Black Holes in the FCC Selection–Stitch Model:
Bekenstein–Hawking Entropy, Geometric Evaporation,
and Primordial-Black-Hole Signatures
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA raghu@idrive.com
July 25, 2026
Abstract
We develop the black-hole sector of the Face-Centered Cubic (FCC) Selection–Stitch Model
(SSM), a discrete-spacetime framework in which the vacuum is a close-packed lattice carry-
ing the [[192, 130, 3]] CSS stabilizer code, treated here as a working assumption rather than
an established result. Two structural results are established first. (i) We show that the
four-dimensional D
4
root lattice — whose constant-time slice is the FCC lattice — supports
emergent linearized gravity when treated as an intrinsic (Regge) simplicial geometry: the rank-
four bond tensor is exactly isotropic, the graviton is the incompatible sector of the edge-length
field, the Regge action is itself selected — it is the unique member of the local frustration-
linear family that admits a flat vacuum with a covariant linearized limit — and on an explicit
simplicial subdivision the continuum kinetic operator equals the linearized Einstein operator
term by term, a symbolic identity with the ghost-free Fierz–Pauli coefficients. (ii) Integrat-
ing out the lattice modes generates a Sakharov-induced Newton constant proportional to the
squared lattice spacing; matching it to the observed Newton constant fixes the bond length
L
0
at the Planck scale up to a single O(1) coefficient, with no horizon input, and match-
ing the bond-level Ryu–Takayanagi relation to the Bekenstein–Hawking normalization then
pins that one number, L
0
≃ 1.843 ℓ
P
; applying the same relation at the horizon scale gives
S = A/(4ℓ
2
P
) with no further free parameter. The coefficient rests on that postulate and the
calibration rather than on the lattice; what the lattice supplies is the compatibility of the two,
its orientation-averaged bond count reducing to the RT-cell count by a computed ratio. An
SSM black hole is a region in which the lattice has crossed its metric wall: viewed from outside
it is a K = 0 topological vacancy, viewed from inside an inert Z
2
codespace, the two descrip-
tions dual across the horizon. The same horizon supports a non-thermal evaporation channel:
the intact lattice exerts a surface tension on the vacancy boundary, giving a lifetime τ ∝ M
2
.
A faceted, curvature-driven boundary then has a classical terrace-nucleation barrier propor-
tional to the horizon radius, producing an exponential freeze-out once the horizon exceeds a
scale ξ ∼ 1 fm; the error-correcting code may supply a quantum modification, but the corre-
sponding dynamical conversion sequence is not derived, and we show by direct computation
that static code quantities do not grow with horizon size. Phenomenologically this is a two-
parameter evaporation law — an unsuppressed τ ∝ M
2
channel below ξ and an exponential
freeze-out above it — with two testable consequences. For the hypothesis ξ ∼ 1 fm it predicts
a hard lower edge to the surviving PBH mass function near M
cut
≃ 10
16.5
g, and it shifts
the constraints on lighter PBHs from present-day γ-rays to BBN and CMB energy-injection
bounds, limiting their transient early-Universe abundance at f
PBH
≲ 10
−2
–10
−6
against BBN
and COBE/FIRAS at the order-of-magnitude level. The existence and sharpness of the cutoff
survive the O(1) uncertainties and the undetermined exponent geometry, which moves it by
under a decade; its absolute location does not survive the unexplained hierarchy behind ξ,
1
which we state as a postulate. Both outputs are testable — through a PIXIE-class spectral-
distortion measurement correlated with a BBN anomaly along the mass–epoch map — without
committing to the SSM. The phenomenology is conditional on a formation mechanism that
evades the same suppression; we conjecture that primordial holes are relic pockets of the vac-
uum’s crystallization transition — never assembled, hence never barrier-gated — but derive
no mass function from this.
Contents
1 Introduction 3
1.1 Scope of the claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 The SSM black hole 5
2.1 The FCC vacuum and the Z
2
/U(1) horizon . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The interior duality: K = 0 vacancy ≡ Z
2
saturation . . . . . . . . . . . . . . . . . 5
2.3 No singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 The spacetime lattice and its spatial slice . . . . . . . . . . . . . . . . . . . . . . . 6
3 The linearized continuum limit of the lattice 6
3.1 Exact rank-four isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 The graviton as the incompatible edge-length mode . . . . . . . . . . . . . . . . . . 8
3.3 The kinetic operator equals linearized Einstein . . . . . . . . . . . . . . . . . . . . 8
3.4 Induced Newtonian coupling and the equivalence principle . . . . . . . . . . . . . . 10
3.5 What this section does and does not license . . . . . . . . . . . . . . . . . . . . . . 11
4 Newton’s constant and the lattice scale 11
4.1 The magnitude of L
0
: induced coupling matched to the observed G
N
. . . . . . . . 11
4.2 Fixing the O(1) coefficient by the bond-level RT relation . . . . . . . . . . . . . . . 11
4.3 Status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5 The area law S ∝ A 12
6 The coefficient: one bit per RT cell 13
6.1 What is derived, what is calibrated . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7 Geometric evaporation dynamics 15
7.1 Surface tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7.2 Boundary recession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7.3 Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7.4 Robustness of the M
2
exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
7.5 Nature of the emitted energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
8 Boundary locking: a curvature-gated nucleation barrier 17
8.1 Survival cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
8.2 Status of the suppression scale ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2
9 Evaporation history of the sub-cutoff population 20
9.1 Formation and evaporation times . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
9.2 Six evaporation bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
9.3 Inversion relative to Hawking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
10 Cosmological constraints 23
10.1 Energy-injection formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
10.2 Pre-BBN: the safely cleared population . . . . . . . . . . . . . . . . . . . . . . . . 23
10.3 BBN and photodissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
10.4 CMB spectral distortions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
10.5 A shifted constraint landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
10.6 Extended mass functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
11 Information: a conjectural bookkeeping 26
11.1 Four levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
11.2 L1 is radiated; L2–L4 are not . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
12 Comparison with other approaches 27
13 Predictions 27
14 Dynamical black holes: formation, growth, and mergers 28
15 Connection to the companion papers 29
16 Discussion, limitations, and falsifiability 29
16.1 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
16.2 Claim ledger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
16.3 Falsifiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
17 Conclusions 30
A Orientation dependence and the reduction factor 32
B The punctured code: static quantities do not grow with the vacancy 34
C The D
4
subdivision and the closed-form kinetic identity 35
1 Introduction
The Bekenstein–Hawking area law S
BH
= A/(4G
N
) [1, 2] links thermodynamics, quantum mechan-
ics, and gravity through the Planck area, and its microscopic origin is one of the central problems of
quantum gravity. Two programs supply microscopic derivations. String-theory fuzzballs [3] reach
the exact coefficient for extremal D1–D5 holes in ten dimensions but do not address non-extremal
astrophysical holes. Loop Quantum Gravity [4, 5] recovers the area law from spin-network punc-
ture counting but fixes the coefficient only by tuning the Immirzi parameter γ = ln 2/(π
√
3) a
posteriori. Neither connects the entropy to the structure of matter.
In a separate empirical arena, primordial black holes (PBHs) remain a dark-matter candidate
requiring no physics beyond general relativity and an enhanced primordial spectrum [9, 10]. Their
3
viability below ∼ 10
17
g is set by the products of Hawking evaporation [2, 11, 12, 13]: holes below
M
⋆
≃ 5 × 10
14
g have already evaporated, and those up to ∼ 10
17
g are constrained by their
present-day emission. Each such bound assumes the semiclassical rate τ
Hawk
∝ M
3
, derived on a
fixed continuum background and expected to fail once quantum-gravitational effects matter.
This paper treats both problems within one framework. The Selection–Stitch Model (SSM) [23,
24, 25] models the vacuum as a discrete close-packed tensor network — the D
4
root lattice in four
spacetime dimensions, whose spatial slice is the FCC lattice with coordination number K = 12,
the densest sphere packing in three dimensions [8] — carrying an associated stabilizer code [25].
In contrast to the companion papers [23, 24], which take the gravitational language as given,
the present paper begins from the continuum side: Section 3 shows that the linearized long-
wavelength limit of the intrinsic lattice dynamics is the linearized Einstein theory — massless
spin-2 propagation with the ghost-free Fierz–Pauli structure and an induced Newtonian potential
— so that the gravitational language used at the horizon is derived at leading order rather than
assumed wholesale. We then derive the area law and its coefficient from the lattice geometry (
§§
4–
6); we show that the same horizon construction implies a second evaporation channel, geometric
rather than thermal, with τ ∝ M
2
(
§
7), frozen above a nucleation-barrier scale (
§
8); and we trace
the cosmological consequences of that channel for the PBH population (
§§
9–10). The unifying
object is the Z
2
/U(1) horizon: the boundary between the inert Z
2
codespace interior and the
propagating U(1) exterior, which serves both as the surface whose severed bonds carry the entropy
and as the phase boundary whose recession drives the evaporation.
1.1 Scope of the claims
Because the underlying assumption is strong and unestablished, we state plainly what is derived,
what is calibrated, and what is imported, so that the testable phenomenology is cleanly separable
from the speculative microphysics. Throughout the paper, major statements are labeled with one
of six statuses: [derived] (follows from the lattice geometry, with proof or exact computation),
[calibrated] (fixed by matching one number to established physics), [imported] (taken from
general relativity as a consistency input), [assumed] (a phenomenological modeling input with
stated freedom), [conjectured] (an interpretation not required by the results), and [testable]
(an observational consequence). Section 16.2 collects the labels in one table.
Assumed, not derived: the SSM lattice and its stabilizer-code structure, taken from Refs. [23,
24, 25]. We do not argue here that it is preferred over other discrete-spacetime programs, nor
do we re-derive it.
Derived at linearized order: the continuum limit of the intrinsic lattice dynamics — exact rank-
four isotropy, a two-polarization massless spin-2 sector with the Fierz–Pauli coefficients equal
term by term to the linearized Einstein operator on an explicit subdivision, and an induced
Newtonian coupling whose match to the observed Newton constant fixes the lattice scale at
the Planck length (
§
3, Appendix C). This addresses, at leading order, the demand that the
lattice reproduce gravity; the nonlinear regime, gravitational radiation, and the Schwarzschild
geometry are not derived.
Calibrated (one number): the O(1) coefficient of the lattice scale L
0
. Its Planck magnitude
follows from matching the induced coupling to the observed Newton constant (
§
4), with no
horizon input; the residual coefficient is fixed by matching the bond-level Ryu–Takayanagi
relation to the Bekenstein–Hawking normalization. With L
0
fixed, applying the same RT
relation at the horizon scale returns the area-law coefficient with no further parameter (
§
6), and
4
the evaporation law follows without further tuning; the RT relation itself is a stated postulate
used twice.
Imported from general relativity: the Schwarzschild radius and Hawking temperature, used as
consistency inputs at the horizon. The lattice is shown to reduce to linearized Einstein gravity
in the weak-field limit, but it is not shown to reduce to the full curved-spacetime theory, and
we do not claim that the horizon geometry is derived.
Conjectured: the relic-void origin of primordial holes — formation as pockets of uncrystallized
void trapped by the vacuum’s crystallization front rather than by assembled collapse (
§
14)
— which exempts primordial formation from the locking barrier but carries no derived mass
function; and the informational interpretation of
§
11.
Modeling inputs with O(1) freedom: the interface mobility coefficient β in the recession law
(
§
7), the electromagnetic fraction f
EM
of the released energy (
§
10), and the suppression scale
ξ (
§
8). The exponential form of the suppression and its linear-in-R
H
exponent are derived
from classical curvature-gated terrace nucleation under two stated conditions (faceted interface,
bulk-degenerate phases); an areal exponent, the coherent-quantum alternative, would move the
cutoff by under a decade (
§
8). The numerical value ξ ∼ 1 fm is a postulate, discussed critically
in
§
8.2.
The two genuinely load-bearing, testable outputs are the τ ∝ M
2
scaling and the sharp survival
cutoff. The existence of a cutoff is not special to the SSM: any super-exponential suppression
that switches on near a microphysical scale produces one, so a reader who rejects the lattice may
treat
§§
9–10 as the phenomenology of an effective two-parameter evaporation law.
2 The SSM black hole
2.1 The FCC vacuum and the Z
2
/U(1) horizon
In the SSM vacuum the FCC bond network acts as a quantum error-correcting code with K = 12
bonds per node; the FCC lattice has been established explicitly as a [[192, 130, 3]] CSS code
with a 67% encoding rate from weight-12 stabilizers [25]. Physical particles are propagating U (1)-
charged topological defects requiring genuinely complex bond phases. Staggered Z
2
configurations
(e
ik·ˆn
j
∈ {+1, −1} on every bond) carry no U(1) charge and are inert codespace states.
Definition 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
configurations. The event horizon ∂V is the 2D boundary surface of area A = 4πR
2
separating the Z
2
interior from the U(1) exterior.
A propagating excitation in the Z
2
codespace cannot couple to the U(1) physical sector with-
out acquiring U(1) electric charge, which requires crossing the horizon. We take this Z
2
/U(1)
separation as definitional: a staggered Z
2
configuration carries no propagating U(1) charge by
construction, so the saturated interior is causally sequestered from the physical exterior. This
provides an intrinsic no-hair structure with no singularity.
2.2 The interior duality: K = 0 vacancy ≡ Z
2
saturation
These two faces are dual descriptions of one object: the entropy calculation below uses the sat-
urated Z
2
interior, while the evaporation dynamics of
§
7 use the severed-bond K = 0 vacancy.
5
From the exterior U (1) perspective every horizon bond terminates at the boundary, so the interior
reads as a K = 0 void. From the interior those same bonds are frozen Z
2
states carrying no U (1)
charge, so the interior reads as a saturated codespace. The Z
2
picture is the natural language for
the holographic entropy count; the K = 0 picture is natural for the boundary-recession dynamics.
We use each where it is simplest and rely on their equivalence throughout.
2.3 No singularity
Gravitational collapse drives topological-defect density upward; once a region reaches maximum
defect density the K = 12 lattice is locally saturated and cannot be compressed further. The
saturated region must grow to accommodate infalling information, rendering the hole a finite,
densely packed region of Z
2
codespace rather than a density divergence. A singularity is prohibited
by the finite information capacity of the discrete graph, with no recourse to extra dimensions or
wrapped branes. [conjectured] (the collapse dynamics themselves are not modeled here; see
§
14).
2.4 The spacetime lattice and its spatial slice
Two lattices appear in this paper and their roles must be kept distinct. The spacetime lattice of the
SSM is the four-dimensional D
4
root lattice, D
4
= {x ∈ Z
4
:
P
i
x
i
≡ 0 (mod 2)}, with 24 minimal
vectors saturating the four-dimensional kissing-number bound [29]. Its constant-time slice D
4
∩
{x
4
= 0} is the FCC lattice. Constructions that live on a spatial slice — the [[192, 130, 3]] stabilizer
code, the horizon bond count of
§
5, and the RT cell of
§
6 — are FCC statements. Statements
about propagation and emergent dynamics — isotropy of kinetic operators, the graviton, Lorentz
invariance — are D
4
statements, and must be: we show in
§
3 that the FCC slice by itself fails
the rank-four isotropy test that the full D
4
lattice passes exactly. Nothing in the entropy or
evaporation sections requires four-dimensional isotropy; they count static spatial structure on
the slice. Nothing in the gravity section requires the code; it uses only the intrinsic simplicial
geometry. The two sectors meet at the single scale L
0
, the nearest-neighbor bond length common
to both.
3 The linearized continuum limit of the lattice
A recurring and justified objection to discrete-spacetime models of this type is that the gravi-
tational language — horizons, Schwarzschild radii, the Bekenstein–Hawking normalization — is
imported rather than derived. This section establishes the strongest continuum statement we can
currently prove: treated as an intrinsic, background-independent simplicial geometry, the D
4
lat-
tice supports emergent linearized gravity, with the linearized Einstein operator recovered exactly.
The construction is weak-field and leading-order; it does not produce the nonlinear theory or the
Schwarzschild solution, and we continue to import those (
§
1.1). Full technical details, including
the explicit subdivision and the closed-form evaluation that makes the identity exact, are collected
in Appendix C; every quantitative diagnostic is reproduced by the verification scripts listed in the
Data Availability statement.
3.1 Exact rank-four isotropy
Theorem 1. The rank-four bond tensor T
µνρσ
=
P
n∈N
n
µ
n
ν
n
ρ
n
σ
over the 24 minimal vectors
N of D
4
equals 4 (δ
µν
δ
ρσ
+ δ
µρ
δ
νσ
+ δ
µσ
δ
νρ
) exactly. The hypercubic lattice Z
4
and the three-
6
D
4
(24 nn)
4
(8 nn)
FCC slice
(12 nn)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
T
1111
/
T
1122
isotropic value = 3
3.00
2.00
degenerate
(
T
1122
= 0)
(a) rank-four bond-tensor isotropy
TT trace gauge
1.0
0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
C
/|
k
|
2
(b) polarization sectors
D
4
Regge (computed)
linearized Einstein
0 20 40 60 80
angle of
k
: (0001) (1110) [deg]
1.100
1.075
1.050
1.025
1.000
0.975
0.950
0.925
0.900
C
(
k
)/|
k
|
2
spread = 1.5 × 10
9
(c) isotropy of the TT kinetic coefficient
TT +
TT ×
20 10 0 10 20
q
FP
( ,
k
) (lin. Einstein)
20
10
0
10
20
C
Regge
( ,
k
)
60 random ( ,
k
)
max rel. dev. = 4 × 10
8
(d) generic polarizations
Figure 1: Diagnostics of
§
3, all computed directly from the lattice constructions (verification
scripts in the Data Availability statement). (a) Rank-four isotropy: D
4
attains the isotropic
value exactly; the hypercubic and FCC-slice lattices do not. (b) Sector coefficients of the kinetic
operator on the explicit D
4
subdivision of Appendix C: the transverse-traceless polarizations are
degenerate, the trace sector carries the ghost-free Fierz–Pauli coefficient, and the gauge sector
is null. (c) The TT kinetic coefficient C(
ˆ
k)/|k|
2
as k sweeps from a coordinate axis to a body-
type diagonal: isotropic to a fractional spread of 1.5 × 10
−9
. (d) For 60 random symmetric
polarizations and directions, the Regge coefficient agrees with −q
FP
(Eq. (1)) to a few parts in
10
8
— the numerical shadow of the exact symbolic identity of
§
3.3.
dimensional FCC sublattice both fail this isotropy.
The diagnostic ratio T
1111
/T
1122
equals 3 for an isotropic tensor; direct enumeration (Fig. 1a)
gives 3 for D
4
, a degenerate value for Z
4
(the cross component T
1122
vanishes, leaving cubic
anisotropy), and 2 for the FCC slice. The structural reason is that the D
4
minimal vectors form a
spherical design of strength sufficient to render all moment tensors through rank four isotropic [29];
this is what makes the isotropy exact rather than asymptotic, and it is the condition for the leading
long-wavelength operators of the emergent theory to carry no anisotropic correction. [derived]
7
3.2 The graviton as the incompatible edge-length mode
The decisive methodological commitment is that the lattice is intrinsic (Regge-like): its fundamen-
tal dynamical data are the edge lengths {ℓ
e
}, with no embedding into an external flat space [26, 28].
By the Saint-Venant decomposition, edge-length perturbations split into a compatible part — re-
alizable by displacing vertices in flat space, carrying no curvature — and an incompatible part,
which cannot be removed by any repositioning.
Proposition 1. For any smooth displacement field u
µ
, the linearized Riemann tensor of h
µν
=
∂
µ
u
ν
+ ∂
ν
u
µ
vanishes identically.
A metric built by displacing points in flat space is pure gauge: the two transverse-traceless
polarizations of a wave along x
3
— the modes h
11
− h
22
and h
12
— are orthogonal to the entire
image of u 7→ ∂
(µ
u
ν)
. A metric built from a vector does not contain the spin-2 degrees of freedom.
The intrinsic edge-length field does: the 12 independent D
4
edge-direction dyads span the full
ten-dimensional space of symmetric rank-two tensors, so the edge strains determine a general h
µν
including the TT modes, and the TT modes carry genuine curvature (R
0101
= ω
2
/2 for the “+”
mode). The graviton is the incompatible sector of the edge-length field. That this family of lattices
supports non-removable intrinsic curvature is concrete: five regular tetrahedra sharing an edge
subtend 5 arccos(1/3) = 352.64
◦
, leaving the Regge deficit δ = 2π − 5 arccos(1/3) ≈ 0.128 rad
that no flat embedding can close — precisely the geometric frustration whose relief drives the
K=4 → K=12 crystallization in the companion matter analysis [23]. [derived]
3.3 The kinetic operator equals linearized Einstein
The dynamics of the intrinsic metric follow the Regge action S =
P
h
A
h
δ
h
, with A
h
the area of a
triangular hinge and δ
h
its deficit angle [26, 27]. On the flat background the Schl¨afli identity holds,
so the first variation vanishes and the physics is in the second variation — the graviton kinetic
operator. That linearized Regge calculus reproduces linearized general relativity is a classical
result of Roˇcek and Williams [27]; the content added here is D
4
-specific: the closed-form rational
kinetic tensor, the symbolic (rather than perturbative) identity, the exact rank-four isotropy
of Theorem 1, and the failure of the natural alternative lattices reported below. Rather than
assume the continuum limit has the required structure, we compute the operator on a concrete
D
4
Delaunay subdivision (169 vertices, 1696 four-simplices; Appendix C) and verify, term by term,
the hypotheses that identify it as Fierz–Pauli:
Gauge structure, no extra modes. The linearized deficit map has, about a fully interior vertex,
a kernel of dimension exactly 4 with a clean spectral gap; the null directions are exactly the
vertex translations, i.e., the linearized diffeomorphisms. This is the discrete origin of δh
µν
=
∂
µ
χ
ν
+ ∂
ν
χ
µ
.
Two-derivative, isotropic. The kinetic coefficient C(
ˆ
k) = S
(2)
/|k|
2
converges with O(k
2
) cor-
rections and is direction-independent to a fractional spread below 10
−8
(Fig. 1c); the naive
rank-four moment of the full edge set is not isotropic, but the physical Regge weights restore
the spherical-design isotropy of Theorem 1 in the weighted operator.
Exact identity. Because the Regge Hessian is finite-range, the leading kinetic tensor is a finite
lattice sum over a single vertex star, and closed-form evaluation of the dihedral-angle derivatives
renders all of its entries exactly rational. Contracting against general polarization ε
µν
and
wavevector k, the difference from the linearized Einstein quadratic form
q
FP
(ε,
ˆ
k) =
1
2
|ε|
2
− |
ˆ
k·ε|
2
+ (tr ε)(
ˆ
k·ε·
ˆ
k) −
1
2
(tr ε)
2
(1)
8
expands to exactly zero — a symbolic polynomial identity, not a numerical match. The sector
coefficients are C
TT
: C
trace
: C
gauge
= −1 : +3 : 0 (Fig. 1b), the linearized-Einstein values up to
the overall sign convention, with the ghost-free Fierz–Pauli trace coefficient; for generic random
polarizations the agreement with −q
FP
holds to a few parts in 10
8
(Fig. 1d), the numerical
shadow of the symbolic identity.
On this subdivision, the continuum kinetic operator of linearized D
4
Regge dynamics is the
linearized Einstein operator, identically: a single-cone, two-polarization graviton. [derived] The
identity is established on the stated subdivision; subdivision independence is a natural further
check and is listed among the open questions (
§
16.1).
A control on the alternatives. Rank-four isotropy is a moment condition on the bond vectors,
and it is fair to ask whether it is genuinely necessary or merely a convenient diagnostic. We
therefore ran the entire construction of this section, unchanged, on the two natural alternatives,
each with its own Delaunay decomposition at comparable patch size (Table 1). Both fail, and the
first failure is not anisotropy but flatness: the Delaunay decompositions of Z
4
and FCC ×Z carry
hinge deficits of order unity at the lattice scale, so there is no flat background to perturb about
and the linearization is ill posed before the kinetic operator is reached. Where the expansion
is nonetheless carried through formally, the transverse-traceless coefficient varies by factors of
two to four across directions, against a fractional spread of 1.5 × 10
−9
for D
4
: an O(1) Lorentz
violation in the gravitational sector rather than a small correction. We report the spread rather
than the coefficient itself, since on a curved background the small-k expansion is not valid and
the magnitude is an artifact of extrapolating it. The gravitational sector is therefore a statement
about the four-dimensional lattice that the natural alternatives cannot reproduce, not a property
any dense lattice would share. [derived]
Lattice Background flatness max |δ
h
| TT coefficient spread Outcome
D
4
(24 minimal vectors) 5.3 × 10
−15
1.5 × 10
−9
Fierz–Pauli
Z
4
(8) 3.8 4.3 no flat background
FCC ×Z (12+2) 4.9 2.2 no flat background
Table 1: The same Regge construction run on three lattices. Only D
4
has a flat Delaunay
background and a direction-independent kinetic coefficient. The comparison uses each lattice’s
own decomposition at comparable but not identical patch size (89, 169, and 215 vertices), so it is
a diagnostic on the natural alternatives rather than a uniqueness proof.
Selection of the Regge action within the lattice. The construction above adopts the
Regge action; we now report a computed result that substantially reduces the arbitrariness in
that choice, within the leading local frustration-linear family defined next. Assume only that
the effective action of the intrinsic geometric sector is local, supported on hinges, symmetric
under the lattice point group, and linear in the flatness obstruction — the leading order of an
elastic expansion in the frustration, the same expansion that organizes the companion matter
analysis [23]. On the subdivision of Appendix C the star hinges fall into exactly two congruence
classes, C
1
(squared edge lengths (2, 2, 2); 96 in the origin star) and C
2
((2, 2, 4); 29), so the most
general such functional is
S(w
1
, w
2
) = w
1
X
h∈C
1
A
h
δ
h
+ w
2
X
h∈C
2
A
h
δ
h
, (2)
9
with one weight per class; only the ratio matters, and the Regge action is the equal-weight point.
Two computations then select that point.
Proposition 2 (Selection of the Regge weights). Within the family (2), on the stated subdivision,
every origin-edge orbit carries a nonzero first variation of the action about flat space,
∂S
∂ℓ
2
e
flat
= 6 |w
1
− w
2
| for every origin edge e, (3)
so flat space is a solution if and only if w
1
= w
2
: the Regge action is the unique member of the
family that admits a flat vacuum.
The statement is independent of any metric identification, and the normalization is contained
in the derivative itself: since A
h
∼ (length)
2
and δ
h
is dimensionless, ∂S/∂ℓ
2
e
is a pure number,
independent of the lattice scale. At the flat background the (∂A
h
) δ
h
term vanishes (δ
h
= 0),
so Eq. (3) is exactly
P
h
w
c(h)
A
h
∂δ
h
/∂ℓ
2
e
. Its computation is not a symmetry shortcut: the per-
simplex Schl¨afli identity splits by congruence class into sums of ±
1
2
(short-edge variations) and
∓
3
4
(long-edge variations), and the signs vary across the simplices sharing an edge — 16 of one
sign against 4 of the other for a short edge — so cancellation must be computed rather than
assumed. It is incomplete: summed over all simplices containing the edge, the total is exactly
6(w
1
−w
2
) on the 24-edge short orbit and −6(w
1
−w
2
) on the long orbit, constant on each orbit by
the point group; the sign is orbit-dependent and carries no content, and either orbit alone already
forces w
1
= w
2
, since stationarity is a per-edge condition. Second, as corroboration at the fixed
edge-strain-to-metric identification: the linearized kinetic operator has the Fierz–Pauli form only
at equal weights, the relative covariance violation growing linearly as ≃ 4.3 ∆w, against 7 ×10
−9
at the Regge point (verified per class, at several asymmetries, and for random weight vectors).
Within the stated family, therefore, the lattice selects the Regge action; it is no longer a free choice.
[derived] within the family; the family itself — locality, hinge support, frustration linearity —
is an input [assumed]. One structural negative belongs with this result: the stabilizer layer is
strictly local, and the edge lengths are not code degrees of freedom, so the code cannot itself fix
the geometric dynamics; it enters the gravitational sector only through the entanglement census
that the RT relation calibrates (
§
4). Gravity is a property of the lattice, matter and entropy of
the code, and the two meet at the single scale L
0
— which is the division of labor
§
2.4 asserts,
now with its gravitational half computed rather than assumed.
3.4 Induced Newtonian coupling and the equivalence principle
A localized lattice defect sits at a geometrically forced position and is locally flat: its gravita-
tional influence is not a local curvature but its stress-energy, the elastic self-energy stored in its
compressed bonds, which is also its inertial mass. Classically, a self-equilibrated localized source
produces only a short-range Eshelby elastic field ∼ 1/r
2
; the Newtonian 1/r law is instead an
induced (Sakharov) effect [30, 31, 33]: integrating out the lattice phonon modes generates the
Fierz–Pauli kinetic term with coupling 1/(16πG
ind
), and the defect stress-energy sources it. For
N
b
= 3 minimally coupled transverse modes with a Brillouin-zone-edge cutoff Λ = π/a,
1
16πG
ind
=
N
b
(1/6) Λ
2
32π
2
=⇒ G
ind
=
4
π
a
2
≃ 1.273 a
2
, (4)
with the O(1) prefactor C = 4/π cutoff-scheme dependent. Matching G
ind
to the observed
Newton constant then fixes a = ℓ
P
/
√
C ≃ 0.89 ℓ
P
: Planck-scale spacing follows from the measured
10
strength of gravity, with no horizon input. The static limit gives ∇
2
Φ = 4πG
ind
ρ and Φ =
−G
ind
M/r. Because every excitation propagates on the same emergent metric, matter couples
universally via
1
2
h
µν
T
µν
and the gravitational and inertial masses coincide to leading order —
conditional on the induced matter coupling being exactly generally covariant, which is the keystone
open question of the gravitational sector and is not settled here. [derived] up to the O(1)
prefactor and the stated covariance condition.
3.5 What this section does and does not license
Equation (4) and the Fierz–Pauli identity justify, at leading order, three usages made down-
stream: (i) speaking of a metric and of weak-field gravitational potentials on the lattice; (ii)
taking the gravitational coupling to be of Planck magnitude, which fixes the order of magnitude
of L
0
independently of black-hole physics (
§
4); and (iii) treating localized defects as stress-energy
sources with equal gravitational and inertial mass. They do not license the Schwarzschild geom-
etry, trapped surfaces, or horizon thermodynamics, which remain imported consistency inputs
[imported]. The gap between the linearized theory established here and the strong-field regime
assumed at the horizon is the principal theoretical debt of the paper.
4 Newton’s constant and the lattice scale
Fixing the bond length L
0
by a single calibration against the Bekenstein–Hawking normalization
would leave the recovered entropy coefficient non-predictive: the scale would be set by the very
quantity the model aims to explain. The induced-gravity result of
§
3.4 avoids that circularity,
and the logic below is structured accordingly.
4.1 The magnitude of L
0
: induced coupling matched to the observed G
N
Equation (4) ties the emergent Newton constant to the lattice cutoff, G
ind
= C a
2
with C = O(1)
and cutoff-scheme dependent. Matching to the observed Newton constant, G
ind
= G
N
, fixes
a = ℓ
P
/
√
C: the Planck magnitude of the lattice scale follows from the measured strength of
gravity — a matching, not a derivation from the lattice alone, but one in which black-hole physics
plays no role [calibrated] to the observed G
N
. Equivalently L
0
= c
1
ℓ
P
with c
1
= O(1). What
the scheme-dependent loop cannot fix is the single dimensionless number c
1
.
4.2 Fixing the O(1) coefficient by the bond-level RT relation
We pin c
1
by requiring the bond-level entanglement to satisfy the Ryu–Takayanagi (RT) rela-
tion [6], the discrete analog of Jacobson’s derivation of G
N
from local thermodynamics [7].
Proposition 3 (Lattice scale from FCC entanglement). For an FCC lattice of bond length L
0
in
which each bond is a maximally entangled Bell pair, requiring the RT relation to hold across the
fundamental (111) bilayer cell yields
G
N
=
p
2/3 L
2
0
4 ln 2
!
= ℓ
2
P
. (5)
Proof. (111) interlayer spacing. With conventional cubic side a
cube
= L
0
√
2 (nearest-neighbor
bond a
cube
/
√
2 = L
0
), the closest-packed (111) planes have spacing h = a
cube
/
√
3 = L
0
p
2/3 ≃
0.8165 L
0
. Bilayer cell. The minimal surface separating two adjacent (111) layers is the bilayer
11
cross-section of width L
0
and height h, A
1
= hL
0
=
p
2/3 L
2
0
. Entropy per bond. Each bond is a
Bell pair |Ψ⟩ =
1
√
2
(|+−⟩ + |−+⟩) with reduced state ρ =
1
2
⊮, giving S
bond
= ln 2 (one ebit). RT
at the bond. Imposing S
bond
= A
1
/(4G
N
) gives Eq. (5); setting G
N
= ℓ
2
P
fixes
L
0
= ℓ
P
s
4 ln 2
p
2/3
≃ 1.843 ℓ
P
. (6)
Equivalently, in terms of the plaquette scale a
plaq
= L
0
(8/3)
1/4
=
√
2 L
0
h ≃ 2.355 ℓ
P
,
G
N
=
a
2
plaq
8 ln 2
= ℓ
2
P
. (7)
Physically, G
N
= hL
0
/(4S
bond
): Newton’s constant is the product of the two independent FCC
length scales divided by four bond entropies.
4.3 Status
The division of labor is now explicit: the magnitude of L
0
is fixed by matching the induced
coupling to the observed Newton constant (order of magnitude, scheme-limited, no horizon input);
the coefficient c
1
= 1.843 is the same external constant imposed precisely, through the RT relation
— the model’s one calibration, absorbed here and never adjusted again. The consistency of the
two determinations — the scheme-limited match c
1
≃ 0.89 for the naive cutoff Λ = π/a against
the RT value c
1
= 1.843 — is a nontrivial internal check at the factor-of-two level, which is the
honest precision of a cutoff-dependent one-loop estimate. With L
0
fixed, the area-law coefficient
(
§
6) and the evaporation normalization (
§
7) contain no further freedom beyond the flagged O(1)
modeling inputs.
5 The area law S ∝ A
The information accessible to an external observer is carried by the Z
2
bond phases of bonds
crossing ∂V; each crossing bond contributes one classical bit from its locked interior endpoint.
With 4 atoms per cubic cell of side a
cube
= L
0
√
2, the volume per atom is V
prim
= a
3
cube
/4 =
L
3
0
√
2/2, so the mean area per surface node is σ
A
= V
prim
/L
0
= L
2
0
√
2/2. Of the 12 bonds at
a surface node, K/2 = 6 point outward by isotropy, and an outward bond crosses the boundary
only when its far endpoint lies outside, which on average halves the count; the mean number of
crossing bonds per surface node is thus ν = K/4 = 3, giving
N
bonds
= ν
A
σ
A
=
3
√
2 A
L
2
0
, S = N
bonds
ln 2 =
3
√
2 ln 2
L
2
0
A. (8)
This establishes S ∝ A [derived]; the direct bond count of Fig. 2 confirms the linearity for
R = 4ℓ
P
to 16ℓ
P
and reproduces the coefficient of Eq. (8) with a fitted slope within 1% of 3
√
2
and R
2
= 0.9999. This bulk coefficient is not yet 1/(4ℓ
2
P
); the gap is closed in the next section
by applying the RT relation of
§
4 a second time, at the horizon scale.
12
0 500 1000 1500 2000 2500 3000
horizon area
A
(
2
P
)
0
1000
2000
3000
4000
entropy
S
(units of ln 2)
direct bond count is linear in A
direct FCC bond count (24 radii)
bulk count,
S
= 3
2
ln 2
A
/
L
2
0
(
R
2
= 0.9999)
projected coefficient
S
=
A
/(4
2
P
)
Figure 2: The direct FCC count verifies the linear area scaling of the bulk bond entropy; the
dashed line shows the Bekenstein–Hawking coefficient obtained after the RT-cell reduction of
§
6.
Points: bonds crossing spheres of 24 radii centered at a generic (non-lattice) point; line: Eq. (8).
6 The coefficient: one bit per RT cell
Section 5 counts every bond severed by the horizon. That count is a bulk quantity, and the entropy
accessible on the horizon is not the same thing: the same Ryu–Takayanagi relation that fixed G
N
at the bond scale in
§
4 also fixes how much entropy a unit of horizon area carries. Making the
two applications consistent is what fixes the entropy per unit area, with no additional geometric
input and no second calibration.
The bulk count is already the orientation average. The coefficient of Eq. (8) is not tied to
any lattice plane. The areal crossing density varies by 29% across orientations (from 2
√
3/L
2
0
at
the densest (111) plane to 2
√
5/L
2
0
), but a spherical horizon samples the average, ⟨N ⟩ = 3
√
2/L
2
0
,
which equals the bulk count exactly. Appendix A derives the density, verifies the average against
a direct scan over 2 × 10
6
orientations (agreement to four figures), and shows that faceting the
horizon would not change the count: the Wulff shape severs only 3.4% fewer bonds than the
sphere of equal volume, its area cost almost exactly canceling its lower crossing density. No
screen orientation needs to be selected, and none is.
The bulk-to-RT-cell compatibility ratio is computed. Proposition 3 assigns one ebit to
the (111) bilayer cell of area A
1
= hL
0
=
p
2/3 L
2
0
. The number of bulk bonds crossing one RT
cell is therefore not a free choice but a product of two quantities already fixed:
ν
∗
= ⟨N⟩A
1
=
3
√
2
L
2
0
·
q
2
3
L
2
0
= 2
√
3 ≃ 3.464. (9)
13
The horizon carries one bit per RT cell. Applying the RT relation at the horizon scale
with the same cell that was used at the bond scale, each A
1
of horizon carries one ebit, so
S = ln 2
A
A
1
=
⟨N⟩A
ν
∗
ln 2 =
q
3
2
A ln 2
L
2
0
. (10)
Rearranging Proposition 3, 1/(4ℓ
2
P
) =
p
3/2 ln 2/L
2
0
, so
S =
A
4ℓ
2
P
. (11)
Proposition 4 (Coefficient). Once L
0
is fixed by the bond-level RT matching (
§
4), applying the
same relation at the horizon scale gives one ebit per bilayer cell and returns S = A/(4ℓ
2
P
) with no
further free parameter.
We state the logical content of this precisely, because it is easy to overstate. Fixing A
1
=
4G
N
ln 2 at the bond scale makes S = (A/A
1
) ln 2 = A/(4G
N
) at the horizon scale immediate; the
coefficient 1/4 therefore comes from the RT postulate together with the calibration, and not from
the lattice. What the lattice supplies is the compatibility: the FCC bulk bond count reduces to
the RT-cell count by the ratio ν
∗
of Eq. (9), which is computed rather than chosen, so the same
cell assignment can be used at both scales without a second calibration. The sharp content of the
compatibility is a capacity condition: one ebit per cell requires at least one crossing bond per cell,
ν
∗
≥ 1, and the computed ν
∗
= 2
√
3 satisfies it with margin (it would hold, more narrowly, even
at K = 6, where ν
∗
= 1.73; see Appendix A). Had the lattice furnished fewer than one microscopic
carrier per RT cell, no consistent assignment would have existed. That is a real but limited claim,
and it is the one we make. A seemingly natural alternative obtains the same coefficient through
a bulk-to-screen “projection factor” tied to the (111) plane; that route requires a preferred screen
orientation and a dimensional-reduction prescription selected rather than derived (Appendix A).
The route used here needs neither: the count is the orientation average (computed, Appendix A),
ν
∗
is a consistency ratio (computed), and the one remaining assumption is the RT relation itself,
stated as a postulate in
§
4 and used twice. ν
∗
numerically equals
√
K at K = 12, but that is a
coincidence of this coordination and not a general formula (Appendix A); the structural quantity
is the number of bulk bonds per RT cell. [derived] for ⟨N⟩ and ν
∗
; [assumed] for the RT relation
and its horizon-scale application.
6.1 What is derived, what is calibrated
The entropy result has three logically separate parts, and conflating them overstates the case. (i)
The area scaling S ∝ A is an internal result of the model: it follows from bond counting alone,
requires no external input, and is confirmed by direct enumeration (Fig. 2). (ii) The calibration of
L
0
is a match to known physics: Eq. (6) fixes the one free lattice scale by imposing G
N
= ℓ
2
P
, which
is the Bekenstein–Hawking normalization itself; this step is not predictive of that normalization,
because it uses it (the induced-gravity loop of
§
4 independently fixes the order of magnitude, but
not the coefficient). (iii) The recovery of the coefficient 1/4 is then a consistency statement: with
L
0
fixed by (ii), applying the same RT relation at the horizon scale returns A/(4ℓ
2
P
) with no
further parameter. Its content is that one calibration suffices for both, not that the number 1/4
is predicted from nothing. The absolute normalization is therefore obtained only after matching
to black-hole thermodynamics; fixing c
1
independently — from the stabilizer-code structure, a
microscopic energy scale, or any observable not involving horizon entropy — remains the principal
open problem of the entropy sector.
14
7 Geometric evaporation dynamics
The same horizon supports a non-thermal decay channel. The intact K = 12 lattice surrounding
the vacancy exerts a surface tension on the boundary; modeling the horizon as a phase boundary
that recedes by curvature-driven re-stitching gives a lifetime scaling as M
2
. The ingredients of this
section are physically motivated interface dynamics, not a microscopic derivation from the lattice
Hamiltonian; each input is flagged, and the robustness of the M
2
exponent against alternative
interface laws is treated in
§
7.4.
7.1 Surface tension
Each removed stabilizer carries entropy ln 2 over horizon area L
2
0
, and dissolves into Bell-pair
halves of characteristic energy E
bond
= ℏc/(4L
0
) [24]. The surface energy per unit horizon area
is therefore
σ =
E
bond
L
2
0
=
ℏc
4L
3
0
. (12)
7.2 Boundary recession
A mobility relating interface velocity to pressure has dimensions m
4
(J s)
−1
; the only such com-
bination from {L
0
, ℏ} is µ
int
= β L
4
0
/ℏ with β = O(1) (we set β = 1 and flag it as the single
undetermined dynamical coefficient, which rescales τ
Geo
but not its M
2
scaling) [assumed]. The
overdamped law
˙
R = −µ
int
P with Laplace pressure P = 2σ/R
H
gives
˙
R
H
= −
L
4
0
ℏ
·
2
R
H
·
ℏc
4L
3
0
= −
c L
0
2R
H
, (13)
subluminal for every R
H
> L
0
/2, i.e., for every black hole.
7.3 Lifetime
With R
H
= 2GM/c
2
,
˙
M = (c
2
/2G)
˙
R = −c
5
L
0
/(8G
2
M), and integrating from M to 0,
τ
Geo
=
4G
2
c
5
L
0
M
2
=
4ℓ
P
L
0
t
P
M
m
P
2
. (14)
With the calibrated L
0
≃ 1.843 ℓ
P
from Eq. (6), the prefactor is 4ℓ
P
/L
0
≃ 2.17, so
τ
Geo
≃ 2.17 t
P
M
m
P
2
, τ
Geo
(10
15
g) ≃ 0.25 ms (unsuppressed), (15)
up to the O(1) mobility constant β. For comparison, the canonical Hawking mass M
⋆
≃ 5×10
14
g
completes thermal evaporation near the present age of the Universe; the geometric channel, where
it operates unsuppressed, is faster by more than twenty orders of magnitude. This raw figure
applies only to the unsuppressed channel; the physically relevant lifetime includes the barrier
suppression of
§
8, already non-negligible at this mass (R
H
≈ 1.5 fm).
15
7.4 Robustness of the M
2
exponent
The recession law assumes overdamped, curvature-driven interface motion. Since the exponent
is the paper’s main structural prediction, we ask what other interface dynamics would give, and
why they are disfavored within the adopted phase-boundary model. One caution first: labels like
“inertial” and “diffusive” are underspecified until one says what carries the inertia or performs
the transport — the interface alone, or the bulk it moves through — and the two readings give
different exponents. Table 2 therefore lists both readings where they differ.
Interface dynamics Equation of motion Lifetime Status here
overdamped, curvature-driven (model A)
˙
R = −A/R τ ∝ M
2
adopted
interface diffusive wander R
2
∼ Dt τ ∝ M
2
same exponent
conserved order parameter (model B, Ostwald) R
2
˙
R = −A τ ∝ M
3
excluded: requires con-
served transport
interface-inertial (underdamped)
¨
R = −A/R τ ∝ M excluded: interface carries
no momentum
bulk-inertial (Rayleigh collapse) R
¨
R +
3
2
˙
R
2
= −P/ρ τ ∝ M
3/2
excluded: no bulk inertia
to carry
ballistic, constant rate
˙
R = −v
0
τ ∝ M excluded: ignores the cur-
vature drive
tunneling-controlled rate ∝ e
−S(R)
super-polynomial absorbed into the barrier of
§
8
Table 2: Lifetime scaling for the standard interface-dynamics classes; the inertial and diffusive
rows are split by whether the interface or the bulk carries the relevant quantity. The exponent is a
diagnostic of the dynamical class rather than a generic consequence of surface tension, which makes
the prediction more specific rather than weaker. (The inertial integrals are verified numerically;
e.g.,
¨
R = −A/R from rest gives τ = R
0
p
π/2A.)
Three physical properties of the lattice boundary select the first row. The order parameter
is not conserved: the saturated region is radiated away as the boundary recedes (
§
7.5), not
redistributed to other parts of the interface, which is what a conserved (Ostwald, model B)
evolution would require. This excludes τ ∝ M
3
, which is worth noting because that exponent
coincides with the Hawking scaling and would make the geometric channel indistinguishable from
the thermal one. The boundary carries no momentum, and there is no bulk medium to carry it
either: advance by one cell is a discrete stochastic stabilizer flip (
§
8), a transition rate rather
than an acceleration, which removes both inertial rows — the interface-only reading (τ ∝ M)
and the Rayleigh-collapse reading (τ ∝ M
3/2
), the latter requiring an inertia-bearing bulk fluid
that the vacancy does not contain. The driving is curvature: the Laplace pressure 2σ/R
H
on a
closed boundary is what sets the rate, so the recession speed cannot be independent of R
H
, which
excludes the ballistic case. What survives is τ ∝ M
2
, from either the deterministic curvature-
driven law or diffusive interface wander, which give the same exponent with different prefactors.
We stress the epistemic status: this is an argument from the qualitative character of the
boundary dynamics, not a derivation from a lattice Hamiltonian or a master equation for the
single-cell transition probability, which we have not constructed [assumed]. A microscopic treat-
ment establishing the transition rate from the code dynamics would settle the exponent rather
than motivate it, and is listed in
§
16.1. Because the exponent discriminates sharply between
classes, a reconstructed PBH mass function is in principle a measurement of which class operates;
the cosmological analysis of
§
10 is presented for the M
2
law, with the understanding that the
16
qualitative shift of constraints from present-day to early-Universe probes survives any accelerated
channel, while the quantitative mass–epoch map does not.
7.5 Nature of the emitted energy
Three questions about this channel bear on the cosmological analysis below. What is emitted. As
the boundary recedes it severs the Bell pairs of the cells it sweeps; each severed bond releases an
elementary lattice excitation of characteristic microscopic energy E
bond
= ℏc/(4L
0
) (Eq. (12)). We
do not claim that the naive product N
severed
E
bond
equals the rest energy: with N
severed
∝ A ∝ M
2
and a mass-independent E
bond
, that product scales as M
2
rather than as Mc
2
, and it exceeds
the rest energy by many orders of magnitude for any astrophysical mass. Energy conservation
instead requires the mean energy of the independently emitted quanta to fall with mass; with the
RT-cell count of
§
11 it must be ¯ε = M c
2
A
1
/A ∝ 1/M, of order the horizon temperature. The
microscopic reconfiguration occurs at the bond-energy scale, but the asymptotic outgoing quanta
must be far softer; the conversion between these scales is not constructed, and we treat it as open
[assumed]. Spectral character. The emission is non-thermal at the lattice level: it is set by the
discrete bond energy and the recession kinematics rather than by a horizon temperature, and it
carries no presumption of a Planck spectrum. Coexistence with Hawking. The two channels are
independent and additive. The semiclassical Hawking process, τ
Hawk
∝ M
3
, continues to operate
on the same horizon; the shorter of the two lifetimes dominates. For R
H
≲ ξ the geometric channel
is unsuppressed and faster, so it sets the lifetime; for R
H
≫ ξ it is frozen by the nucleation barrier
(
§
8) and Hawking emission is recovered unchanged. The geometric channel does not modify or
contradict the semiclassical result; it is a Planck-scale lattice effect switched on only near the
metric wall and exponentially absent for macroscopic horizons, so standard astrophysical black-
hole phenomenology is left intact. Because the channel acts in the early Universe by depositing
energy, the cosmological bounds of
§
10 depend on the total electromagnetic energy released and
the epoch of release; deposition efficiencies are in general species-, energy-, and redshift-dependent,
which
§
10 folds into an O(1) effective fraction and states as its precision class [assumed].
8 Boundary locking: a curvature-gated nucleation barrier
Boundary advance by one cell converts a region of U (1) exterior into Z
2
interior. Whether that
conversion is fast or slow is an interface-kinetics question, and we treat it as one: the horizon of
§
7 is a phase boundary under curvature-induced pressure, and the rate at which such a boundary
advances is set by the barrier to nucleating a new layer on it. This section derives that barrier,
finds it linear in R
H
, and states the two conditions it requires. We then ask separately what the
error-correcting structure of the vacuum contributes, and report — by direct computation on the
code (Appendix B) — that no static code quantity supplies the same growth.
The barrier from curvature-gated terrace nucleation. The lifetime acquires a multiplica-
tive factor exp(∆G
∗
/ε
0
), where ∆G
∗
is the barrier to advancing the boundary and ε
0
the fluctu-
ation scale, so the scaling of ∆G
∗
with R
H
is the load-bearing quantity of the cosmology and it
deserves an argument rather than an analogy. The natural classical worry is Peierls–Nabarro kink
motion [35], whose kink-pair nucleation barrier is independent of the interface size; were that the
mechanism here, the suppression would be a constant and the cutoff would carry no prediction.
That worry, however, applies to an interface advancing under a fixed driving force. The boundary
of
§
7 is driven by curvature: the Laplace pressure P = 2σ/R
H
falls with the horizon radius, and
17
for a faceted interface at small driving the advance is nucleation-limited in the Burton–Cabrera–
Frank sense [32]. A terrace of the new phase of radius ρ on the facet costs step free energy γ
st
per unit rim length and gains P h per unit area, with h =
p
2/3 L
0
the interlayer spacing, so
∆G(ρ) = 2πρ γ
st
− πρ
2
P h, ∆G
∗
=
πγ
2
st
P h
=
πγ
2
st
2σh
R
H
, (16)
a barrier linear in R
H
from purely classical interface kinetics: the same curvature that drives the
recession in
§
7 gates it here. The corresponding suppression is exp(∆G
∗
/ε
0
) = exp(R
H
/ξ) with
ξ = 2σhε
0
/(πγ
2
st
), where we take the fluctuation scale ε
0
= E
bond
; a quantum-nucleation action
scale at T → 0 [34] would change the required γ
st
by orders of magnitude without altering the
R
H
-dependence.
Two conditions underwrite Eq. (16), and both are themselves constrained. First, the interface
must be faceted, below its roughening transition: a kinetically rough interface advances with
no barrier at any radius, which would evaporate the entire asteroid-mass window, in 2.5 s at
10
17
g and 2.5 × 10
10
s at 10
22
g (without any suppression, the M
2
law reaches t
univ
only at
10
25.6
g; the suppression is what preserves the window). We are careful about the logical status of
this: no asteroid-mass primordial population has been observed, so it is a conditional consistency
requirement rather than an observational constraint; if PBHs populate that dark-matter window,
their survival requires the faceted regime within this model. (Stellar holes are not the discriminant:
even unsuppressed, the M
2
law gives them ∼ 10
33
s.) This is compatible with the orientation-
averaged bond count of
§
6, which concerns a macroscopically spherical horizon: a sphere that
is smooth on large scales is microscopically terraced, and it is the terraces, not a global Wulff
facet, that Eq. (16) requires. Second, the driving must be purely interfacial: a bulk free-energy
difference ∆g between the saturated interior and the U(1) vacuum would add an R
H
-independent
term, P → ∆g + 2σ/R
H
, and saturate the barrier at large R
H
at the constant πγ
2
st
/(∆g h),
restoring exactly the fatal size-independent case. Whether the two phases are bulk-degenerate
is a defined question on the complex that we have not settled; it is the sharp classical falsifier
(
§
16.3). Equation (16) also exposes what the femtometer input means microphysically: with σ,
h, and ε
0
at the lattice scale, ξ ∼ 1 fm is equivalent to an anomalously small step free energy,
γ
2
st
≃ 2σhε
0
/(πξ), which the framework does not explain: the hierarchy of
§
8.2, now located in a
definite interfacial quantity. [derived] given the two stated conditions; the conditions themselves
are open.
Writing the suppression in the form used below,
τ
eff
Geo
= τ
Geo
exp
R
H
ξ
, (17)
with ξ as above, identified numerically with the QCD-confinement scale, ξ ∼ 1 fm (
§
8.2).
Two routes, not one mechanism. Equation (16) is a classical suppression, controlled by
the step free energy, the surface tension, the layer height, and the fluctuation scale. The error-
corrected vacuum suggests a second and logically independent route: a partially advanced patch is
syndrome-bearing, so below threshold the code restores it before it can spread, and net recession
requires a conversion that evades detection, giving (p/p
th
)
d
eff
/2
with d
eff
the weight of the cheapest
undetectable conversion. We stress that d
eff
is not the code distance. The FCC code has d = 3 at
every system size [25], and a constant distance yields a suppression independent of n; but d is a
minimum over the entire logical group, and asserting that some operator has weight three places
no constraint on the weight of any particular operation. The quantity relevant here is the weight
18
of one specific reconfiguration, not the cheapest operation the code admits. The two routes have
the same functional form when d
eff
∝ R
H
, but they are not the same mechanism: no mapping
between ∆G
∗
/ε
0
and (d
eff
/2) ln(p
th
/p) has been established, and their controlling quantities are
disjoint. We keep them separate, and only the classical route is derived.
The code route is not supported by any static quantity. We computed the static code-
theoretic content of a vacancy explicitly (Appendix B; a finite-size study at L = 6 and L = 8
over wrap-free vacancy radii): puncturing the [[3L
3
, 2L
3
+ 2, 3]] code with a spherical vacancy
leaves the minimum bulk logical weight at exactly 3 at every radius studied — explicit weight-3
logicals exist arbitrarily far from the vacancy — and one boundary-node conversion removes at
most K = 12 edge qubits, affects at most 13 vertex checks and 6 octahedral checks, and carries
Z
2
-syndrome weight 12, all independent of R. No static code quantity grows with the vacancy.
If the code route contributes a growing exponent at all, its origin must be dynamical, in the
requirement that intermediate configurations along a conversion sequence evade detection; that
minimization, over sequences with syndrome extraction and correction interleaved, is not carried
out here. The vacuum’s operating point p/p
th
≃ 0.13 (
§
8.1) makes the below-threshold picture
applicable but does not fix how any weight scales. Accordingly the growing barrier we adopt is
the classical one of Eq. (16), and the code enters as a possible modification of its dynamics rather
than as its source.
Linear or areal, and why it matters less than it appears. Two geometries for a growing
exponent are natural: linear, R
H
/ξ (the classical terrace, whose transition state is a single critical
rim), or areal, (R
H
/ξ)
2
(coherent conversion over the whole boundary, the form a code-dynamical
treatment could in principle select instead). The classical mechanism selects the linear form under
the stated conditions; the quantum case is unsettled and is the defined calculation flagged in
§
16.3.
The choice matters less than the exponential dependence suggests:
exponent geometry M
cut
R
H
(M
cut
) d log τ /d log M at M
cut
constant (no growth) free parameter — 2
linear, R
H
/ξ 10
16.45
g 42 fm 44
areal, (R
H
/ξ)
2
10
15.66
g 6.8 fm 94
The two growing geometries place the cutoff 0.8 decades apart, both below the asteroid-mass
window, and both give an edge far sharper than the Hawking roll-off, whose slope at the same
mass is 3. The existence, approximate location, and sharpness of the cutoff are therefore robust to
this ambiguity; only its precise position is not. The one case that would destroy the prediction is
the constant barrier, which in the classical analysis requires either a kinetically rough interface (in
tension with the survival of an asteroid-mass population) or a bulk driving term ∆g = 0 (open).
We adopt the linear geometry below and quote M
cut
≃ 10
16.5
g, with the understanding that an
areal geometry would move it to ≃ 10
15.7
g. [assumed] for the adopted linear geometry beyond
the classical regime.
8.1 Survival cutoff
Setting τ
eff
Geo
= t
univ
with R
H
= 2GM/c
2
gives a sharp cutoff at
M
cut
≈ 10
16.5
g, R
H
(M
cut
) ≈ 42 fm. (18)
19
Reproducing this requires R
H
/ξ ≃ 42. (In the code-route reading of
§
8, the same exponent
corresponds, via p
L
∼ (p/p
th
)
d
eff
/2
, to p/p
th
≃ 0.13, the vacuum operating about an order of
magnitude below its own error threshold — the expected regime for a stable code; this number
parametrizes the below-threshold picture but plays no role in the classical derivation.) For a
10
15
g hole, R
H
/ξ ≃ 1.5: the recession rate is suppressed by e
−1.5
≃ 0.22, equivalently the lifetime
is enhanced by e
1.5
≃ 4.5, giving τ
eff
Geo
≃ 1.1 ms, still far below t
univ
. For a stellar-mass hole,
R
H
/ξ ∼ 3 ×10
19
and the suppression is indistinguishable from zero: such objects are boundary-
locked and decay only through the Hawking channel.
This cutoff sits below the asteroid-mass dark-matter band 10
17
–10
22
g, which it therefore
leaves intact; the geometric channel does not destroy the asteroid-mass scenario but predicts a
hard lower edge to the surviving PBH mass function near 10
16.5
g. The location depends on the
order-of-magnitude ξ (in the exponent) and the mobility β (in the prefactor) only logarithmically:
over a factor-of-four range in ξ and a factor-of-ten range in β, log
10
(M
cut
/g) spans just 16.2–16.8
(Table 3).
ξ = 0.5 fm ξ = 1 fm ξ = 2 fm
β = 0.3 16.15 16.44 16.73
β = 1 16.17 16.45 16.74
β = 3 16.18 16.46 16.75
Table 3: Sensitivity of the survival cutoff, log
10
(M
cut
/g), to the two uncertain inputs. The cutoff
varies by < 0.6 decade across the box and is essentially insensitive to β.
8.2 Status of the suppression scale ξ
We are explicit about the weakest input of the scenario. The microscopic bond length is L
0
∼
ℓ
P
; the suppression scale that freezes the channel is taken to be ξ ∼ 1 fm, twenty orders of
magnitude larger, and we have identified it with the QCD-confinement scale. No mechanism
within the present framework connects the interfacial step free energy — equivalently ξ — to the
confinement scale, and a Planck-scale vacuum has no obvious reason to develop a femtometer-
scale interfacial hierarchy; the identification is a hypothesis, motivated only by the observation
that the confinement scale is the longest microphysical length in the vacuum’s matter sector
[conjectured]. Two honest readings are therefore available. In the SSM-committed reading,
ξ ∼ 1 fm is a falsifiable hypothesis and Eq. (18) is its consequence. In the effective-law reading, ξ is
simply the second free parameter of the two-parameter evaporation law, M
cut
floats logarithmically
with it, and the observational program of
§
16.3 measures ξ rather than testing it. All downstream
numbers quote the ξ = 1 fm benchmark; the mapping to other values is given by Table 3 and by
the log-linear relation R
H
(M
cut
)/ξ ≃ 42.
9 Evaporation history of the sub-cutoff population
The geometric channel changes not only how fast sub-cutoff PBHs evaporate but when, moving
their energy release from the present epoch into the early Universe.
20
10
15
10
16
PBH mass
M
(g)
10
0
10
2
10
4
10
6
10
8
10
10
redshift of evaporation
z
evap
BBN (
z
10
9
)
thermalized (no distortion)
-distortion
y
-distortion
post-recombination / present
Figure 3: Redshift at which a sub-cutoff PBH evaporates through the geometric channel (ξ = 1 fm
benchmark). The lightest holes disappear well before BBN; the band approaching M
cut
injects
across the spectral-distortion and recombination epochs. Shaded regions mark the distortion eras.
9.1 Formation and evaporation times
A PBH forms with mass of order the horizon mass, t
form
∼ GM/c
3
≲ 10
−23
s for the masses
of interest; in the relic-void picture of
§
14 formation instead coincides with the crystallization
transition itself. Either way t
form
is negligible beside the geometric lifetime. A sub-cutoff PBH
therefore evaporates at cosmic time t
evap
≃ τ
eff
Geo
(M). In the radiation era, T ≃ 1 MeV (t/s)
−1/2
and 1 + z = T /T
0
(T
0
= 2.35 × 10
−4
eV) fix the evaporation redshift z
evap
(M) (Fig. 3): a 10
15
g
hole evaporates at z ∼ 10
11
(well before BBN), a 10
16
g hole at z ∼ 10
7
, and a 2 × 10
16
g hole
only at z ∼ 5 × 10
3
, just before recombination.
9.2 Six evaporation bands
Sweeping M from the lightest holes surviving to the radiation era up to M
cut
, z
evap
crosses
every major epoch, dividing the population into the bands of Table 4. The boundaries follow
from equating t
evap
(M) to each epoch: BBN onset (1 s) at 3.8 × 10
15
g, BBN end (300 s) at
6.8×10
15
g, the photodissociation threshold (10
4
s) at 8.8×10
15
g, double-Compton thermalization
(z
dc
≃ 2×10
6
) at 1.3×10
16
g, the µ/y boundary (z ≃ 5×10
4
) at 1.75×10
16
g, and recombination
(z ≃ 1100) at 2.2 × 10
16
g.
9.3 Inversion relative to Hawking
Under Hawking evaporation a 10
16
g hole has τ
Hawk
∼ 10
37
s and is observationally stable, con-
strained by its present-day γ-ray flux. Under the geometric channel the same hole evaporated at
z ∼ 10
7
and is long gone, its energy deposited into the pre-recombination plasma. The observable
consequence shifts in kind, from a present-day photon flux to an early-Universe energy injection,
21
Mass range (g) z
evap
Dominant cosmological probe
M ≲ 4 × 10
15
≳ 10
9
none (thermalizes before BBN)
4–7 × 10
15
∼ 10
8
–10
9
BBN light-element abundances
0.9–1.3 × 10
16
∼ 10
6.5
–10
7.5
D/He photodissociation
1.3–1.8 × 10
16
∼ 10
5
–10
6.3
CMB µ-distortion (FIRAS)
1.8–2.2 × 10
16
∼ 10
3
–10
4.7
CMB y-distortion (FIRAS)
2.2 × 10
16
–M
cut
≲ 10
3
recombination / present γ-rays
Table 4: Geometric-channel evaporation epoch vs. sub-cutoff PBH mass, and the probe constrain-
ing each band, for the ξ = 1 fm benchmark. Boundaries carry the O(1) normalization uncertainty
of Eq. (14) with the barrier suppression, and shift logarithmically with ξ.
10
10
10
4
10
2
10
8
10
14
10
20
10
26
10
32
10
38
PBH mass
M
(g)
10
50
10
38
10
26
10
14
10
2
10
10
10
22
10
34
10
46
lifetime (s)
age of universe
Planck time
asteroid-mass PBH
dark-matter window
M
cut
10
16.5
g
Hawking:
M
3
geometric (unsuppressed):
M
2
geometric + nucleation barrier
Figure 4: Black-hole lifetime vs. mass. Blue: standard Hawking, τ ∝ M
3
. Orange dashed:
unobstructed geometric channel, τ
Geo
∝ M
2
(overall coefficient uncertain at O(1)). Orange solid:
geometric channel with nucleation-barrier suppression exp(R
H
/ξ), active above ξ ∼ 1 fm. The
barrier-suppressed geometric lifetime equals the age of the Universe near M
cut
∼ 10
16.5
g (R
H
≈
42 fm): below this mass PBHs evaporate geometrically in well under a second; above it (including
the asteroid-mass dark-matter window) only Hawking decay survives. The qualitative content is
the change in scaling and the sharp lower cutoff, not the precise coefficient.
and the relevant constraints shift from γ-ray telescopes to BBN and the CMB spectrum. Fig-
ure 4 summarizes the lifetime structure across the full mass spectrum, and Table 5 the fate of
representative objects.
22
Black hole M (g) R
H
Geometric fate (barrier-suppressed)
PBH (below cutoff) 10
15
1.5 fm τ
eff
Geo
≈ 1.1 ms; evaporates
PBH (survival cutoff) ∼ 10
16.5
∼ 42 fm τ
eff
Geo
∼ t
univ
PBH (DM window) 10
17
–10
22
0.1–10
3
pm barrier-locked; survives
Stellar (Cyg X-1) 4 × 10
34
62 km barrier-locked
SMBH (Sgr A*) 8 × 10
39
1.2 × 10
7
km barrier-locked
Table 5: Geometric-channel fate across the BH mass spectrum including the nucleation barrier.
PBHs below M
cut
∼ 10
16.5
g evaporate geometrically in well under a second; those above, including
the asteroid-mass DM window, are barrier-locked.
10 Cosmological constraints
The bounds in this section are order-of-magnitude estimates: they use standard analytic injection
formalism with O(1) effective deposition fractions, published BBN exclusions rather than a ded-
icated reaction network, and the COBE/FIRAS limits rather than modern numerical distortion
calculations. The constraints are derived for a monochromatic mass function;
§
10.6 relaxes this
with lognormal convolutions. Within those limits, the qualitative conclusion — constraints move
from the present-day sky to the early Universe — is stable.
10.1 Energy-injection formalism
A population of PBHs of mass M and would-be dark-matter fraction f
PBH
carries comoving
energy density ρ
PBH
= f
PBH
ρ
DM
. When the population evaporates at z
evap
(M), a fraction f
EM
of the released rest energy is deposited electromagnetically; global energy conservation over the
hole’s history gives E
EM
= f
EM
Mc
2
per hole [assumed], with f
EM
≃ 0.5 our benchmark and
the assumption that high-energy lattice excitations thermalize with standard electromagnetic
deposition efficiencies flagged in
§
16.1. The fractional energy injected relative to the CMB photon
energy density at the injection epoch is
∆E
E
γ
≃ f
EM
f
PBH
ρ
DM
ρ
γ
z
evap
= f
EM
f
PBH
4.85 × 10
3
1 + z
evap
. (19)
10.2 Pre-BBN: the safely cleared population
Holes below ∼ 4 ×10
15
g evaporate at z ≳ 10
9
, before neutron–proton freeze-out completes; their
energy thermalizes into the radiation bath with no observable trace (the only residuals are a possi-
ble N
eff
contribution and negligible entropy dilution), and this entire band — monochromatically
unconstrained by any Hawking-era bound because the holes no longer exist — is otherwise uncon-
strained up to the total dark-matter density (its high-mass tails are bounded through extended
mass functions,
§
10.6).
10.3 BBN and photodissociation
Holes evaporating during or after nucleosynthesis (1–10
13
s) inject electromagnetic energy that
alters light-element abundances, first through n/p interconversion and hadronic showers, later
through photodissociation of D and
4
He. The published bounds are commonly expressed as
23
limits on ζ
EM
= f
EM
(E/m) Y , the EM energy per background photon; translating Eq. (19),
ζ
EM
≃ 3.1 × 10
−9
f
EM
f
PBH
GeV, (20)
essentially independent of M because both ρ
PBH
and n
γ
dilute together, so M enters only through
t
evap
(M), which selects the applicable published limit on ζ
EM
[14, 15, 16]. Photodissociation is
inactive for t
evap
≲ 10
4
s (injected energy is degraded below nuclear thresholds); deuterium
destruction dominates for 10
4
≲ t
evap
/s ≲ 10
6
, giving way to
4
He photodissociation later, where
the bound is strongest. Inverting Eq. (19) against the published ζ
EM
(t
evap
) exclusions yields the
BBN/photodissociation curve of Fig. 5, running from f
PBH
≲ 1 at the photodissociation onset
(∼ 9 ×10
15
g) to f
PBH
≲ 10
−2
near 2 ×10
16
g, switching off above 2.2×10
16
g. Hadronic injection
during nucleosynthesis (t
evap
< 300 s) would require a reaction-network treatment; it is flagged
and used to support no conclusion [assumed].
10.4 CMB spectral distortions
Energy injected after double-Compton decoupling (z ≲ 2 × 10
6
) cannot fully thermalize and
distorts the CMB spectrum: a µ-type chemical potential for 5 × 10
4
≲ z ≲ 2 × 10
6
and a y-type
distortion for z ≲ 5 × 10
4
[17]. Using the standard visibility approximation,
µ ≃ 1.4
∆E
E
γ
e
−(z/z
dc
)
5/2
, y ≃
1
4
∆E
E
γ
, (21)
the FIRAS limits |µ| < 9 × 10
−5
, |y| < 1.5 × 10
−5
[18] bound the bands of Table 4 at
f
PBH
≲ 10
−2
(µ band), f
PBH
≲ 10
−4
–10
−6
(y band), (22)
tightening toward M
cut
as z
evap
falls and the injection is less diluted. A PIXIE-class mission [17]
improves these by ∼ 10
3
and would either detect the injection pattern of a sub-cutoff population
or push f
PBH
below 10
−7
across the distortion bands [testable].
10.5 A shifted constraint landscape
Figure 5 assembles the resulting exclusion. Its qualitative content, robust to every O(1) input, is an
inversion of kind: constraints that were present-day (γ-ray flux from surviving holes) become early-
Universe (BBN and CMB energy injection from vanished holes), and the band below 4 × 10
15
g,
constrained under Hawking evaporation, becomes monochromatically unconstrained. The γ-ray
limits of Refs. [11, 12, 13] do not apply because the holes they constrain no longer exist; the
reference line in the figure marks where those limits would sit. We are careful about the logic:
one cannot relax a bound that does not operate, and the relaxed abundances describe a transient
early-Universe population that no longer exists today. It is not surviving dark matter, and the
result must not be read as opening an intermediate-mass PBH dark-matter window; the asteroid-
mass band sits above the cutoff (
§
8.1) and is untouched. The sharp lower edge of the surviving
mass function at M
cut
is the model’s distinctive signature: standard Hawking evaporation predicts
a smooth roll-off at M
⋆
≃ 5 × 10
14
g (lifetime slope 3), while the barrier produces an essentially
vertical edge (slope ≳ 44) at 10
16.5
g.
24
10
15
10
16
10
17
PBH mass
M
(g)
10
9
10
8
10
7
10
6
10
5
10
4
10
3
10
2
10
1
10
0
f
PBH
(would-be DM fraction)
survival
cutoff
combined band (all
O
(1) uncertainties)
BBN/photodissociation (reconstructed)
-distortion, FIRAS
y
-distortion, FIRAS
combined exclusion (central)
standard Hawking -ray (does not apply; reference)
Figure 5: Order-of-magnitude constraints on the would-be dark-matter fraction of sub-
cutoff PBHs under the geometric channel (ξ = 1 fm benchmark). Colored curves:
BBN/photodissociation (reconstructed from published ζ
EM
exclusions), µ- and y-distortion (FI-
RAS); black: combined central exclusion; gray band: variation over the O(1) inputs (f
EM
= 0.3–
0.7, mobility 0.5–2). The dotted line marks the present-day Hawking γ-ray bound that would
apply if the holes survived; it does not apply here. Above M
cut
(dashed vertical) the population
is barrier-locked and standard constraints on surviving PBHs take over.
10.6 Extended mass functions
The bounds above assume a monochromatic mass function. For an extended function ψ(M) the
standard prescription f
−1
max
[ψ] =
R
dM ψ(M )/f
max
(M) applies [10]. Convolving lognormal ψ of
widths σ
ln M
= 0.3–1.0 against the combined monochromatic bound of Fig. 5 gives two systematic
effects, both in the direction of robustness. Distributions centered in the thermalization-erased
band (M
c
≲ 1.3 × 10
16
g), unconstrained or weakly constrained in the monochromatic case,
acquire finite bounds from their high-mass tails leaking into the constrained bands: for example
f
PBH
≲ 1.4 × 10
−3
at M
c
= 5 × 10
15
g for σ
ln M
= 0.5, tightening to ∼ 4 × 10
−5
at σ
ln M
= 1.0.
Distributions centered near the cutoff are bounded within a factor of a few of the monochromatic
value. For distributions straddling M
cut
we conservatively require the sub-cutoff part alone to
satisfy the early-Universe bounds; the super-cutoff part is barrier-locked, injects nothing, and is
subject to the separate standard constraints on surviving PBHs. No order-of-magnitude loophole
appears for the tested lognormal family; we do not claim a result for arbitrary extended mass
functions. The monochromatic curve inherits a step where the photodissociation window closes
at t ≃ 10
12
s, an artifact of the hard window edge rather than a feature of the constraint; the
convolution smooths it.
25
11 Information: a conjectural bookkeeping
This section is more interpretive than the observational results above, and nothing in
§§
9–10
depends on it [conjectured].
11.1 Four levels
In the SSM the vacuum is an FCC lattice whose edges carry Bell pairs, and those edges are
the physical qubits of the CSS code [25]. There are therefore several structurally distinct scales
at which the lattice holds information, and L1–L4 are those scales, ordered from least to most
organized — equivalently, by the non-locality of the carrier: one edge, twelve edges, a coordination
shell, the entire complex.
L1 — bond. The bare fact that a Bell pair exists on a given edge. One binary fact per edge,
ln 2 of entropy. It is a counting statistic: it tells you how many bonds there are, not which is
which or how they relate. Purely local, zero relational content.
L2 — check. The ±1 eigenvalue of a weight-12 stabilizer — a vertex Z
2
check on its twelve
incident edges, or an octahedral X-check. One bit, but a joint bit: it is a property of the twelve
edges together and of no edge individually. This is the first relational level.
L3 — cell. The configuration of a full cuboctahedral neighborhood, the twelve-neighbor shell
around a node. This is geometric rather than code-theoretic data: in the intrinsic-Regge
language of
§
3 the local edge-length configuration is what carries deficit angle, so L3 is the
discrete stand-in for local curvature.
L4 — global. The joint pattern of stabilizer eigenvalues across the whole lattice, which selects
which codeword the vacuum is in — the encoded quantum state (n − k = 62 independent
generators reducing the 2
192
-dimensional physical space to the 2
130
-dimensional codespace at
L = 4). This is the information error correction exists to protect.
L1 is the only level with a derived counting in this paper; the higher levels are structural distinc-
tions supplied by the code language.
11.2 L1 is radiated; L2–L4 are not
As the boundary recedes, each swept RT cell releases its content to the exterior. The number of
emitted quanta over the hole’s history is the RT-cell count,
N
quanta
=
A
A
1
=
S
BH
ln 2
, (23)
one per horizon ebit — consistent by construction with
§
6, and carrying mean energy ¯ε =
Mc
2
A
1
/A (
§
7.5); we do not assume each quantum carries exactly one bit of thermodynamic
entropy to infinity, only that the emitted-quantum count equals the ebit count. The higher levels
do not cross the K=12 → K=0 boundary as structure: a stabilizer eigenvalue (L2) is a joint
property of its twelve edges, unrecoverable once they are severed; a cuboctahedral cell (L3) loses
its coordination center; and the global logical state (L4) is undefined once a macroscopic fraction
of stabilizers is severed, any re-crystallized region carrying a fresh, statistically independent L4
state. Indistinguishable constituents cannot carry relational structure once their relations are
broken: the count survives and is radiated, the organization does not. In this interpretation the
26
emitted L1 entropy is monotonic in time — there is no interior-to-exterior information return of
the Page type. Whether an effective Page-curve description [36, 37] nonetheless applies to coarse-
grained exterior observables is left open; the assertion that a changing Hilbert-space dimension
removes the unitarity requirement is not made, and constructing the explicit quantum channel
(or unitary embedding with ancilla) that realizes this bookkeeping is unfinished business. The
monotonic-emission picture and the remnant endpoint are jointly the model’s conjecture, stated
so that it can be confronted rather than defended.
12 Comparison with other approaches
Table 6 positions the construction against the two programs with microscopic entropy counts,
along the axes a referee of any of the three would demand. All three recover the Bekenstein–
Hawking coefficient in some restricted setting: the fuzzball count is exact but only for extremal
holes in ten dimensions; LQG applies to non-extremal holes in 3+1 but fixes the coefficient by
tuning the Immirzi parameter; the present construction applies to non-extremal holes in 3+1 but
fixes its one lattice coefficient by matching to the same normalization it then reproduces. The
distinguishing observable of this model is the evaporation law, not the entropy.
String/fuzzball LQG SSM (this work)
Microscopic d.o.f. D-brane bound states spin-network punctures FCC bond phases / stabilizer
code
Horizon smeared by microstate ge-
ometry
punctured surface Z
2
/U(1) phase boundary
Spacetime dim. 10 3+1 3+1 (spatial slice of D
4
)
Non-extremal holes no (exact count) yes yes
S = A/4G: origin derived (extremal) derived up to Immirzi tun-
ing
calibrated (L
0
) + RT postulate;
compatibility computed
Singularity resolved (fuzzball) resolved (bounce) resolved (saturation)
[conjectured]
Evaporation law Hawking Hawking Hawking + geometric M
2
chan-
nel below cutoff
Information unitary (AdS/CFT) unitary (expected) conjectural monotonic bookkeep-
ing (
§
11)
Continuum limit exact (string theory) semiclassical limit open linearized Einstein derived (
§
3);
nonlinear open
Distinctive observ-
able
none at astrophysical scales none established PBH mass-function edge at
10
16.5
g; shifted constraint land-
scape
Table 6: Comparison along the axes of assumptions, degrees of freedom, horizon treatment, en-
tropy derivation, singularity resolution, evaporation, information, and observability. Memory-
burden scenarios [19, 20, 21, 22] also slow evaporation below ∼ 10
10
g and open a light-PBH
window; they differ observationally from the present model, which accelerates sub-cutoff decay
instead.
13 Predictions
The survival cutoff and the shifted landscape [testable]. The reconstructed PBH mass function
should show a hard lower edge near 10
16.5
g (or 10
15.7
g for the areal exponent geometry), far
sharper than the Hawking roll-off; and the abundance of sub-cutoff holes is bounded by early-
Universe injection rather than present-day flux, testable by a PIXIE-class distortion measurement
correlated with BBN anomalies along the mass–epoch map of Table 4. These two tests stand
27
independently of the lattice hypothesis, as constraints on an effective two-parameter evaporation
law.
Macroscopic holes: null results [derived] given the imported exterior. Two independent
suppressions operate, quantified in Table 7: discreteness corrections to horizon-scale observables
enter at O(ϵ
p
) with ϵ = ℓ
P
/R
H
≲ 10
−39
and p ≥ 1 — shadow diameters, photon-ring radii, and
lensing angles against EHT precision of order 10
−1
, quasinormal frequencies and ringdown times
against LIGO–Virgo ringdown precision of order 10
−2
, and tidal deformabilities identically —
while the geometric channel is frozen by exp(R
H
/ξ) with exponents of 10
19
–10
28
. Deviations are
suppressed uniformly, not selectively.
Source R
H
ϵ = ℓ
P
/R
H
R
H
/ξ
Stellar-mass hole (10 M
⊙
) 3.0 × 10
4
m 5.5 × 10
−40
3.0 × 10
19
Cyg X-1 (21 M
⊙
) 6.2 × 10
4
m 2.6 × 10
−40
6.2 × 10
19
Sgr A* (4.3 × 10
6
M
⊙
) 1.3 × 10
10
m 1.3 × 10
−45
1.3 × 10
25
M87* (6.5 × 10
9
M
⊙
) 1.9 × 10
13
m 8.4 × 10
−49
1.9 × 10
28
Table 7: Discreteness parameter and barrier exponent for representative astrophysical black holes.
Fractional deviations from general relativity enter at O(ϵ
p
), p ≥ 1.
Sub-cutoff PBH transits, femtolensing, and dark-matter capture signatures involving 10
15
–
10
16.5
g objects are all predicted absent, since that population no longer exists. The model is
falsified outright by a confirmed present-day detection of any sub-cutoff PBH (
§
16.3); it is not
testable through electromagnetic or gravitational-wave observations of stellar or supermassive
holes, whose exterior physics it reproduces by construction of the imported Schwarzschild back-
ground, with lattice corrections bounded by (L
0
/R
Schw
)
2
≲ 10
−77
.
14 Dynamical black holes: formation, growth, and mergers
The stationary construction of
§
2 does not by itself describe formation. Worse, taken at face value
the locking barrier of
§
8 acts symmetrically: the same nucleation cost that freezes recession above
ξ freezes advance, so a macroscopic saturated region could not grow by boundary conversion any
more than it can shrink. A formation story is therefore not optional, and we state the one the
framework suggests, at the conjectural level it deserves [conjectured].
Relic voids of the crystallization transition. In the SSM cosmology the vacuum itself crystal-
lizes from a disordered phase into the K = 12 lattice. First-order crystallization generically leaves
behind uncrystallized inclusions — pockets the advancing fronts fail to close. Such a pocket is,
by the duality of
§
2.2, exactly an SSM black hole: a K = 0 vacancy bounded by intact lattice.
On this picture primordial holes are not formed by matter collapse at all; they are relic voids,
present from the transition, whose subsequent history is pure boundary recession — the physics
of
§§
7–8, which never requires growth. The barrier symmetry problem then does not arise for the
primordial population: nothing ever had to grow. Two consistency remarks support the picture
without establishing it. The scales are consistent without adjustment: a trapped void of radius
1–50 fm corresponds to 10
14.8
–10
16.5
g, exactly the mass range the cosmology of
§§
9–10 concerns,
and the surviving population is whatever recession has not yet consumed — everything below
M
cut
is gone, while the shape of the surviving function above it depends on the underived bubble-
size distribution. And a relic void needs an assigned mass: the identification of a void of radius
R with a hole of mass M = Rc
2
/2G presumes that the surrounding lattice reads the vacancy as
28
gravitating positive energy of that magnitude; the elastic energy stored in the terminated-bond
shell is a candidate accounting, but we have not carried it out, and we flag the mass assignment
as an open element of the conjecture alongside the K = 0 ≡ Z
2
duality itself.
Stellar collapse. For holes formed by matter collapse the framework currently has no mech-
anism to override the growth barrier, and we do not hide the tension: either collapse proceeds
through a channel that is not boundary-layer nucleation (bulk saturation of a large region at once,
plausibly barrier-free but unmodeled), or the model is incomplete precisely where astrophysical
holes are formed. The phenomenology of
§§
9–10 does not depend on the resolution, since it con-
cerns holes that formed primordially and are treated as given; the astrophysical nulls of
§
13 rest
on the imported exterior, not on a formation account.
Accretion, mergers, area growth. Infalling matter must convert to saturated codespace at the
boundary, and two merging saturated regions must join with total severed-bond count not less
than the sum, the lattice statement of the Hawking area inequality. Whether the lattice reproduces
that inequality is a defined combinatorial question on the complex and is checkable; it has not
been checked. Causality: the interface velocity of Eq. (13) is subluminal for all R
H
> L
0
/2, and
every propagating branch of the complex shares one isotropic cone (
§
3); this is necessary but not
sufficient, since a full treatment needs the causal structure of the reconfiguration process itself.
15 Connection to the companion papers
The framework used here rests on results established in three companion papers. The FCC lat-
tice as a [[192, 130, 3]] CSS stabilizer code, with the vertex and octahedral check structure used
in Appendix B, is constructed in Ref. [25]. The identification of matter as incomplete crystal-
lization — topological defects of the K=4 → K=12 transition, with the metric-wall exclusion
scale and the geometric frustration used in
§
3.2 — is developed in Ref. [23]. The mass–energy–
information correspondence supplying the bond energy E
bond
= ℏc/(4L
0
) of Eq. (12) is Ref. [24].
The present paper is the black-hole and cosmology sector of that program; it can nonetheless be
read standalone as an effective model, per
§
1.1.
16 Discussion, limitations, and falsifiability
16.1 Limitations
Six limitations are structural, not presentational, and we list them in descending order of how
much of the paper each one touches. (1) The lattice hypothesis itself is assumed from the com-
panion papers. (2) The gravitational sector is established at linearized order only, on one explicit
subdivision; subdivision independence, the nonlinear theory, the Lorentzian continuation of the
Euclidean computation, and the exact general covariance of the induced matter coupling are
open, and the horizon geometry is imported; the Regge action is selected within, not beyond,
the stated frustration-linear family, and the strictly local stabilizer layer cannot itself fix the geo-
metric dynamics, entering the gravitational sector only through the RT-calibrated entanglement
census; in the entropy sector the bond-level RT relation itself remains a postulate, used twice.
(3) The recession law rests on interface phenomenology; the master equation for a single conver-
sion step, which would settle the M
2
exponent rather than motivate it, is not constructed. (4)
The barrier derivation is conditional on faceting and on bulk degeneracy ∆g = 0, neither settled;
the exponent geometry beyond the classical regime (linear vs. areal) is undetermined; and the
scale ξ is a postulate whose hierarchy the framework does not explain. (5) The cosmological
29
bounds are order-of-magnitude: analytic injection formulas with O(1) deposition fractions, pub-
lished ζ
EM
exclusions rather than a reaction network, the visibility approximation and FIRAS
rather than modern numerical distortion calculations, hadronic injection neglected, and thermal-
ization of high-energy lattice excitations with standard efficiencies assumed. (6) Formation: the
relic-void mechanism is a conjecture without a derived mass function or mass assignment, and
stellar-collapse formation is unresolved.
16.2 Claim ledger
The inline status tags are collected here so the paper’s epistemic map can be read at a glance.
16.3 Falsifiability
The scenario fails under any of the following, listed from observational to theoretical.
F1. A surviving sub-cutoff PBH. Any secure present-day detection of a primordial black hole below
the cutoff range 10
15.7
–10
16.5
g (allowing for the exponent-geometry ambiguity) falsifies the
geometric channel under its stated assumptions. The test has force because the standard
Hawking picture permits holes above M
⋆
≃ 5 × 10
14
g to survive to the present if they formed.
F2. A smooth reconstructed mass function. A PBH mass function continuing smoothly through
10
16.5
g with no lower edge excludes the freeze-out. A measured lifetime exponent other than
M
2
discriminates among the classes of Table 2: M
3
is degenerate with Hawking, M indicates
inertial or ballistic dynamics, M
3/2
a bulk-inertial interface.
F3. Distortions inconsistent with the mass–epoch map. A PIXIE-class µ or y measurement iden-
tifying injection at an epoch incompatible with the mass function inferred from other probes
falsifies the mapping, independently of the lattice hypothesis.
F4. A size-independent barrier. A demonstration that the boundary advances by local nucleation
with an R
H
-independent cost — classically, a kinetically rough interface or a bulk driving
term ∆g = 0; code-theoretically, net recession by accumulated correctable drift — removes the
cutoff as a prediction. The static preconditions are now computed (Appendix B): no static
code quantity grows with the vacancy, so a size-dependent barrier, if present, is dynamical in
origin. Settling the dynamical minimization is the decisive theoretical test, and it requires no
observation.
F5. Failure of the gravitational sector. A demonstration that the Fierz–Pauli identity of
§
3.3 is an
artifact of the tested subdivision, or that no generally covariant induced matter coupling exists,
removes the derived status of the linearized limit and returns the model to a bookkeeping of
imported gravity.
F6. A macroscopic deviation. Conversely, since the model predicts no deviation from general rela-
tivity for astrophysical holes at any foreseeable precision (
§
13), a confirmed horizon-scale depar-
ture from GR at EHT or LIGO/Virgo precision would falsify the imported-exterior structure
on which the astrophysical sector rests.
17 Conclusions
Within the FCC Selection–Stitch Model, a black hole is the Z
2
/U(1) phase boundary of a satu-
rated lattice region. Treating the parent D
4
lattice as an intrinsic simplicial geometry — with an
30
Statement Status
Rank-four isotropy of D
4
; flatness of the displacement metric; edge
dyads span Sym
2
(
§
3.1–3.2)
derived
Alternatives fail: Z
4
, FCC×Z have no flat background; FCC slice
not rank-four isotropic (Table 1)
derived (diagnostic, not unique-
ness)
Regge kinetic operator = linearized Einstein, symbolic identity on
the stated subdivision (
§
3.3)
derived; subdivision independence
open
Regge action unique in the local frustration-linear family: flat vac-
uum and covariant limit fail for any unequal class weighting, tadpole
of Prop. 2 (
§
3.3)
derived within the family; the fam-
ily is an input
Induced Newtonian limit, G
ind
= C a
2
with C = O(1) scheme-
dependent (
§
3.4)
derived up to C; conditional on co-
variant coupling
Nonlinear Einstein equations; Schwarzschild geometry; R
H
=
2GM/c
2
; T
H
imported
Area law S ∝ A from bond counting (
§
5) derived
Planck magnitude of L
0
: induced coupling matched to observed G
N
,
no horizon input; coefficient c
1
= 1.843 (RT)
calibrated (order of magnitude) /
calibrated (precise)
Entropy coefficient S = A/(4ℓ
2
P
) (
§
6) calibrated (RT postulate used
twice)
⟨N⟩ = 3
√
2/L
2
0
; ν
∗
= 2
√
3; capacity condition ν
∗
≥ 1 (
§
6, App. A) derived (consistency ratio, not the
origin of 1/4)
Surface tension σ = ℏc/4L
3
0
; mobility β = O(1) (
§
7) assumed
Overdamped curvature-driven recession; τ
Geo
∝ M
2
(
§
7.4) assumed; robust across dynamical
classes
Terrace barrier ∆G
∗
∝ R
H
(Eq. (16)) derived, conditional on faceting
and ∆g = 0
Bulk degeneracy ∆g = 0 of the two phases open; the sharp classical falsifier
d
eff
distinct from d = 3; no static code quantity grows with R
(App. B)
derived (statics); conversion dy-
namics open
Exponent geometry: linear (classical) vs. areal (quantum alterna-
tive)
derived vs. undetermined; moves
M
cut
by < 1 decade
ξ ∼ 1 fm (equivalently, anomalously small step free energy) conjectured postulate (
§
8.2)
Global EM release E
EM
= f
EM
Mc
2
; standard deposition efficiencies assumed
M
cut
≃ 10
16.5
g; mass–epoch map; f
PBH
≲ 10
−2
–10
−6
derived given the above; testable,
order-of-magnitude
Z
2
interior; saturation in place of a singularity; interior duality (
§
2) conjectured
Information levels L1–L4; monotonic radiated entropy; remnant end-
point (
§
11)
conjectured
Relic-void origin of primordial holes; mass assignment of a void (
§
14) conjectured; mass assignment
open
Accretion, mergers, area theorem on the lattice; stellar-collapse for-
mation
open
Table 8: Status of each major statement. The testable rows do not depend on the conjectured
rows, but they do depend on the imported and assumed rows above them, and on the unresolved
formation question.
action the lattice itself selects, flat-vacuum stationarity and the covariant limit both failing for
any unequal hinge weighting — its linearized continuum dynamics are those of Einstein gravity
— an exact rank-four isotropy, a two-polarization graviton in the incompatible edge-length sector,
a symbolic Fierz–Pauli identity on an explicit subdivision, and an induced Newton constant pro-
portional to the squared lattice spacing — so the gravitational vocabulary the horizon sections
rely on is, at leading order, derived rather than assumed, with the nonlinear regime imported.
Counting the severed Z
2
bond phases gives an entropy proportional to area; the induced coupling,
31
matched to the observed Newton constant, fixes the Planck magnitude of the bond length, and
one RT calibration pins its coefficient, and applying the same RT relation at the horizon returns
S = A/(4ℓ
2
P
) with no further parameter — a compatibility statement whose sharp content is the
computed capacity ratio ν
∗
= 2
√
3 ≥ 1, not a derivation of 1/4 from the lattice. The same hori-
zon supports a geometric evaporation channel, τ
Geo
∝ M
2
, frozen above M
cut
≃ 10
16.5
g (linear
exponent; 10
15.7
g areal) by a classical curvature-gated terrace-nucleation barrier, conditional on
a faceted interface and bulk-degenerate phases; direct computation shows no static code quantity
supplies that growth, locating any quantum contribution in the undetermined conversion dynam-
ics. Tracing the sub-cutoff evaporation history maps PBH mass to cosmological epoch: the lightest
holes vanish before BBN without trace, while a band up to the cutoff injects across nucleosyn-
thesis, photodissociation, and the CMB spectral-distortion eras, bounding the intermediate-mass
abundance at f
PBH
≲ 10
−2
–10
−6
against BBN and COBE/FIRAS at the order-of-magnitude
level. These early-Universe limits replace, rather than add to, the present-day γ-ray bounds that
apply under standard evaporation. The two load-bearing outputs — the M
2
scaling and the
sharp cutoff — are stable against the O(1) inputs and can be tested as an effective two-parameter
law, independently of the lattice hypothesis, by a PIXIE-class experiment correlated with a BBN
anomaly along the mass–epoch map. The phenomenology remains conditional on a formation
mechanism that evades the growth barrier; the relic-void conjecture supplies one for primordial
holes and none yet for stellar collapse, which we identify as the principal unresolved problem of
the dynamical sector.
Data availability. All quantitative results are reproducible from scripts at https://github.
com/raghu91302/ssmtheory/: the evaporation history, cutoff, sensitivity table, and constraint
curves by compute pbh.py; every diagnostic of
§
3 (isotropy, displacement-metric flatness, deficit-
map kernel, the closed-form Regge-to-Fierz–Pauli identity, and Fig. 1) by linearized gravity verify.py
and pbh revised make fig isotropy4.py; the alternative-lattice control of Table 1 by lattice test.py,
and the Regge-selection computation of
§
3.3 by pbh revised regge uniqueness.py, both of
which load the construction from linearized gravity verify.py; the punctured-code statics
of Appendix B, in both CSS sectors, by pbh revised vacancy analysis.py; the direct area-law
count of Fig. 2 by pbh revised arealaw.py, and the orientation scan and Wulff evaluation of
Appendix A by pbh revised verify parallel.py; the extended-mass-function convolution of
§
10.6 by pbh revised extended mf.py. All six figures, including Fig. 6 from the punctured-code
output, are regenerated by pbh revised make figs.py.
Declaration of competing interest. The author declares no competing financial interests or
personal relationships that could have influenced this work.
A Orientation dependence and the reduction factor
Section 6 reduces the bulk crossing count to the horizon entropy by the ratio ν
∗
= ⟨N⟩A
1
. This
appendix establishes the two ingredients — that ⟨N⟩ = 3
√
2/L
2
0
is the correct count for a spherical
horizon, and that no screen orientation needs to be selected — and records why an earlier route
through the densest (111) plane, which a reader may find natural, does not work.
Orientation dependence is real. Writing N(ˆn) = n
v
P
b
|b·ˆn| for the areal density of horizon-
crossing bonds on a plane of normal ˆn, with n
v
=
√
2/L
3
0
the site density and the sum over the
six independent bond directions, direct evaluation gives
32
ˆn (111) (211) (311) (100) (110) sphere average
N(ˆn) L
2
0
3.4641 4.0825 4.2212 4.0000 4.2426 4.2426
The spread over the sphere is 29%, from 2
√
3 = 3.4641 at (111) to 2
√
5 = 4.4721, and a scan
over 2 ×10
6
orientations confirms (111) as the global minimum, as expected for the densest plane.
Orientation cannot be left implicit.
For a spherical horizon the average is the answer. The number of bonds crossing a surface
is a flux, so it depends only on the local orientation of the surface and not on microscopic roughness.
For a sphere the total is
H
N(ˆn) dA = ⟨N⟩A with ⟨N⟩ = n
v
K
2
⟨|cos θ|⟩L
0
= n
v
K
2
L
0
2
= 3
√
2/L
2
0
.
The bulk count of Eq. (8) equals this exactly, and the numerical scan reproduces it to four figures
(4.24264 against 4.24264). Section 5 is therefore already orientation-correct.
Why the (111) route fails. It is tempting to argue that a horizon under surface tension should
facet to the orientation that severs fewest bonds, which is (111), and to use N
(111)
in place of ⟨N ⟩.
The argument does not survive its own logic. The Wulff shape for a crystal whose minimal face
is (111) is a regular octahedron, and at equal enclosed volume
A
oct
A
sph
= 1.1826,
N
(111)
A
oct
⟨N⟩A
sph
= 0.9656. (24)
The faceted horizon severs only 3.4% fewer bonds than the sphere of equal volume, not the
22.5% that substituting N
(111)
for ⟨N ⟩ would imply: the 18.3% area cost almost exactly cancels
the 18.4% density gain. This is not an accident, since the Wulff shape is by construction the
minimizer of total severed-bond number, and that minimum lies only slightly below the sphere.
Faceting cannot supply a
p
3/2 reduction, and we do not invoke it; the construction of
§
6 uses
the sphere average throughout and needs no shape assumption.
The reduction factor, and a coincidence to avoid. With ⟨N⟩ fixed, the reduction to one
ebit per RT cell is the computed ratio ν
∗
= ⟨N⟩A
1
= 2
√
3 of Eq. (9), and the capacity condition
ν
∗
≥ 1 of
§
6 is satisfied with margin. At K = 12, ν
∗
equals
√
K, which invites writing the factor as
a formula in the coordination number. That would be misleading. Carrying the lattice geometry
through, the general expression is ν
∗
= n
v
KL
0
A
1
/4, and Table 9 shows it coincides with
√
K
only at K = 12. The structural content is the number of bulk bonds crossing one RT cell, not a
power of the coordination.
K 6 8 12 24
bonds per RT cell ν
∗
1.732 2.309 3.464 6.928
√
K 2.449 2.828 3.464 4.899
coincide no no yes no
Table 9: The identity ν
∗
=
√
K holds only at the FCC coordination and is not a general rule.
Status. The count ⟨N ⟩ is derived; ν
∗
is computed from it and from the RT cell fixed in
§
4.
What is assumed is the RT relation itself and its application at the horizon scale with the same
cell. No orientation, faceting, or dimensional-reduction prescription enters.
33
B The punctured code: static quantities do not grow with the
vacancy
Section 8 distinguishes the boundary-conversion weight d
eff
from the code distance and identifies
the scaling of d
eff
with R
H
as the open assumption. This appendix computes what can be
computed today: the static code-theoretic quantities of the vacuum with a vacancy, in both CSS
sectors, which bound what any future dynamical treatment must respect.
Construction. All results below are a finite-size study at L = 6 and L = 8; headline numbers
are quoted at L = 8. We build the [[3L
3
, 2L
3
+ 2, 3]] FCC code of Ref. [25] on a periodic
box and verify the intact parameters (k = 434 at L = 6 and k = 1026 at L = 8, matching
2L
3
+ 2), then carve a spherical vacancy about a lattice node: every edge with an endpoint
inside the cut radius is removed, vertex (Z
2
) stabilizers of removed nodes are dropped, boundary
vertex stabilizers are truncated to their surviving edges, and octahedral stabilizers containing any
removed node are dropped (dropping, rather than truncating, preserves the CSS commutation
condition). Because the FCC shells are discrete, the distinct vacancies at L = 8 have volume-
equivalent radii R
eff
/L
0
= 0.55, 1.30, 1.47, 1.94, 2.10, 2.37 (1 to 79 removed nodes); all are
wrap-free on the L = 8 torus, whose half-width is 2.83 L
0
. The result is the punctured code
[[n
′
(R), k
′
(R), ·]] describing the intact exterior; we quote areas via the volume-equivalent radius
of the removed-node set throughout.
Results. Five facts, each computed exactly and shown in Fig. 6.
1. The bulk sectors develop no low-weight logicals. In the X-type sector (kernel of the truncated
vertex-check matrix) exhaustive zero-column and duplicate-column tests find no weight-1 or
weight-2 logicals at any radius. In the Z-type sector (kernel of the surviving octahedral-check
matrix, modulo the vertex-check row space) the same tests do find low-weight elements — up
to 72 of weight 1 and up to 335 of weight 2 across the tested radii — but every one of them is
boundary-localized: each lives on edges whose octahedral checks were dropped in the carving,
all within one bond length of the vacancy surface (maximum midpoint depth 1.0 L
0
over all
radii). These are not bulk logical operators; they are the degrees of freedom the puncture itself
creates at the horizon — the code-theoretic face of the severed-bond boundary bits counted in
§
5 — and their number grows with the vacancy area, as item 5 verifies for the severed-bond
count. We therefore report bulk and boundary sectors separately rather than quoting a single
punctured-code distance.
2. The minimum bulk logical weight is 3, independent of R. Bond triangles (3-cycles of the FCC
graph) lie in the X-type kernel by parity; at every radius we exhibit an explicit exterior triangle
outside the octahedral row space, located maximally far from the vacancy (4.1 L
0
from its center
on the L = 8 torus, at every radius). The constant distance of the parent code is inherited by
the bulk, and there is no growing protected distance: cheap bulk logicals exist arbitrarily far
from the vacancy at every radius. This is the computational content behind the statement of
§
8 that d = 3 constrains no particular operation.
3. One conversion step affects an R-independent set of objects. Advancing the vacancy by one
boundary node removes that node’s surviving edge qubits (at most K = 12; boundary mean
degree 9.1–11.0 across radii); it affects at most 13 vertex checks (the node’s own plus its
surviving neighbors’; boundary maximum 12) and at most 6 octahedral checks (boundary
34
0.5 1.0 1.5 2.0
vacancy radius
R
eff
(
L
0
)
0
10
20
30
40
weight / count
(a) no static growth with
R
min. bulk logical weight
d
0
(
R
)
checks per conversion step (max)
linear law
R
, for contrast
0 200 400 600
edges removed
0
100
200
300
400
logicals lost
k
0
k
0
(b) logical deficit
code-rate slope 2/3
0 20 40 60
vacancy area
A
(
L
2
0
)
0
50
100
150
200
250
300
severed bonds
(c) severed bonds vs. area
severed bonds
3
2
A
/
L
2
0
(App. A)
Figure 6: The punctured FCC code vs. vacancy radius (L = 8; radii are volume-equivalent, in units
of L
0
). (a) The minimum bulk logical weight is exactly 3 at every radius (exhaustive weight-≤ 2
exclusion in the bulk sectors plus explicit exterior weight-3 logicals), and one boundary-conversion
step affects at most 16 checks; neither grows with R. A linear law ∝ R is shown for contrast: if
the locking exponent grows, its origin is dynamical, not static. (b) Logicals lost on puncturing vs.
removed edges, approaching the code-rate slope 2/3. (c) Severed bonds vs. vacancy area, following
the orientation-averaged crossing density 3
√
2 A/L
2
0
of Appendix A.
maximum 4); and simply flipping the incident edges carries Z
2
-syndrome weight 12. Every one
of these counts is independent of R. It is the check counts, not the edge count, that answer
the request for the stabilizer generators involved in one step of horizon recession.
4. Logical deficit tracks the code rate. The deficit k
0
−k
′
per removed edge rises from 0.50 at the
single-node vacancy toward the code rate 2/3 (0.66 at the largest vacancy), as it must once
the removed region is bulk-dominated (Fig. 6b).
5. Severed bonds track the orientation-averaged density. The number of severed bonds scales
linearly with the vacancy area at a density within 8% of the orientation average 3
√
2/L
2
0
of
Appendix A for every vacancy larger than a single node (Fig. 6c) — independent numerical
support for the bond counting of
§
5, now from the code side.
What this does and does not establish. No static quantity relevant to bulk locking grows
with R: the bulk logical weight is constant, the single-step conversion cost is constant, and the
only R-dependent code structure created by the puncture is boundary-localized and areal — that
is, it is the entropy, not a barrier. If the boundary-locking exponent grows with R
H
, as
§
8 adopts
and as the classical terrace barrier of Eq. (16) independently gives, its origin is dynamical: the
detectability of intermediate configurations along a conversion sequence, not a static distance
of the punctured code. The dynamical minimization, over sequences with syndrome extraction
and correction interleaved, remains open, exactly as stated in
§
8 and
§
16.3. These are finite-size
statements over the tested radii, not theorems in the thermodynamic limit.
C The D
4
subdivision and the closed-form kinetic identity
This appendix records the construction behind
§
3.3 in enough detail to reproduce it; the verifica-
tion scripts in the Data Availability statement implement every step.
35
Subdivision. The patch is the set of D
4
sites with |x|
2
≤ 8 (169 vertices), triangulated into
1696 four-simplices by a deterministic Delaunay construction with symbolic perturbation to resolve
the cross-polytope degeneracies. The background is flat: every interior hinge deficit vanishes to
machine precision (max |δ
h
| < 6 × 10
−15
).
Gauge kernel. Linearizing the deficit angles in the squared edge lengths about a fully interior
vertex (25 incident edges, 125 incident hinges) gives a map of rank 21 with kernel dimension
exactly 4 and a clean spectral gap; the kernel is spanned by the four vertex-translation modes,
i.e., the linearized diffeomorphisms, with no extra zero modes.
Closed form. The second variation of the Regge action about flat space is a finite quadratic
form in the squared-edge perturbations, supported on the star of one vertex. Each dihedral-
angle derivative has a Cayley–Menger expression whose only irrationality is the simplex-volume
surd, and at the integer D
4
edge lengths that surd cancels on each hinge against the matching
surd in the triangle-area derivative; all 1100 entries of the resulting kinetic tensor are exactly
rational. Contracting against a general symmetric polarization and wavevector and subtracting
the linearized-Einstein quadratic form q
FP
of Eq. (1) yields the zero polynomial — the symbolic
identity quoted in
§
3.3. The two-derivative form of the leading term is forced by the exact nullity
of uniform strains; isotropy of the weighted operator follows from the spherical-design property of
the minimal vectors (Theorem 1) even though the raw edge set of the subdivision, which contains
a second and longer edge class, is not isotropic on its own.
Numerical shadows. Independent numerical checks reproduce the identity: the TT kinetic
coefficient is direction-independent to a fractional spread of 1.5 × 10
−9
over a sweep from a
coordinate axis to a body-type diagonal; the sector ratios are −1 : +3 : 0 to 10
−8
; and 60
random polarization/direction pairs match −q
FP
to a few parts in 10
8
(Fig. 1). The gauge sector
annihilates the kinetic tensor to 8 × 10
−9
. The small dimensionless residuals are float round-off
on exactly rational quantities.
Consistency at second order. As a control on the linearization, the quadratic action evalu-
ated on finite-amplitude TT waves matches the continuum value with the expected O(k
2
) conver-
gence (S
(2)
/k
2
ratios 1.0004, 1.0001, 1.00003 at |k| = 0.2, 0.1, 0.05), and a localized point defect
inserted at a forced position (d
s
=
p
3/8 L) is locally flat, its hinge deficits vanishing identically,
consistent with the source analysis of
§
3.4.
References
[1] J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973), doi:10.1103/PhysRevD.7.2333.
[2] S. W. Hawking, Commun. Math. Phys. 43, 199 (1975), doi:10.1007/BF02345020.
[3] S. D. Mathur, Class. Quantum Grav. 26, 224001 (2009), doi:10.1088/0264-9381/26/22/224001,
arXiv:0909.1038.
[4] C. Rovelli, Phys. Rev. Lett. 77, 3288 (1996), doi:10.1103/PhysRevLett.77.3288.
[5] A. Ashtekar, J. Baez, A. Corichi, K. Krasnov, Phys. Rev. Lett. 80, 904 (1998),
doi:10.1103/PhysRevLett.80.904.
[6] S. Ryu, T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006), doi:10.1103/PhysRevLett.96.181602.
[7] T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995), doi:10.1103/PhysRevLett.75.1260.
[8] T. C. Hales, Ann. Math. 162, 1065 (2005), doi:10.4007/annals.2005.162.1065.
36
[9] B. J. Carr, F. K¨uhnel, Annu. Rev. Nucl. Part. Sci. 70, 355 (2020), doi:10.1146/annurev-nucl-050520-
125911.
[10] B. Carr, K. Kohri, Y. Sendouda, J. Yokoyama, Rep. Prog. Phys. 84, 116902 (2021), arXiv:2002.12778.
[11] B. J. Carr, K. Kohri, Y. Sendouda, J. Yokoyama, Phys. Rev. D 81, 104019 (2010), arXiv:1604.05349.
[12] M. Korwar, S. Profumo, JCAP 05, 036 (2023), arXiv:2302.04408.
[13] M. Boudaud, M. Cirelli, Phys. Rev. Lett. 122, 041104 (2019), arXiv:1807.03075.
[14] R. H. Cyburt, J. R. Ellis, B. D. Fields, K. A. Olive, Phys. Rev. D 67, 103521 (2003), arXiv:astro-
ph/0211258.
[15] M. Kawasaki, K. Kohri, T. Moroi, Y. Takaesu, Phys. Rev. D 97, 023502 (2018), arXiv:1709.01211.
[16] V. Poulin, P. D. Serpico, Phys. Rev. D 91, 103007 (2015), arXiv:1503.04852.
[17] J. Chluba, R. A. Sunyaev, Mon. Not. R. Astron. Soc. 419, 1294 (2012), arXiv:1109.6552.
[18] D. J. Fixsen et al., Astrophys. J. 473, 576 (1996), doi:10.1086/178173.
[19] G. Dvali, Fortschr. Phys. 66, 1800007 (2018), arXiv:1810.02336.
[20] V. Thoss, A. Burkert, K. Kohri, Mon. Not. R. Astron. Soc. 532, 451 (2024), arXiv:2402.17823.
[21] A. Alexandre, G. Dvali, E. Koutsangelas, Phys. Rev. D 110, 036004 (2024), arXiv:2402.14069.
[22] G. Montefalcone, D. Hooper, K. Freese, C. Kelso, F. K¨uhnel, P. Sandick, (2025), arXiv:2503.21005.
[23] R. Kulkarni, “Matter as incomplete crystallization,” Phys. Open 27, 100423 (2026),
doi:10.1016/j.physo.2026.100423.
[24] R. Kulkarni, “The Mass–Energy–Information Equivalence: A Bottom-Up Identification of the Particle
Spectrum via FCC Lattice Error Correction,” Phys. Open (2026), doi:10.1016/j.physo.2026.100414.
[25] R. Kulkarni, “A 67%-Rate CSS Code on the FCC Lattice: [[192, 130, 3]] from Weight-12 Stabilizers,”
(2026), arXiv:2603.20294.
[26] T. Regge, Nuovo Cimento 19, 558 (1961), doi:10.1007/BF02733251.
[27] M. Roˇcek, R. M. Williams, Phys. Lett. B 104, 31 (1981), doi:10.1016/0370-2693(81)90848-0; Z. Phys.
C 21, 371 (1984), doi:10.1007/BF01581603.
[28] H. Kleinert, Gauge Fields in Condensed Matter, Vol. II, World Scientific (1989), doi:10.1142/0356.
[29] J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Springer (1999),
doi:10.1007/978-1-4757-6568-7.
[30] A. D. Sakharov, Dokl. Akad. Nauk SSSR 177, 70 (1967); reprinted Gen. Rel. Grav. 32, 365 (2000),
doi:10.1023/A:1001947813563.
[31] M. Visser, Mod. Phys. Lett. A 17, 977 (2002), arXiv:gr-qc/0204062.
[32] W. K. Burton, N. Cabrera, F. C. Frank, Philos. Trans. R. Soc. Lond. A 243, 299 (1951),
doi:10.1098/rsta.1951.0006.
[33] G. E. Volovik, The Universe in a Helium Droplet, Oxford University Press (2003),
doi:10.1093/acprof:oso/9780199564842.001.0001.
[34] I. M. Lifshitz, Yu. Kagan, Sov. Phys. JETP 35, 206 (1972).
[35] F. R. N. Nabarro, Proc. Phys. Soc. 59, 256 (1947), doi:10.1088/0959-5309/59/2/309.
[36] D. N. Page, Phys. Rev. Lett. 71, 3743 (1993), doi:10.1103/PhysRevLett.71.3743.
[37] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, A. Tajdini, Rev. Mod. Phys. 93, 035002
(2021), arXiv:2006.06872.
37
