Bekenstein-Hawking Entropy from the FCC
Selection Stitch Model:
A Geometric Derivation via Z
2
/U(1) Horizon
Phase Boundary
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
April 2026
Abstract
We derive the Bekenstein-Hawking (BH) area law
S
BH
= A/(4ℓ
2
P
)
directly from the ge-
ometry of the Face-Centred Cubic (FCC) Selection Stitch Model (SSM) lattice, without
invoking string theory, a free Immirzi parameter, or a thermodynamic limit. In the SSM
framework, a black hole is a macroscopic region of space in which every K
= 12
FCC
node is maximally saturated with topological defects, rendering the interior a static Z
2
codespace. The event horizon is the 2D boundary separating this Z
2
interior from the
physical U(1) exterior.
We prove that: (i) the entropy is exactly proportional to the horizon areanot the
volumeby counting the Z
2
bond phases severed at the boundary (veried numerically
for horizon radii
R = 4ℓ
P
16ℓ
P
); (ii) the coecient is
exactly
1/4
in Planck units, fol-
lowing from the SSM relation
G
N
= a
2
plaq
/(8 ln 2)
, where
a
plaq
= L(8/3)
1/4
≃ 2.355 ℓ
P
is
the FCC plaquette scale and
L ≃ 1.843 ℓ
P
is the bond length; (iii) the geometric factor
ν
eff
= 3/(2
√
2)
, arising from the interplay of the K
= 12
kissing number, spatial dimen-
sion
D = 3
, and the (111) triangular surface structure, resolves the 2D/3D counting
discrepancy to an exact result.
Unlike fuzzball/string constructionswhich require 10D supergravity and exact charge
extremalityor Loop Quantum Gravitywhich requires a freely tuned Immirzi parameter
this derivation uses only the geometry of the 3D FCC lattice and the Z
2
/U(1) phase
theorem established in the companion fermion paper [7]. The same lattice that gener-
ates a single chirally symmetric Dirac fermion simultaneously encodes the correct black
hole entropy.
1 Introduction
The Bekenstein-Hawking area law [1, 2]
S
BH
= A/(4G
N
)
is one of the deepest results in
theoretical physics, linking thermodynamics, quantum mechanics, and gravity through the
Planck area. Its microscopic derivation remains one of the central challenges of quantum
gravity.
Two main frameworks have produced microscopic derivations: string theory fuzzballs [3] and
Loop Quantum Gravity (LQG) [4, 5]. Fuzzball calculations achieve the exact coecient for
1
extremal D1-D5 black holes in 10-dimensional Type IIB supergravity, but require string-scale
physics and do not address non-extremal astrophysical black holes. LQG derives the area law
from spin-network puncture counting, but requires the Immirzi parameter
γ = ln 2/(π
√
3)
to be tuned
a posteriori
to match the coecient. Neither framework connects the black hole
entropy to the chiral structure of fermions.
In this paper we derive the BH area law from the 3D FCC Selection Stitch Model (SSM)
lattice, which in a companion paper [7] was shown to produce a single chirally symmetric
Dirac fermion at the
Γ
-point via the Irrational Doubler Theorem. The same K
= 12
FCC
structure that sequesters fermion doublers at irrational BZ coordinates also generates the
exact BH entropy coecient from rst principles, with no free parameters.
Physical framework.
This work operates under the paradigm of discrete holographic
spacetime: the FCC bond network is the fundamental physical reality, not a computational
regulator. The continuum limit
L → ∞
does not correspond to a physical process; the
lattice spacing is xed at the Planck scale. Within this framework, the doublers of the
fermion paper are kinematically sequestered outside the physical Hilbert space, and black
hole thermodynamics is encoded in the 2D horizon boundary geometry.
2 The SSM Black Hole
2.1 Denition
In the SSM vacuum, the FCC bond network acts as a quantum error-correcting (QEC) code
with K
= 12
bonds per node. The code parameters of the FCC lattice have been established
explicitly: the [[192, 130, 3]] CSS code on the FCC lattice achieves a 67% encoding rate
from weight-12 stabilizers, 24
×
higher than the cubic 3D toric code [6]. Physical particles
are propagating U(1)-charged topological defects that require genuinely complex (U(1)) bond
phases. Staggered Z
2
congurations (
e
ik·
ˆ
n
j
∈ {+1, −1}
for all bonds) carry no U(1) charge
and are inert codespace states.
Denition 1
(SSM Black Hole)
.
An SSM black hole of radius
R
is a compact region
V
of the
FCC lattice in which every node is maximally saturated: all K
= 12
bond phases are locked
into staggered Z
2
congurations. The event horizon
∂V
is the 2D boundary surface of area
A = 4πR
2
separating the Z
2
interior from the U(1) exterior.
We propose that the interior is causally disconnected from the U(1) exterior: a propagating
excitation in the Z
2
codespace cannot couple to the U(1) physical sector without acquiring
U(1) electric charge, which requires crossing the horizon boundary. This is consistent with
the Z
2
/U(1) phase theorem established in the companion fermion paper [7]: Z
2
congurations
carry no propagating U(1) charge by construction. This provides an intrinsic no-hair structure
with no singularity: the interior is a nite region of saturated code, not an innite density
divergence.
1
1
The interior may equivalently be described as a
topological vacancy
(K
= 0
void) from the exterior U(1)
perspective. These descriptions are dual: from outside, all horizon bonds terminate at the boundary (K
= 0
view); from inside, those same bonds are frozen Z
2
states carrying no U(1) charge (Z
2
saturation view). The
Z
2
picture used here is the natural language for the holographic projection; the K
= 0
picture is natural for
deriving the geometric evaporation dynamics.
2
2.2 Derivation of Newton's Constant from FCC Geometry
We derive
G
N
from rst principles using two inputs only: the FCC (111) surface geometry
and the entanglement entropy per bond.
Proposition 2
(Newton's Constant from FCC Entanglement)
.
For an FCC lattice with bond
length
L
in which each bond is a maximally entangled Bell pair, the Ryu-Takayanagi relation
applied to the fundamental (111) bilayer unit cell yields:
G
N
=
p
2/3 L
2
4 ln 2
= ℓ
2
P
.
(1)
Proof.
The proof proceeds in four steps.
Step 1: The (111) interlayer spacing.
The FCC conventional cubic cell has side length
a
cube
= L
√
2
, since the nearest-neighbour bond length is
a
cube
/
√
2 = L
. The (111) crystallo-
graphic plane is the closest-packed plane, with interplanar spacing:
h =
a
cube
√
3
=
L
√
2
√
3
= L
r
2
3
.
(2)
Numerically,
h = L
p
2/3 ≃ 0.8165 L ≃ 1.505 ℓ
P
.
Step 2: The fundamental bilayer unit cell.
Consider two adjacent (111) layers of the
FCC lattice separated by the interplanar distance
h
. The
bilayer unit cell area
A
1
is the
product of the interplanar spacing
h
and the nearest-neighbour bond length
L
, which is the
characteristic in-plane length scale of the triangular (111) surface lattice. The correct minimal
surface separating the two (111) layers is the bilayer cross-section: a rectangle of width
L
(bond length, setting the in-plane scale) and height
h
(interlayer spacing, the out-of-plane
scale). This is the product of the two independent FCC length scales in the (111) geometry:
A
1
= h × L = L
r
2
3
× L =
r
2
3
L
2
.
(3)
Numerically,
A
1
≃ 2.773 ℓ
2
P
. Other surface choices (e.g.
σ
111
/ν
111
=
√
3L
2
/6
) would give
dierent
L/ℓ
P
ratios; the bilayer choice is selected because it is the unique area that yields
G
N
= ℓ
2
P
consistent with the Schwarzschild radius and Hawking temperature in the SSM
framework [6].
Step 3: Entanglement entropy per bond.
In the SSM vacuum, each FCC bond is a
maximally entangled two-qubit Bell state:
|Ψ
bond
⟩ =
1
√
2
|+1⟩ ⊗ |−1⟩ + |−1⟩ ⊗ |+1⟩
.
(4)
The reduced density matrix for either endpoint qubit is
ρ =
1
2
1
2×2
, giving:
S
bond
= −Tr[ρ ln ρ] = −2 ×
1
2
ln
1
2
= ln 2.
(5)
This is the maximum entanglement for a qubit pair: one ebit per bond.
3
Step 4: Ryu-Takayanagi applied to the fundamental bond.
The Ryu-Takayanagi
(RT) formula [12]
S = A
min
/(4G
N
)
was derived for macroscopic surfaces in AdS/CFT. In the
SSM, we adopt it as a
founding postulate
at the microscopic level: we require that the RT
formula hold for the most fundamental entangled pair in the lattice, namely a single bond.
This is the SSM analogue of the Jacobson [10] derivation, which obtains
G
N
by requiring the
rst law of thermodynamics to hold locally for each Rindler horizon. Here we require the RT
relation to hold locally for each bond:
S =
A
min
4G
N
.
(6)
The minimal surface separating the two layers of the bilayer cell has area
A
min
= A
1
=
p
2/3 L
2
, and the entanglement across it is
S = S
bond
= ln 2
. Substituting and solving for
G
N
:
ln 2 =
p
2/3 L
2
4G
N
=⇒ G
N
=
p
2/3 L
2
4 ln 2
.
(7)
Setting
G
N
= ℓ
2
P
determines the bond length:
L = ℓ
P
s
4 ln 2
p
2/3
≃ 1.843 ℓ
P
.
(8)
The RT postulate at the bond level is self-consistent: it gives
G
N
= ℓ
2
P
and, applied macro-
scopically in Section 4, recovers the exact BH coecient without additional assumptions.
□
Physical interpretation.
Newton's constant is the product of the (111) interlayer spacing
and the bond length, divided by four times the bond entanglement entropy and the FCC
structure tensor eigenvalue (
c
F
= 4
, from
S
µν
=
P
j
ˆn
µ
j
ˆn
ν
j
= 4δ
µν
established in the companion
fermion paper [7]):
G
N
=
h · L
4 S
bond
=
h · L
4 ln 2
.
(9)
Gravity is not imposed; it is the thermodynamic consequence of the entanglement structure
of the FCC bond network.
Two FCC length scales.
The bond length
L ≃ 1.843 ℓ
P
determines a second natural scale
via the plaquette:
a
plaq
= L
8
3
1/4
=
√
2Lh ≃ 2.355 ℓ
P
,
(10)
where
h = L
p
2/3
is the (111) interlayer spacing from (2). The Regge decit angle at
each FCC node,
δ = 2π − 5 arccos(1/3) ≃ 0.1284
rad, provides an independent geometric
characterisation of the curvature encoded in
G
N
. In terms of
a
plaq
, equation (7) reads:
G
N
=
a
2
plaq
8 ln 2
= ℓ
2
P
.
(11)
Both forms encode the same Newton's constant; the plaquette form is used in Section 4.
4
3 S
∝
A: The Area Law
3.1 Counting Boundary Degrees of Freedom
The information accessible to an external observer is encoded in the Z
2
bond phases of bonds
that cross the horizon
∂V
. Each such bond has one endpoint in the Z
2
interior (phase locked
to
{+1, −1}
) and one endpoint in the U(1) exterior. The interior endpoint carries exactly
one classical bit.
Surface node density.
The FCC lattice has 4 atoms per conventional cubic cell of side
a
cube
= L
√
2
, giving a volume per atom
V
prim
= a
3
cube
/4 = L
3
√
2/2
. For a macroscopic sphere
(averaging over all surface orientations), the mean area per surface node is:
σ =
V
prim
L
=
L
2
√
2
2
.
(12)
Crossing bonds per node.
For a node at position
r
just inside the horizon with outward
normal
ˆ
r
, bond
ˆ
n
j
crosses if
ˆ
n
j
·
ˆ
r > 0
. By the isotropy of the FCC bond directions over the
sphere:
P (
ˆ
n
j
·
ˆ
r > 0) = 1/2
, giving
ν = K/2 = 6
crossing bonds per surface node.
Total crossing bonds:
N
bonds
= ν ·
A
σ
= 6 ·
A
L
2
√
2/2
=
6
√
2 A
L
2
.
(13)
Entropy:
S = N
bonds
ln 2 =
6
√
2 ln 2
L
2
A.
(14)
This establishes
S ∝ A
exactly. Figure 1 conrms the linear relationship numerically for
R = 4ℓ
P
to
16ℓ
P
.
Figure 1: Left: Entropy
S
(in units of
ln 2
) vs horizon area
A
(in units of
ℓ
2
P
) from SSM bond
counting on the FCC lattice for 24 values of
R
. The linear relationship
S ∝ A
is exact. Right:
Comparison of how SSM, LQG, and fuzzball/string theory each reproduce
S
BH
= A/(4ℓ
2
P
)
.
5
4 The Exact Coecient
4.1 From Newton's Constant
From (11), the BH entropy formula gives:
S
BH
=
A
4G
N
=
A
4ℓ
2
P
=
A · 8 ln 2
4 a
2
plaq
=
2 ln 2
a
2
plaq
A.
(15)
This is exact: the coecient
2 ln 2
per
a
2
plaq
follows directly from
G
N
= a
2
plaq
/(8 ln 2)
, which
was derived in Proposition 2 above.
4.2 Geometric Origin: 2D Holographic Projection
To resolve the apparent discrepancy between the 3D bulk bond-counting coecient (14) and
the exact BH value, we apply the holographic principle directly. The event horizon is not a
3D volumetric boundary; it is natively a 2D projected holographic screen.
When the 3D FCC lattice is projected onto its densest 2D boundarythe (111) planeit
forms a triangular lattice. The physical area occupied by a single node on this 2D holographic
screen is exactly:
σ
111
=
√
3
2
L
2
.
(16)
For a macroscopic event horizon of area
A
, the total number of native 2D nodes comprising
the holographic screen is
N
surf
= A/σ
111
.
In the 3D bulk, nodes share bonds symmetrically in all directions. Each crossing bond is an
edge in both the 3D bulk graph (where it connects two nodes) and in the 2D surface graph
(where it connects a surface node to a subsurface node). The eective bit content of each
crossing bond from the 2D perspective is reduced by the ratio of the 2D surface embedding
to the 3D bulk: specically, the bond is shared among
p
K(D − 1)/D
eective surface sites,
where the
√
·
arises from the geometric mean of the
D − 1 = 2
surface dimensions and the
D = 3
bulk coordination [6]. For the FCC lattice (
K = 12
,
D = 3
):
Projection factor
=
r
K(D − 1)
D
=
r
12 · 2
3
=
√
8 = 2
√
2.
(17)
Each node on the (111) surface natively possesses
ν
111
= 3
bonds crossing the horizon.
Applying the projection factor to translate 3D bulk bonds to eective 2D surface bits:
ν
eff
=
ν
111
2
√
2
=
3
2
√
2
.
(18)
Because the horizon marks the Z
2
/U(1) phase boundary, every eective crossing bond is
locked into a purely real Z
2
state (
±1
), contributing exactly one bit of entropy (
ln 2
). The
total entropy is therefore:
S = ν
eff
N
surf
ln 2 =
3
2
√
2
·
A
√
3
2
L
2
· ln 2 =
r
3
2
A ln 2
L
2
.
(19)
6
From Proposition 2 (equation (1)), we established
G
N
=
p
2/3 L
2
/(4 ln 2) = ℓ
2
P
. Rearranging
this fundamental relation gives exactly:
1
4ℓ
2
P
=
r
3
2
ln 2
L
2
.
(20)
Substituting (20) into (19):
S =
A
4ℓ
2
P
.
(21)
This projection factor is not a free parameter. It is the rigid topological translation from the
3D Regge geometry of the FCC bulk to its 2D holographic boundary, determined solely by
the FCC kissing number
K = 12
and the spatial dimension
D = 3
.
Figure 2: Left: The FCC (111) surface forms a triangular lattice. Each surface node has
3 bonds crossing the horizon (red arrows, into the Z
2
interior) and 6 in-plane tangential
bonds (grey, carrying no horizon information). Right: the topological projection factor
p
K(D − 1)/D = 2
√
2
translates the 3D bulk coordination (K
= 12
,
D = 3
) to the 2D
holographic screen (
D − 1 = 2
), yielding
ν
eff
= 3/(2
√
2)
and the exact BH coecient.
Proposition 3
(Exact BH coecient)
.
The SSM bond-counting entropy on the (111) FCC
holographic screen, with projection factor
p
K(D − 1)/D = 2
√
2
, gives
S = A/(4ℓ
2
P
)
exactly,
with no free parameters.
Proof.
Direct substitution:
ν
eff
= 3/(2
√
2)
,
σ
111
=
√
3L
2
/2
, so
ν
eff
/σ
111
= 3/(2
√
2)·2/(
√
3L
2
) =
p
3/2/L
2
. Then
S =
p
3/2 · (ln 2/L
2
) · A = A/(4ℓ
2
P
)
by (20).
□
5 Connecting to the Companion Fermion Paper
The companion paper [7] established:
1.
Irrational Doubler Theorem:
Every non-
Γ
zero of the FCC bond-direction Dirac
operator has FCC fractional coordinate
f
i
= ±1/(2
√
2)
irrational, hence kinemati-
cally sequestered from any nite integer-
L
lattice.
7
2.
Z
2
/U(1) phase boundary:
Type-1 (isolated Z
2
) and Type-2 (at-band U(1)) doubler
zeros both reside at the same irrational coordinates. Physical electrons require complex
U(1) bond phases; Z
2
congurations are inert codespace states.
These are precisely the structures used in this paper. The Z
2
/U(1) boundary of the black
hole event horizon is the macroscopic realisation of the microscopic phase boundary that
sequesters fermion doublers. The FCC lattice achieves:
One chiral fermion
at
Γ
(no doublers in the U(1) physical sector).
BH area law
with exact coecient
1/4
from the same K
= 12
geometry.
No other known lattice achieves both simultaneously. The same
K = 12
FCC framework has
also been shown to generate the particle mass spectrum, ne structure constant, and weak
mixing angle from quantum error-correction principles [8].
6 Information Preservation
In the fuzzball programme, information is preserved because the interior is lled with string
microstates all the way to the horizon there is no information loss because there is no
interior void. In the SSM framework, the mechanism is dierent but equivalent: the interior
is a nite Z
2
codespace in which all bond congurations are enumerable. The
2
N
bonds
congu-
rations of the horizon-crossing bonds constitute a nite set of microstates; no information is
destroyed during Hawking evaporation because the bond phase congurations are preserved
in the outgoing U(1) radiation.
Specically: when a horizon bond evaporates (transitions from Z
2
/U(1) boundary to pure
U(1) exterior), its
{+1, −1}
phase is transferred to the outgoing radiation as a U(1) phase
modulation. This is the SSM analogue of fuzzball leakage of information into Hawking
radiation.
7 Comparison with Existing Approaches
Table 1: Comparison of microscopic BH entropy derivations.
Feature SSM (this work) LQG Fuzzball/string
Spacetime dimension 3+1 3+1 9+1
Free parameters 0 1 (Immirzi
γ
) 0 (extremal only)
Exact coecient
yes
yes (after tuning) yes (extremal)
Non-extremal BH yes yes open
Chiral fermion unied
yes
no no
Singularity resolved yes (Z
2
saturation) no yes (fuzz)
Unique predictions
τ ∝ M
2
; FCC spectrum area gap echoes
vs LQG.
LQG counts spin-network punctures on the horizon, each carrying
ln(2j + 1)
bits
for spin
j
. The area law follows but the coecient requires tuning
γ = ln 2/(π
√
3)
. The SSM
coecient is xed by the FCC kissing number
K = 12
and dimension
D = 3
with no tuning.
vs Fuzzball.
Mathur's programme derives the correct entropy for extremal D1-D5 black
holes via explicit microstate counting in 10D. The SSM derives the same entropy coecient
in 3D from a lattice that also generates the fermion spectrum. The fuzzball programme has
no connection to chiral fermions or the Standard Model.
8
Furthermore, the mechanisms resolving the central singularity dier profoundly. In the
fuzzball paradigm, the singularity is avoided because fundamental strings pu up due to
their extended 1D nature, creating a horizon-sized tangle. In the SSM framework, a singular-
ity is prohibited by the strict nite information capacity of the discrete graph. Gravitational
collapse induces topological defects; when a region achieves maximum defect density, the
K
= 12
lattice is locally saturatedit cannot be compressed further into a singularity. The
saturated region must therefore grow macroscopically to accommodate infalling information,
rendering the black hole a nite, densely packed region of Z
2
codespace. This eliminates the
singularity natively in 3D, without invoking the 10 dimensions or wrapped branes required
by string theory.
vs Bekenstein.
Bekenstein's original argument was thermodynamic. The SSM derivation
is microscopic and combinatorial: it enumerates the Z
2
bond phase congurations at the
horizon and recovers BH thermodynamics as a consequence.
8 Unique Predictions
The SSM framework makes two falsiable predictions that dier qualitatively from both
standard Hawking radiation and fuzzball scenarios.
Prediction 1: Geometric evaporation (
τ ∝ M
2
)
Because the event horizon is a discrete boundary of severed bonds, the surrounding intact
K
= 12
lattice exerts a mechanical surface tension on the vacancy boundary. The energy
to create one unit of new horizon area is the Planck energy
E
P
= ℏc/ℓ
P
divided by the
fundamental area
a
0
= 4G
N
= 4ℓ
2
P
(one Planck area per bit, from
S = A/(4G
N
)
with
G
N
= ℓ
2
P
):
σ =
E
P
a
0
=
ℏc
4ℓ
3
P
.
(22)
The Young-Laplace pressure on a spherical horizon of radius
R
H
is
P = 2σ/R
H
. By absolute
quantum rate theory, the work done per Planck volume
W = Pℓ
3
P
= ℏc/(2R
H
)
drives a
horizon recession:
˙
R
H
= −
W
ℏ
ℓ
P
= −
c ℓ
P
2R
H
.
(23)
Using
R
H
= 2GM/c
2
and integrating gives the geometric lifetime:
τ
geo
= 4 t
P
M
m
P
2
,
(24)
where
t
P
is the Planck time and
m
P
the Planck mass. This
τ ∝ M
2
scaling is parametrically
faster than the standard Hawking channel (
τ
Hawk
∝ M
3
).
For primordial black holes (PBHs) at the critical observational threshold
M = 10
15
g, the
geometric channel gives
τ
geo
≃ 0.45
ms, while the Hawking channel predicts
τ
Hawk
≈ 4 ×
10
17
s (
≈ 13.8
Gyr, the current age of the universe) using the standard multi-species formula
with
g
∗
≈ 210
eective Standard Model degrees of freedom [11]. Fermi-LAT and Planck
observations constrain the abundance of such long-lived PBHs [9]; the geometric channel
removes them entirely by evaporating them in
0.45
ms in the early universe. Macroscopic
black holes are stabilised by Peierls-Nabarro lattice locking at the QCD connement scale
L
corr
∼ 1
fm, restricting the geometric channel to microscopic PBHs.
9
This prediction is strictly falsiable: the
τ ∝ M
2
scaling predicts a
sharp cuto
at
M ∼ 10
15
g
with zero surviving PBH relics, rather than the smooth tail from Hawking evaporation.
Figure 3: Black hole lifetime comparison: Hawking (
τ ∝ M
3
, blue) vs SSM geometric
evaporation (
τ ∝ M
2
, red). At
M = 10
15
g the Hawking channel predicts
τ
Hawk
≈ 4×10
17
s (
≈
13.8
Gyr, exploding today) while the geometric channel gives
τ ≃ 0.45
ms, cleanly resolving
the gamma-ray abundance constraint [9]. Macroscopic black holes (
R
H
≫ 1
fm) are Peierls-
locked (purple dashed line) and stable against geometric evaporation.
Prediction 2: FCC spectral modulation of Hawking radiation
The FCC reciprocal lattice has high-symmetry points at
Γ
, X, W, L, K with energy scales
E
X
≃ 5.45/a
plaq
,
E
K
≃ 0.014/a
plaq
. The Hawking emission spectrum should show fractional
deviations from the perfect blackbody spectrum at these discrete frequencies, at a level
∆E/E ∼ (a
plaq
/R
Schw
)
2
potentially observable for Planck-scale micro-black holes.
9 Conclusion
We have derived the Bekenstein-Hawking entropy
S = A/(4ℓ
2
P
)
from the FCC Selection
Stitch Model lattice with no free parameters. The key results are:
1.
Area law:
S ∝ A
exactly, from counting Z
2
bond phases at the horizon boundary
(numerical verication for
R = 4ℓ
P
16ℓ
P
).
2.
Exact coecient:
The SSM relation
G
N
= a
2
plaq
/(8 ln 2)
gives
S = 2 ln 2 · A/a
2
plaq
=
A/(4ℓ
2
P
)
exactly. The geometric factor
ν
eff
= 3/(2
√
2) = 3/
p
K(D − 1)/D
encodes the
2D nature of the horizon in 3D FCC geometry.
3.
Unication:
The same K
= 12
FCC lattice that generates a single chirally symmetric
Dirac fermion (companion paper [7]) simultaneously encodes the correct black hole
entropy.
10
4.
No free parameters:
Unlike LQG, the coecient requires no Immirzi tuning. Unlike
fuzzball, no extremality condition is assumed.
The connection between the Z
2
/U(1) phase theorem for fermion doubling and the Z
2
/U(1)
horizon boundary for black holes suggests a deep structural unity: the same geometric mech-
anism that sequesters fermion doublers at irrational BZ coordinates also encodes black hole
thermodynamics at the Planck scale.
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18410364
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11
