solid (such as defect nucleation
and domain boundary stress) without violating the continuous symmetries of the 2D bound-
ary with which it is in perfect isometry. The Event Horizon is therefore modeled physically
as a discrete domain wall separating the ordered, fully coordinated vacuum (ϕ = 1) from
the topologically disordered interior (ϕ = 0). Unlike purely continuous space, this isometric
discrete lattice possesses a finite, mechanical surface tension σ.
2
2.2. Allen-Cahn Dynamics
Because the event horizon is a genuine discrete domain boundary within the polycrys-
talline bulk substrate, its spatial evolution is governed by the Allen-Cahn equation for non-
conserved order parameters [5]:
∂ϕ
∂t
= −µ
δF
δϕ
(1)
where µ is the lattice mobility and F is the free energy functional. In materials science, this
equation describes how polycrystalline domain walls migrate to minimize surface energy—a
process known as curvature-driven grain growth. Applied to the vacuum, it describes how
the physical fabric of spacetime mechanically attempts to heal topological punctures.
3. Derivation of the Geometric Decay Rate
3.1. Holographic Surface Tension (σ)
The thermodynamic pressure acting to close a spherical void of radius R
H
is given by the
Young-Laplace equation:
P
healing
= σ
1
R
1
+
1
R
2
=
2σ
R
H
(2)
To determine the absolute surface tension σ of the vacuum, we rely on the isometric holo-
graphic principles of the network. The Bekenstein-Hawking entropy bound dictates that one
fundamental degree of freedom occupies exactly 4l
2
P
of boundary area. If the maximal en-
ergy of a single discrete node is the Planck energy (E
P
= ℏc/l
P
), then the geometric surface
tension is the energy per fundamental area:
σ =
E
P
4l
2
P
=
ℏc
4l
3
P
(3)
3.2. Quantum Rate Theory and Domain Wall Mobility (µ)
The radial velocity of the domain wall
˙
R is the product of the lattice mobility and the
inward pressure P . To rigorously derive the mobility µ, we invoke absolute quantum rate
theory.
The drift velocity of a domain wall shifting by one discrete lattice step (l
P
) is v = (W/ℏ)l
P
,
where W is the mechanical work done per quantum jump. The work done by the physical
Young-Laplace pressure over a Planck volume is W = P l
3
P
. Therefore, the drift velocity is:
˙
R = −
P l
3
P
ℏ
l
P
= −
l
4
P
ℏ
P (4)
This establishes the exact domain wall mobility as µ = l
4
P
/ℏ. Substituting the Young-Laplace
pressure (Eq. 2) and our derived surface tension (Eq. 3) yields:
˙
R = −
l
4
P
ℏ
2
R
H
ℏc
4l
3
P
= −
c
2
l
P
R
H
(5)
The recession speed is directly proportional to the curvature of the black hole, tightly con-
strained by an exact coefficient of 1/2.
3
3.3. Mass Loss Rate and the τ ∝ M
2
Scaling
Converting this spatial recession into a mass loss rate using the standard Schwarzschild
relation R
H
= 2GM/c
2
gives:
dM
dt
=
c
2
2G
˙
R = −
c
3
l
P
4GR
H
= −
c
5
l
P
8G
2
M
(6)
This confirms the linear inverse scaling
˙
M ∝ −M
−1
. Integrating this differential equation
from an initial mass M to 0 strictly yields the lifetime of the black hole under geometric
evaporation:
τ
Geo
=
4G
2
c
5
l
P
M
2
= 4t
P
M
m
P
2
(7)
This τ ∝ M
2
scaling is fundamentally distinct from—and significantly faster than—the
standard Hawking radiation scaling (τ ∝ M
3
). As the defect heals, the immense localized
mass-energy is released back into the continuous bulk as high-frequency metric perturbations
and standard model decay products.
4. Quantitative Evaluation of the PBH Constraint
To demonstrate the power of this geometric channel, we calculate the exact lifetime of a
Primordial Black Hole at the critical observational threshold of M = 10
15
g.
Under standard Hawking radiation, the lifetime of this PBH is roughly 4 × 10
17
seconds
(∼ 13.8 billion years), meaning these black holes should be actively exploding in the present
epoch.
Applying our derived geometric decay law (Eq. 7), where M/m
P
≈ 4.6 × 10
19
and the
Planck time t
P
≈ 5.4 × 10
−44
s, the geometric lifetime evaluates to:
τ
Geo
= 4(5.4 × 10
−44
)(4.6 × 10
19
)
2
≈ 4.5 × 10
−4
seconds (8)
A 10
15
g PBH does not survive for 14 billion years; under the mechanical tension of the iso-
metric lattice, it completely evaporates in 0.45 milliseconds. This rapid geometric channel
cleanly and quantitatively eradicates the PBH gamma-ray abundance constraint by prevent-
ing low-mass PBHs from surviving the early universe.
5. Stability Analysis: Peierls Locking and L
corr
If the τ ∝ M
2
geometric decay law applied universally to all black holes, macroscopic
astrophysical objects (like Cygnus X-1) would evaporate almost instantly. We resolve this
via the Peierls-Nabarro stress.
In a discrete lattice, a domain wall cannot glide continuously; it must physically un-stitch
and re-stitch as it hops between discrete lattice sites. This requires overcoming a static
geometric energy barrier known as the Peierls potential. For macroscopic black holes with
very large radii, the spatial curvature is incredibly shallow, meaning the physical Young-
Laplace healing pressure is extremely small. If this inward pressure drops below the innate
Peierls stress σ
P
required to un-stitch a layer of the vacuum, the horizon becomes structurally
"pinned" or locked.
4
We define the effective mobility of the horizon, µ
eff
, to account for this exponential
suppression:
µ
eff
= µ
0
· exp
−
R
H
L
corr
(9)
where L
corr
is the structural correlation length of the polycrystalline lattice domains.
A 10
15
g black hole possesses a Schwarzschild radius of R
H
≈ 1.5 fm. For the geometric
evaporation channel to remain unlocked for 10
15
g PBHs but become exponentially suppressed
for macroscopic black holes, the intrinsic vacuum correlation length must be on the order of
L
corr
∼ 1 fm.
This geometric parameter elegantly converges with the characteristic confinement scale
of the strong nuclear force (the topological mass gap of the vacuum). Microscopic PBHs
(R
H
≪ 1 fm) easily overpower the Peierls barrier and rapidly evaporate. Conversely, stellar-
mass and supermassive black holes (R
H
≫ 1 fm) are permanently locked into the discrete
lattice valleys, decaying only via the negligible, continuous Hawking rate.
6. Conclusion
We have derived a geometric decay law τ ∝ M
2
for black holes residing in an isometric
holographic polycrystalline vacuum. By explicitly calculating the vacuum surface tension
from holographic principles (ℏc/4l
3
P
) and deriving the domain wall mobility via quantum
rate theory, we find that physical lattice tension drives horizon evaporation at a strict rate of
˙
R = −(c/2)(l
P
/R
H
). Quantitatively, this mechanism forces a 10
15
g Primordial Black Hole
to evaporate in just 0.45 milliseconds, completely bypassing the 14-billion-year Hawking
lifespan and cleanly resolving modern gamma-ray abundance constraints. A discrete Peierls
locking mechanism at the femtometer scale ensures that this rapid geometric channel strictly
targets microscopic PBHs, leaving macroscopic astrophysical black holes structurally stable
and immune to geometric collapse.
References
[1] S. W. Hawking, “Particle creation by black holes,” Commun. Math. Phys. 43, 199 (1975).
[2] B. Carr et al., “Primordial Black Holes as Dark Matter: Recent Developments,” Annu.
Rev. Nucl. Part. Sci. 70, 355 (2020).
[3] R. Kulkarni, “Constructive Verification of K = 12 Lattice Saturation: Exploring Kine-
matic Consistency in the Selection-Stitch Model,” Zenodo: 10.5281/zenodo.18294925
(2026).
[4] R. Kulkarni, “Exact Lorentz Invariance from Holographic Projection: Explicit RT Ver-
ification and the Boundary Origin of Bulk Symmetry in the Selection-Stitch Model,”
Zenodo: 10.5281/zenodo.18856415 (2026).
[5] S. M. Allen and J. W. Cahn, “A microscopic theory for antiphase boundary motion and
its application to antiphase domain coarsening,” Acta Metall. 27, 1085 (1979).
[6] R. Kulkarni, “Geometric Horizon Inflation: A Universal Prediction for Binary Black
Hole Mergers,” Zenodo: 10.5281/zenodo.18411594 (2026).
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