Geometric Evaporation and Hierarchical Information Reduction
in Black Holes:
The K = 12 → K = 0 Lattice Phase Transition in the SSM Framework
Raghu Kulkarni
∗
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
May 2026
Abstract
We develop a general theory of black-hole evaporation and information dynamics in the
Selection-Stitch Model (SSM): a discrete K = 12 FCC tensor-network vacuum [1, 2] with
[[192, 130, 3]] CSS stabilizer code [3] and G = a
2
/(8 ln 2) fixed by Bekenstein–Hawking en-
tropy matching [4]. A black hole is a topological vacancy where gravitational compression
has driven the lattice past the metric wall r
min
= L
0
/
√
3 [1], shattering all entanglement
bonds and dropping the local coordination from K = 12 to K = 0. Treating the horizon as a
2D phase boundary with surface tension σ = ℏc/(4L
3
0
) and applying absolute quantum rate
theory to boundary re-stitching, we derive a recession velocity
˙
R = −(c/2)(L
0
/R
H
) and a
geometric lifetime τ
Geo
= 4G
2
M
2
/(c
5
L
0
) ∝ M
2
, holding for all black holes. Peierls locking
at correlation length L
corr
∼ 1 fm suppresses this channel exponentially for R
H
≫ L
corr
,
so stellar-mass and supermassive black holes evaporate only through the standard Hawking
channel; the geometric channel is observable only for primordial black holes (PBHs) with
R
H
≲ L
corr
. A 10
15
g PBH evaporates in 0.27 ms, resolving the long-standing gamma-ray
constraint that requires τ
Hawk
∝ M
3
relics today. We frame the information question hier-
archically: information in the SSM exists at four levels (bond, plaquette, cuboctahedral cell,
global stabilizer pattern). Across the K = 12 → K = 0 boundary the hierarchy itself does not
survive: it reduces to the lowest level. Bond-level information is preserved as the bit count
of severed bonds, emerging as thermal Hawking radiation and reproducing S
BH
= A/(4G);
higher-level structure collapses into that thermal flux. Hawking’s exact-thermal result is
recovered, and the absence of Page-curve recovery of higher-level structure is a falsifiable
prediction.
1 Introduction
The Hawking radiation mechanism [5] predicts a thermal emission rate that gives a black-
hole lifetime τ
Hawk
∝ M
3
, derived from quantum field theory on a fixed curved background.
This calculation suffers from the trans-Planckian problem: the outgoing radiation modes are
traced to field fluctuations with sub-Planckian wavelengths, where the spacetime continuum
approximation must break down. The Selection-Stitch Model (SSM) framework [1–4] replaces
the spacetime continuum with a Face-Centered Cubic (FCC) lattice of coordination number
K = 12 and lattice spacing a ≈ 2.355 ℓ
P
. The trans-Planckian assumption becomes a hard
ultraviolet cutoff at |
k| ≲ π/a [4], and Hawking’s calculation must be re-examined given the
underlying discrete dynamics.
This Letter develops a general theory of black-hole evaporation and information dynamics
in the SSM framework, applicable to all black holes regardless of mass. The two main results—a
geometric evaporation channel and a hierarchical reduction of information at the horizon—hold
∗
raghu@idrive.com
1
universally. Whether they are observationally distinguishable from standard Hawking physics
depends on a single dimensionless ratio: R
H
/L
corr
, where L
corr
∼ 1 fm is the QCD-set structural
correlation length of the discrete vacuum. For R
H
≫ L
corr
(stellar-mass and supermassive black
holes), Peierls locking suppresses the geometric channel exponentially and only the standard
Hawking decay survives; the framework reproduces standard astrophysical-BH phenomenology.
For R
H
≲ L
corr
, the geometric channel dominates, giving observable differences. The latter
regime corresponds to primordial black holes (PBHs) near the historically constrained 10
15
g
threshold [6], providing both the observational motivation for the present analysis and a concrete
falsifiable target.
Geometric evaporation. A black hole is a topological vacancy where the FCC lattice has
been compressed past the metric wall r
min
= L
0
/
√
3 [1], shattering all entanglement bonds
and reducing the local coordination to K = 0. The horizon is a discrete 2D phase boundary
between the K = 12 vacuum and the K = 0 interior, with calculable surface tension. Boundary
re-stitching dynamics, derived from absolute quantum rate theory [7], gives a recession velocity
˙
R = −(c/2)(L
0
/R
H
) and a lifetime τ
Geo
= 4G
2
M
2
/(c
5
L
0
) holding for any M. For 10
15
g
PBHs this gives 0.27 ms, in contrast to Hawking’s prediction of ∼ 10
20
s; for stellar-mass
and supermassive black holes, Peierls locking shuts the channel down, and standard Hawking
lifetimes survive.
Hierarchical information reduction. We address the information question by distinguish-
ing four levels of information content in the SSM framework: (L1) bond-level—the existence of
individual indistinguishable Bell pairs; (L2) plaquette-level—local correlations between the four
bonds of a CSS stabilizer; (L3) cell-level—the configuration of the twelve bonds in a cuboctahe-
dral K = 12 neighborhood; and (L4) global-level—the pattern of stabilizer eigenvalues defining
the logical-qubit state of the [[192, 130, 3]] code. This applies to every black hole, regardless
of size: any K = 12 → K = 0 horizon transition reduces the information hierarchy to its lowest
level. Bond-level information (L1) is preserved as the bit count of severed bonds, emerging as
exactly thermal Hawking radiation; higher-level structure (L2–L4)—the combinatorial pattern
of which Bell pairs occupied which edges—does not propagate across the boundary as struc-
ture but collapses into the L1 thermal flux. The Bekenstein–Hawking entropy S = A/(4G) is
recovered as the L1 bit count of the horizon. Hawking’s exact-thermal radiation calculation is
reproduced naturally: the emitted quanta carry maximal entropy per energy, which is precisely
L1 information. The absence of Page-curve-style recovery of higher-level structure is a sharp
falsifiable prediction holding for every black hole, but observable in evaporation products only
for PBHs near the 10
15
g threshold where geometric evaporation is fast enough to study.
2 Black Holes as Topological Vacancies
2.1 The metric wall and lattice shattering
The FCC vacuum has K = 12 nearest neighbors per site at bond length L
0
= a/
√
2 [1]. The
space between four mutually touching FCC sites (a tetrahedral void) has inscribed-sphere radius
r
min
=
L
0
√
3
. (1)
This is the minimum internode distance compatible with FCC topology: below r
min
, the tetrahe-
dral voids collapse and the lattice cannot maintain its K = 12 connectivity. The matter paper [1]
establishes r
min
as the absolute exclusion limit of the discrete vacuum; gravitational compression
past this wall is geometrically impossible within the K = 12 phase.
2
2.2 The K = 0 topological vacancy
Definition 1 (Topological vacancy). A black hole in the SSM framework is a connected region
V ⊂ R
3
in which the local gravitational compression has forced the internode spacing below r
min
.
Within V, all tetrahedral voids have collapsed and all entanglement bonds passing through V
have been severed. The local coordination number drops from K = 12 (intact vacuum) to K = 0
(vacancy interior). The boundary ∂V between the intact lattice and the vacancy is the event
horizon.
The K = 12 → K = 0 transition is a discrete topology change: not a smooth deformation but
a destruction of network connectivity. The interior of V contains no surviving lattice degrees
of freedom and is causally isolated from the exterior. The boundary ∂V consists of dangling
Bell-pair halves—bonds whose interior partners have been destroyed and which radiate into the
exterior with characteristic energy E
bond
= ℏc/(4L
0
) [4].
2.3 Recovery of standard thermodynamic black-hole properties
The topological-vacancy model reproduces the three central thermodynamic properties of black
holes from the underlying CSS-code structure.
Proposition 2 (Bekenstein–Hawking entropy). The entropy of a topological vacancy of horizon
area A is
S =
A
4G
, G =
a
2
8 ln 2
. (2)
Proof. Each removed Ovoid X-stabilizer of the CSS code [3] on the horizon contributes ∆S =
ln 2 to the entanglement entropy between interior and exterior, and occupies area A
plaq
= L
2
0
=
a
2
/2 in the 2D triad-sheet plaquette structure [4]. Bekenstein–Hawking entropy matching at
the level of a single stabilizer gives G = A
plaq
/(4 ln 2) = a
2
/(8 ln 2) (see [4], eq. 12). For a
horizon of area A comprising N = A/L
2
0
severed stabilizers, S = N ln 2 = A ln 2/L
2
0
= A/(4G)
via the algebraic identity L
2
0
= 4G ln 2.
Proposition 3 (Schwarzschild radius). The boundary of the topological vacancy coincides with
the Schwarzschild radius R
H
= 2GM/c
2
.
Proof. The vacancy nucleates at the center of the gravitating mass, where the energy density of
gravitational self-compression first reaches the metric-wall density. Once nucleated, the vacancy
propagates outward through every shell whose enclosed mass M(r) satisfies the trapped-surface
condition 2GM(r)/(rc
2
) ≥ 1. The outermost such shell is precisely r = 2GM/c
2
= R
H
.
Proposition 4 (Hawking temperature). The dangling-bond emission spectrum from the horizon
is thermal with temperature T
H
= ℏc
3
/(8πGMk
B
).
Proof. The dangling Bell-pair halves on ∂V are in detailed balance with the intact K = 12
lattice exterior at the local Tolman-rescaled vacuum temperature. The Unruh relation T =
ℏκ/(2πk
B
c), with surface gravity κ = c
4
/(4GM) at the Schwarzschild horizon, gives the stated
T
H
.
The SSM topological-vacancy model is therefore consistent with the three thermodynamic
black-hole laws of General Relativity, with all three numerical coefficients fixed by the single
CSS-code parameter G = a
2
/(8 ln 2).
3
3 Hierarchical Information Reduction
We now address the central conceptual question raised by black-hole evaporation: what happens
to the quantum information thrown into the black hole? Our answer is that information itself is
conserved in its lowest form, while the hierarchical structure of information is not. This requires
distinguishing four levels of information content in the SSM framework, all four of which are
present in the K = 12 vacuum but which transform differently across the K = 12 → K = 0 phase
boundary: the hierarchy collapses, leaving only the L1 thermal flux that emerges as Hawking
radiation. This is the same pattern that governs every irreversible thermodynamic process—a
burning library preserves its atoms (lowest level) while losing the organized hierarchy of ink,
words, books, and catalog—and the black-hole case is a clean realization of it.
3.1 Four levels of information
In the SSM framework, the vacuum is a tensor network: a specific decoration of the FCC lattice
with Bell-pair bonds on each of the K = 12 edges per site, organized into a [[192, 130, 3]] CSS
stabilizer code [3]. The information content of this state is hierarchical, with each higher level
encoding more complex combinatorial structure built from lower-level constituents.
• Level 1 (bond-level): the existence of an individual Bell pair on a given edge. Each bond carries
ln 2 of entanglement entropy. Bell pairs are intrinsically indistinguishable: every |Φ
+
⟩
vv
′
is
identical to every other, with no individual label. L1 information is the cardinality of the set
of present bonds—a counting statistic, not a structural one.
• Level 2 (plaquette-level): the local correlations between the four bonds of a CSS stabilizer
plaquette. Each weight-4 stabilizer A
p
= X
e
1
X
e
2
X
e
3
X
e
4
enforces a constraint on the joint
state of four neighboring edges; its eigenvalue (±1) is one bit of L2 structural information. L2
information distinguishes between configurations with the same L1 bond count but different
local correlation patterns.
• Level 3 (cell-level): the configuration of the twelve bonds in the cuboctahedral K = 12 neigh-
borhood of a single FCC site. Each cell encodes the local correlation pattern among its
twelve edges, plus the relations between adjacent stabilizer plaquettes. This is the natural
unit of geometric information in the framework—the discrete analog of a curvature or strain
measurement at a point.
• Level 4 (global-level): the pattern of stabilizer eigenvalues across the entire [[192, 130, 3]]
CSS code. With n = 192 physical qubits and k = 130 logical qubits, the L4 information
content is up to 2
130
logical states distinguished by the global eigenvalue pattern of the
n − k = 62 stabilizers. This is the most complex form of information: the macroscopic
quantum-information state of the encoded computational sector.
Each level is built from constituents of the previous level, but the higher-level information
is not reducible to enumeration of the lower-level constituents: it lives in their relational struc-
ture. The library analogy is direct: ink (L1), letters (L2), words and sentences (L3), and the
meaningful text and the catalog as a whole (L4) are increasingly complex structural levels built
from progressively richer relations among the simpler ingredients.
3.2 Bond-level information is preserved as thermal Hawking radiation
When the K = 12 lattice shatters at the metric wall, every Bell pair passing through the vacancy
region is severed. The number of severed bonds is the bit count of bond-level information present
in the interior, plus the bit count of bond-level information on the boundary itself.
4
The bit count is preserved. Each severed bond becomes a dangling Bell-pair half on ∂V, which
subsequently radiates one quantum of thermal Hawking radiation into the exterior (Proposi-
tion 4). The total number of thermal quanta emitted over the lifetime of the black hole equals
the total number of bonds severed during its formation and existence, which equals the horizon’s
L1 information content:
N
thermal quanta
= N
severed bonds
=
A
horizon
L
2
0
=
S
BH
ln 2
=
A
horizon
4G ln 2
. (3)
The thermal nature of the radiation is the signature of L1-only information. A thermal
distribution at temperature T
H
has maximum entropy for given total energy: it carries one bit
per emitted quantum (the existence of that quantum) and nothing more. This is exactly L1
information, propagating outward—no structural information at L2 or higher is encoded in the
spectrum. Hawking’s exact-thermal calculation [5] is recovered as the L1 information channel
of the SSM framework.
The Bekenstein–Hawking entropy is therefore the bit count of L1 information stored at the
horizon:
S
BH
= N
severed
ln 2 =
A
4G
, (4)
counting one ln 2 per severed bond. This is the L1 bit count of the horizon, neither more nor
less.
3.3 Higher-level structure reduces to the L1 thermal flux
L2, L3, and L4 information do not propagate across the K = 12 → K = 0 phase boundary as
structure: they reduce to the L1 thermal flux, becoming inaccessible as higher-level information.
The bit count is preserved (Section 3.2); the hierarchical structure is not.
L2 reduces. The CSS stabilizer eigenvalues are joint properties of four edges sharing a plaque-
tte. When all four edges are severed simultaneously, the eigenvalue ceases to be a well-defined
quantity: the operator A
p
= X
e
1
X
e
2
X
e
3
X
e
4
acts on a Hilbert space whose dimension has
changed, and the pre-severance eigenvalue is not recoverable from the post-severance dangling-
bond ensemble. The dangling halves are statistically independent thermal states whose corre-
lations with the original stabilizer eigenvalue are washed into the L1 thermal flux.
L3 reduces. The cuboctahedral cell structure—the geometric arrangement of twelve bonds
around a single FCC site—requires the FCC site itself to exist as a coordination center. When
the site is shattered (its inscribed-sphere radius forced below r
min
), the cell ceases to exist as
a structural unit. The twelve bonds it formerly organized are scattered into the boundary
dangling-bond ensemble; their identities as constituents of a particular cell are not retained.
L4 reduces. The global logical-qubit state of the [[192, 130, 3]] code is determined by the
joint eigenvalues of all 62 stabilizers [3]. When a macroscopic fraction of stabilizers (everything
in the vacancy region) is severed, the global logical state is no longer defined: its information
content reduces to the L1 bit count of the severed bonds. The post-evaporation lattice may re-
crystallize into a new K = 12 configuration (Section 3.5), but the new configuration’s L4 state is
statistically independent of the predecessor—a fresh draw from the ensemble of possible vacuum
configurations, not a unitary transformation of the original.
Higher-level information reduces to L1 because indistinguishable constituents cannot main-
tain relational structure once their relations are broken. The same indistinguishability that
makes L1 a pure counting statistic makes L2, L3, and L4 information dependent on bond-by-
bond relationships that do not survive bond severance. What survives is the count—and the
5
count is exactly what emerges as thermal Hawking radiation. The library analogy is precise:
burning an ordered library preserves all the atoms (L1 of that system) while collapsing the en-
tire hierarchy of letters, words, books, and catalog (L2–L4 of that system) into thermal motion.
The atoms in the smoke and ash are exactly the atoms that were in the books; what is gone
is the organization. The K = 12 → K = 0 transition is the same phenomenon at the level of
vacuum bonds.
3.4 Unitarity at each level
The hierarchical structure refines the meaning of “unitarity” in the framework:
L1 unitarity is fundamental and globally preserved. The total bit count of bond-level
information is conserved across the K = 12 → K = 0 transition: every severed bond emerges as
one thermal quantum, with no L1 bits created or destroyed. By the mass-energy-information
equivalence established in the MEI paper [2], L1 conservation is not merely analogous to energy
conservation but identical to it—the bit count of severed bonds and the energy carried away
as thermal Hawking radiation are two expressions of the same conserved quantity in the SSM
framework. This conservation holds exactly even at the phase boundary. The Banks–Susskind–
Peskin argument [10] that local non-unitarity violates energy conservation or locality does not
apply: L1 unitarity is preserved locally, energy is conserved by the MEI equivalence, and the
dynamics is local within the K = 12 phase.
L2–L4 unitarity is emergent within the K = 12 phase. The CSS-stabilizer-code dy-
namics [3, 4] preserves L2, L3, and L4 information by construction within the K = 12 phase:
stabilizer measurements and code operations are unitary maps on the protected logical Hilbert
space. All standard physics—atomic, nuclear, particle, quantum-information experiments—
lives within this regime, and sees full unitarity at all four levels. The strict unitarity assumed in
standard quantum mechanics is, in the SSM framework, the L4 unitarity of the K = 12 phase.
Higher-level unitarity does not extend across phase boundaries. The K = 12 → K = 0
transition is a non-equilibrium phase transition where the CSS-code Hilbert space itself changes
dimension. Unitarity at L2–L4, defined as a norm-preserving map on a fixed Hilbert space,
simply does not apply across this boundary: the hierarchy on which L2–L4 are defined has
collapsed. What remains is L1 unitarity, which holds globally and which is what conventional
energy/probability conservation requires. The standard “information paradox” arises only from
assuming strict L4 unitarity in a regime where the lattice phase boundary makes that assumption
inapplicable.
3.5 Phase-boundary processes: black holes and the Big Bang
The SSM framework predicts that two physical processes reduce the hierarchical information
structure to its lowest level:
Black-hole formation and evaporation (K = 12 → K = 0 → K = 12). Gravitational
collapse drives the K = 12 vacuum into K = 0 vacancy, reducing interior L2–L4 information to
the L1 bit count of severed bonds. Subsequent geometric evaporation (Section 4) re-stitches
K = 12 lattice across the former vacancy region, producing a new specific configuration whose
L4 state is statistically independent of the predecessor. The radiation carries away the L1 bit
count; the L2–L4 organization of the infallen matter is reduced to that thermal flux, and a new
L4 sector is freshly crystallized in the re-stitched vacuum.
6
Big Bang crystallization (phased K = 0 → K = 6 → K = 4 → K = 12). The matter
paper [1] establishes that the FCC vacuum does not crystallize from disorder in a single tran-
sition: the K = 12 vacuum emerges through an ordered sequence of intermediate phases. From
the initial high-temperature K = 0 disordered phase, two-dimensional triangular sheets crys-
tallize first (giving in-plane coordination K = 6), with occasional out-of-plane lifts producing
transitional K = 4 sites, and three such sheets eventually register into the FCC stacking that
yields full cuboctahedral K = 12 coordination. The Big Bang in this picture is specifically the
final K = 4 → K = 12 sheet-projection event that establishes the asymptotic vacuum in which
all subsequent physics operates.
The hierarchical information content emerges in step with the geometric phases:
• L1 (bond existence) appears as soon as the first 2D sheet forms (K = 6).
• L2 (plaquette correlations) becomes well-defined within each sheet, with full four-edge corre-
lations only present at K = 12 once cross-sheet bonds register.
• L3 (cuboctahedral cell) requires the K = 12 coordination and only exists in the final phase.
• L4 (global [[192, 130, 3]] logical state) stabilizes after the full FCC registration, when the
complete CSS-stabilizer structure first exists.
Each level of information is therefore created at a different stage of cosmological crystallization,
with no precursor in earlier phases. The L4 state of the present cosmic vacuum was selected
stochastically during the final K = 4 → K = 12 transition; it is not unitarily related to any
pre-Big-Bang quantum state.
Both processes are non-equilibrium phase transitions in the lattice coordination structure.
Neither is described by Hamiltonian evolution within a fixed Hilbert space; both lie outside the
regime of validity of standard L4 unitarity.
3.6 Response to standard objections
Three standard objections to non-unitary BH-evaporation scenarios deserve direct response
within the hierarchical-reduction framework.
Banks–Susskind–Peskin [10]. The argument is that local non-unitary evolution generically
violates energy conservation or locality. Resolution: L1 unitarity is preserved everywhere,
including at the phase boundary; the bit count of severed bonds equals the bit count of emitted
thermal quanta, and this counting is local at every step. The reduction of L2–L4 structure
to the L1 thermal flux is not local non-unitarity in the BSP sense—it is a topology change in
the underlying Hilbert space, accompanied by exact L1 conservation. Energy conservation and
locality at L1 are preserved, so the BSP argument does not apply.
Page curve [11]. The argument is that if BH evaporation is unitary at all levels, the entangle-
ment entropy of the emitted radiation must follow a specific curve (rising then falling). Recent
replica-wormhole derivations [12, 13] reproduce this curve in toy models. Resolution: the Page
curve assumes L4 unitarity at every step. In the SSM framework, L4 structure does not sur-
vive the K = 12 → K = 0 reduction, and the radiation entropy rises monotonically without ever
falling. The replica-wormhole derivations rely on a smooth gravitational path integral across the
horizon, which does not exist in the SSM framework (the horizon is a discrete phase boundary,
not a smooth Riemannian region). The Page curve is not a prediction of our framework, and
its experimental absence would be a confirmation.
7
AdS/CFT and holography. If a black hole in AdS is dual to a unitary boundary CFT, L4
unitarity holds by construction. Resolution: AdS/CFT applies in a specific setting (asymptot-
ically AdS spacetimes), but the SSM framework is asymptotically flat with a discrete K = 12
FCC vacuum. The AdS/CFT duality does not extend to our setting without modification, and
the SSM vacuum does not have an obvious holographic CFT description. The framework is not
in conflict with AdS/CFT results; it lies outside their domain of applicability.
3.7 Summary of the hierarchical picture
Level Information content Carrier Survives as structure?
L1 (bond) Existence of one Bell
pair; 1 bit (ln 2) per bond
Severed-bond thermal
quantum
Yes (as Hawking radn)
L2
(plaquette)
CSS stabilizer eigenvalue;
weight-4 correlation
Stabilizer plaquette (A
p
) No (reduces to L1)
L3 (cell) Cuboctahedral K = 12
neighborhood
configuration
Local 12-bond pattern No (reduces to L1)
L4 (global) [[192, 130, 3]] logical-qubit
state
Joint stabilizer pattern No (reduces to L1)
Table 1: Four levels of information in the SSM framework and their fate across the K = 12 →
K = 0 phase boundary. L1 information is preserved and emerges as thermal Hawking radiation
with total bit count A/(4G ln 2), recovering the Bekenstein–Hawking entropy. L2–L4 do not
survive as structural information: they reduce to the L1 thermal flux. The total information is
conserved at the lowest level; the hierarchy is not.
The hierarchical picture summarized in Table 1 reconciles three apparently conflicting re-
quirements: (i) Hawking’s exact-thermal radiation calculation, recovered as the L1 information
channel; (ii) the Bekenstein–Hawking entropy S = A/(4G), which is the L1 bit count of the hori-
zon; and (iii) the experimentally absent Page-curve signature, correctly predicted to be absent
because L2–L4 structure reduces to L1 and is not recoverable from the thermal flux. Hawking’s
original 1975 conclusion [5] that the radiation is exactly thermal is precisely the statement that
only L1 emerges from the horizon. The framework’s substantive position is therefore: total
information is preserved at the lowest level, but the information hierarchy itself collapses. This
is the same pattern that governs every irreversible thermodynamic reduction in nature.
4 Geometric Evaporation Dynamics
4.1 Horizon surface tension
Theorem 5 (Discrete surface tension). The surface tension of the K = 12/K = 0 phase boundary
is
σ =
ℏc
4L
3
0
. (5)
Proof. We do not require a microscopic model of how a cuboctahedral cell dissolves: the order
in which individual bonds sever, and any intermediate excited configurations of the cell, are
unknown. What is fixed by the framework is the total bond content and the total energy
released per unit horizon area.
The horizon ∂V is a 2D surface intersecting the lattice. By the Bekenstein–Hawking match-
ing (Proposition 2), the entropy per unit horizon area is ln 2/L
2
0
= 1/(4G), corresponding to one
removed CSS stabilizer per area L
2
0
of the 2D triad-sheet plaquette structure [4]. Each removed
8
unit dissolves into Bell-pair halves whose characteristic energy is E
bond
= ℏc/(4L
0
) [4]. The
total surface energy per unit horizon area is therefore
σ =
E
bond
L
2
0
=
ℏc
4L
3
0
, (6)
independent of the microscopic dissolution sequence.
4.2 Boundary re-stitching dynamics
The K = 0 vacancy is unstable: the surrounding intact lattice exerts mechanical pressure P =
2σ/R
H
inward on the boundary, driving spontaneous re-stitching of K = 12 lattice across ∂V.
The dynamics is set by the quantum transition rate for a single K = 0 boundary node to acquire
its K = 12 coordination through re-bonding with neighbors.
Theorem 6 (Recession velocity). The continuum limit of the discrete quantum transition rate
for boundary re-stitching gives a horizon recession velocity
˙
R = −
cL
0
2R
H
. (7)
Proof. By absolute quantum rate theory [7], the drift velocity of a phase boundary in response
to free-energy difference W per quantum step is v = (W/ℏ) λ, where λ is the elementary step
length. In a discrete lattice, the natural step is one bond: λ = L
0
, with corresponding step
volume V
step
= L
3
0
. The thermodynamic work done by the boundary pressure P = 2σ/R
H
over
one step volume, with σ given by Theorem 5, is
W = P L
3
0
=
2σL
3
0
R
H
=
2L
3
0
R
H
·
ℏc
4L
3
0
=
ℏc
2R
H
. (8)
The radial recession velocity is therefore
˙
R = −W L
0
/ℏ = −(c/2)(L
0
/R
H
), with the minus sign
indicating inward motion.
This radial recession is mathematically isomorphic to mean-curvature flow [8] (the Allen–
Cahn equation describing antiphase-boundary motion), but derived here from first-principles
quantum rate theory on the discrete lattice rather than from a continuum free-energy variational
principle.
4.3 Mass-loss rate and lifetime
Combining the Schwarzschild relation R
H
= 2GM/c
2
(Proposition 3) with the recession veloc-
ity (7) of Theorem 6:
˙
M =
c
2
2G
˙
R = −
c
5
L
0
8G
2
M
. (9)
Separating and integrating from initial mass M to M = 0:
Z
M
0
M
′
dM
′
=
c
5
L
0
8G
2
τ
Geo
=⇒ τ
Geo
=
4G
2
c
5
L
0
M
2
. (10)
Using the algebraic identity L
0
=
√
4G ln 2 from G = a
2
/(8 ln 2), and expressing in Planck units
G = ℏc/m
2
P
, ℓ
P
= ℏ/(m
P
c), t
P
= ℓ
P
/c, with L
0
/ℓ
P
=
√
4 ln 2:
τ
Geo
=
4ℓ
P
L
0
t
P
M
m
P
2
=
2
√
ln 2
t
P
M
m
P
2
≈ 2.40 t
P
M
m
P
2
. (11)
9
This τ ∝ M
2
scaling is parametrically faster than Hawking’s τ ∝ M
3
.
For a PBH at the observational threshold M = 10
15
g = 10
12
kg, with M/m
P
≈ 4.6 ×10
19
:
τ
Geo
(10
15
g) = 2.40 × (5.4 × 10
−44
s) × (4.6 × 10
19
)
2
≈ 0.27 ms. (12)
Hawking’s prediction for the same PBH gives τ
Hawk
≈ 2700 Gyr, i.e. ∼ 200 times the age of
the universe. The geometric channel is faster by a factor ∼ 10
23
. PBHs in the SSM framework
do not survive into the present era; they evaporate within a fraction of a millisecond of their
formation, leaving no detectable gamma-ray signal at the threshold mass.
10
10
10
3
10
4
10
11
10
18
10
25
10
32
10
39
Black-hole mass
M
(g)
10
50
10
37
10
24
10
11
10
2
10
15
10
28
10
41
10
54
Lifetime
τ
(s)
Age of universe
Planck time
m
P
10
15
g
10
M
¯
4
×
10
6
M
¯
τ
Geo
= 0
.
27
ms
τ
Hawk
≈
2700
Gyr
Hawking:
τ
Hawk
∝
M
3
Geometric:
τ
Geo
= 4
G
2
M
2
/
(
c
5
L
0
)
Geometric (with Peierls):
∝
e
R
H
/L
corr
Figure 1: Black-hole lifetime as a function of mass under three scenarios. Blue: standard
Hawking radiation, τ
Hawk
∝ M
3
. Orange dashed: unobstructed geometric channel, τ
Geo
=
4G
2
M
2
/(c
5
L
0
), parametrically faster. Orange solid: geometric channel with Peierls suppression
exp(R
H
/L
corr
) active above the QCD correlation length L
corr
∼ 1 fm. At the observational
PBH threshold M = 10
15
g, the geometric channel gives τ
Geo
= 0.27 ms (against Hawking’s
∼ 2700 Gyr), resolving the gamma-ray constraint. Above ∼ 10
15
g the Peierls factor switches
on sharply, locking stellar-mass and supermassive black holes against geometric evaporation;
these decay only through the standard Hawking channel.
5 Peierls Locking and Macroscopic Stability
Equation (11) as stated would predict rapid geometric evaporation of all black holes, including
stellar-mass and supermassive ones, in conflict with astrophysical observation. This is averted
by the Peierls–Nabarro mechanism for discrete domain walls; the resulting sharp suppression
of geometric evaporation above R
H
∼ L
corr
is visible in Fig. 1 as the rapid upturn of the solid
orange curve.
5.1 The Peierls suppression
In any discrete lattice, a moving phase boundary cannot glide continuously: it must hop between
lattice sites, traversing a periodic potential (the Peierls stress) [9]. For an extended interface of
10
characteristic curvature radius R
H
in a polycrystalline matrix with structural correlation length
L
corr
, the boundary mobility is exponentially suppressed:
µ
eff
= µ
0
exp
−
R
H
L
corr
. (13)
For microscopic boundaries R
H
≲ L
corr
, the exponential is O(1) and the boundary moves freely;
for R
H
≫ L
corr
, the exponential is catastrophic and the boundary is locked in place.
5.2 Identification L
corr
∼ 1 fm
The structural correlation length of the SSM lattice is set by the QCD confinement scale [1]: the
natural length over which the FCC-vacuum stabilizer-code structure remains coherent before
being disrupted by gauge-field fluctuations is L
corr
∼ 1 fm.
For the 10
15
g PBH, R
H
≈ 1.5 fm, so R
H
/L
corr
≈ 1.5 and the Peierls suppression factor is
mild (e
−1.5
≈ 0.22). Geometric evaporation proceeds with only a modest reduction in rate.
For a stellar-mass black hole (M = 10 M
⊙
, R
H
≈ 30 km), R
H
/L
corr
≈ 3 × 10
19
, giving a
suppression factor of order exp(−3 × 10
19
), indistinguishable from zero for any astrophysical
purpose. Stellar-mass and supermassive black holes are therefore geometrically stable in this
framework, and decay only via the standard Hawking channel.
Black hole M (g) R
H
Geometric lifetime
PBH (Planck mass) 2.2 × 10
−5
3.2 × 10
−33
cm ∼ 2.4 t
P
PBH (threshold) 10
15
1.5 fm 0.27 ms
Stellar (e.g. Cyg X-1) 4 × 10
34
62 km Peierls-locked
SMBH (e.g. Sgr A*) 8 × 10
39
1.2 × 10
7
km Peierls-locked
Table 2: Geometric evaporation lifetimes across the BH mass spectrum. Microscopic PBHs
evaporate rapidly; macroscopic BHs are Peierls-locked and effectively stable on this channel.
6 Falsifiable Predictions
The SSM framework predicts four distinctive signatures, applicable across the full BH mass
range:
Universal: no higher-level information recovery (no Page curve). The framework
predicts that any black-hole evaporation—PBH, stellar-mass, or supermassive—reduces L2–L4
structure to the L1 thermal flux; higher-level information is not recoverable from the radiation.
Observable consequences hold for all black holes:
• The Hawking radiation spectrum is exactly thermal at all times, with no Page-curve-style
decrease in entanglement entropy past any time scale.
• Evaporation products are characterized entirely by the macroscopic conserved charges (mass,
angular momentum, possibly U(1) charge) of the initial BH; the L2–L4 quantum-state details
of infalling matter do not survive as recoverable information in the radiation.
• Any future experimental or astrophysical demonstration of Page-curve information recovery
would falsify the framework.
11
Macroscopic black holes: Hawking-only decay. For R
H
≫ L
corr
, Peierls locking sup-
presses the geometric channel to negligible levels (Table 2). Stellar-mass and supermassive
black holes therefore evaporate exclusively through the standard Hawking channel, with life-
times τ ∝ M
3
that exceed the age of the universe by many orders of magnitude. The framework
is consistent with all existing astrophysical observations of black holes; no deviation from stan-
dard GR + Hawking phenomenology is expected for R
H
≫ 1 fm.
PBH-specific: sharp mass cutoff at ∼ 10
15
g. The τ ∝ M
2
scaling, combined with
Peierls locking at L
corr
∼ 1 fm, predicts that the present-day PBH mass spectrum should
have a sharp lower cutoff near M ∼ 10
15
g (where R
H
∼ L
corr
), with zero surviving relics
below this threshold. This is distinct from the gradual roll-off predicted by Hawking τ ∝ M
3
scaling. Future PBH-search experiments sensitive to gamma-ray relic signals at sub-threshold
masses would distinguish the two scaling laws and provide the cleanest observational test of the
geometric channel.
Lorentz-invariant evaporation at leading order. The geometric channel obeys exact
SO(3) rotation isotropy at leading lattice order [1,2], with the first lattice anisotropies entering
at O((R
H
/a)
−4
) via the degree-six fundamental invariant of the FCC point group [4]. For PBHs
at R
H
∼ fm, this anisotropy is suppressed by a factor of ∼ (ℓ
P
/fm)
4
∼ 10
−80
, undetectable in
any plausible observation.
7 Discussion and Conclusion
The topological-vacancy framework reproduces all standard black-hole thermodynamic proper-
ties from the discrete CSS-stabilizer-code structure of the SSM vacuum (Section 2), derives a
geometric evaporation channel from absolute quantum rate theory at the K = 12/K = 0 phase
boundary (Section 4), and provides a hierarchical-reduction resolution of the BH information
question (Section 3). These results hold for black holes of any mass. Their observational con-
sequences split sharply at the QCD-set scale L
corr
∼ 1 fm: for R
H
≫ L
corr
(stellar-mass and
supermassive BHs) the geometric channel is Peierls-locked (Section 5) and only Hawking de-
cay survives, recovering standard astrophysical phenomenology; for R
H
≲ L
corr
(PBHs near
10
15
g) the geometric channel dominates, giving τ
Geo
∝ M
2
and resolving the long-standing
PBH gamma-ray constraint.
The information question is addressed by recognizing that information in the discrete vac-
uum is hierarchical, with four levels (bond, plaquette, cuboctahedral cell, global stabilizer pat-
tern). Across the K = 12 → K = 0 phase boundary the hierarchy itself does not survive: it
reduces to the lowest level. Bond-level (L1) information is preserved as thermal Hawking radia-
tion, reproducing the Bekenstein–Hawking entropy S = A/(4G) as the L1 bit count; higher-level
(L2–L4) structure—the combinatorial configuration of which Bell pairs occupied which edges,
organized into stabilizer plaquettes and cuboctahedral cells—reduces to that thermal flux and is
not recoverable as structure. Unitarity at the bond level is fundamental and globally preserved;
unitarity at higher levels is an emergent property of the K = 12 phase that does not extend
across phase boundaries. The standard information paradox dissolves: there is no contradiction
because the strict L4 unitarity it assumes is an emergent property of the K = 12 phase, which a
black-hole horizon by definition does not respect. The substantive position is: total information
is preserved at the lowest level, but the information hierarchy is not—the same pattern that
governs every irreversible thermodynamic reduction in nature.
Hawking’s original 1975 conclusion [5] of exact-thermal radiation is recovered as the L1 chan-
nel of the framework. Page-curve recovery of higher-level structure [11] is explicitly predicted
to be absent, providing a sharp empirical test.
12
Open questions: the detailed dynamics of L2–L4 reduction at the metric wall (this Letter
treats it as an effective phase transition without specifying the microscopic decoherence path);
the relation between the SSM phase-boundary picture and recent replica-wormhole derivations
of the Page curve in toy models [12, 13], which we have argued lie outside the SSM regime
of applicability but which deserve deeper comparison; and the cosmological-scale information
content of the post-Big-Bang K = 12 vacuum, whose L4 state is freshly crystallized through the
K = 0 → K = 6 → K = 4 → K = 12 sequence of the matter paper [1]. We address these in future
work.
Acknowledgments. This work builds on the FCC lattice studies of [1–4] and the broader
SSMTheory program.
Data availability. A Python script verifying all numerical claims of this paper (FCC ge-
ometry, metric wall, Bekenstein–Hawking entropy recovery, geometric lifetime formulas, PBH
evaporation time, Peierls suppression scales) is available at https://github.com/raghu91302/
ssmtheory/blob/main/verify_pbh.py.
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