Geometric evaporation of primordial black holes

Geometric evaporation of primordial black holes:

early-Universe energy-injection signatures and a

shifted constraint landscape

Raghu Kulkarni

SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA raghu@idrive.com

June 20, 2026

Abstract

Primordial black holes (PBHs) are a long-studied dark-matter candidate whose abundance be-

low ∼ 10

g is tightly bounded by the products of Hawking evaporation. We examine the

observational consequences of a non-standard, geometric evaporation channel that arises if the

vacuum is a discrete close-packed (face-centered-cubic, K = 12) lattice, as proposed in the

Selection–Stitch Model (SSM); we adopt that microphysical picture as a working assumption—

not an established result—and develop its phenomenology in full. A black hole is a topological

vacancy where compression past the lattice metric wall severs the entanglement bonds and drops

the local coordination to K = 0; the bond-counting structure of the underlying stabilizer code

reproduces the Bekenstein–Hawking entropy and ﬁxes the bond length L

√

4 ln 2 ℓ

. Mod-

eling the horizon as a discrete phase boundary that recedes by curvature-driven re-stitching

gives a lifetime τ

Geo

∝ M

, parametrically faster than Hawking’s M

, suppressed exponentially

by the stabilizer code’s distance threshold once the horizon radius exceeds the code-coherence

length ξ ∼ 1 fm. The two eﬀects impose a sharp lower edge on the surviving PBH spectrum at

cut

≃ 10

16.5

g. Tracing when sub-cutoﬀ PBHs evaporate, the lightest (M ≲ 4 ×10

g) disap-

pear before Big-Bang nucleosynthesis (BBN) and leave no trace, while a band up to the cutoﬀ

injects energy across BBN, the photodissociation epoch, and the era of CMB spectral distortions.

Computing the resulting µ- and y-distortions against the COBE/FIRAS limits and the injection

against BBN, this band is bounded at f

PBH

≲ 10

−2

–10

−6

. Because these early-Universe con-

straints replace—rather than add to—the present-day γ-ray bounds that apply under standard

Hawking evaporation (and which here do not apply, since the holes are gone), the geometric sce-

nario admits intermediate-mass primordial abundances that standard evaporation excludes: the

constraint landscape shifts from present-day photons to weaker early-Universe imprints. An ap-

pendix notes, as a more speculative model-level corollary, that if the geometric radiation is read

as a re-emission of severed lattice bonds the bond-level bit count is conserved as the thermal ﬂux

while higher-level structure is not, predicting a monotonic radiation entropy. The mass-resolved

injection signature is falsiﬁable independently of the underlying lattice hypothesis.

1 Introduction

Primordial black holes remain one of the few dark-matter (DM) candidates requiring no physics

beyond general relativity and a suitably enhanced primordial power spectrum [5,6]. Their viability

as DM is set by a patchwork of bounds: microlensing and dynamical limits at high mass, and at

low mass the products of Hawking evaporation [7]. PBHs below M

⋆

≃ 5 × 10

g have already

evaporated; those between M

⋆

and ∼ 10

g are constrained by their present-day emission in the

extragalactic and Galactic γ-ray backgrounds, the 511 keV line, Voyager cosmic-ray e

, and CMB

energy injection [8–10]. These bounds close the sub-10

g range to PBH-dominated DM and set

the lower edge of the asteroid-mass window (10

–10

g).

Each of these bounds assumes the semiclassical Hawking rate, τ

Hawk

∝ M

, derived on a ﬁxed

continuum background and expected to fail once quantum-gravitational eﬀects matter. A growing

literature explores the alternatives. The memory-burden program [11–13] argues that backreaction

stabilizes a hole after roughly half-decay, slowing evaporation and potentially opening a new low-

mass window; subsequent work ﬁnds the window survives only if the transition is abrupt, since

otherwise residual evaporation through BBN and recombination is excluded [14]. These analyses

share a method: posit a modiﬁed evaporation law, then confront the cosmological energy budget

with data. We adopt the same method for a modiﬁcation of opposite sign.

The Selection–Stitch Model (SSM) [27–29] treats the vacuum as a discrete face-centered-cubic

(FCC) tensor network of coordination number K = 12. Discreteness of spacetime at the Planck

scale is a common thread across several approaches to quantum gravity—causal sets [1], causal dy-

namical triangulations [2], spin foams and loop quantum gravity [3], and tensor-network/holographic

constructions [4]—which diﬀer in what discrete structure they posit and how the continuum is re-

covered. The SSM is one such proposal, distinguished by a speciﬁc close-packed lattice and an

associated stabilizer code; we do not argue here that it is preferred over these alternatives, nor

do we re-derive it. We adopt it as a working assumption and ask what it implies for PBHs, then

show that the testable consequences can be stated without committing to the assumption (

1.1).

The consequence is a second evaporation channel, geometric rather than thermal: the horizon is a

phase boundary between the ordered K = 12 vacuum and a disordered interior, and evaporation

proceeds by the boundary re-stitching inward. This channel is faster than Hawking (τ ∝ M

) and

is frozen by lattice pinning for all but the smallest holes, yielding a sharp survival cutoﬀ. The body

develops the full chain: the topological-vacancy construction and the thermodynamic relations it

reproduces (

2); the curvature-ﬂow evaporation dynamics (

3); the code-distance cutoﬀ (

4); the

resulting evaporation history of the PBH population (

5); the cosmological constraints from BBN

and CMB spectral distortions, and how the constraint landscape shifts relative to the standard γ-

ray bounds (

6). A more speculative corollary—a bookkeeping of infalling information in the same

lattice picture—is deferred to Appendix A. The observational content—a mass-resolved mapping of

PBHs to the cosmological epoch at which they deposit their energy—is testable on its own terms,

independently of the lattice hypothesis.

1.1 Scope of the claims

Because this assumption is strong and unestablished, we state plainly what the paper does and does

not claim, to separate the testable phenomenology from the speculative microphysics. Assumed,

not derived: the SSM lattice itself, taken from Refs. [27–29]. Calibrated: the single lattice scale L

ﬁxed by matching the bond-counting entropy to Bekenstein–Hawking (

2)—this is one calibration,

not an independent prediction of the entropy. Imported from general relativity: the Schwarzschild

radius and Hawking temperature, used as consistency inputs at the horizon; the lattice is not

shown to reduce to curved spacetime, and we do not claim it does. Modeling assumptions with

O(1) freedom: the mobility coeﬃcient in the recession law, and the coherence length ξ that sets

the suppression scale (an external input) together with the O(1) coeﬃcient in d

eﬀ

≃ R

/ξ. The

exponential form of the suppression and its linear-in-R

exponent are not free: they follow from

the code-distance threshold (

4). The two genuinely load-bearing, testable outputs are the τ ∝ M

scaling and a sharp survival cutoﬀ. Importantly, the existence of a cutoﬀ is not special to the

SSM: any super-exponential suppression of the channel that switches on near a microphysical (here

QCD-conﬁnement) scale produces one, as we show in

4. What the SSM contributes is the speciﬁc

functional form (M

) and the speciﬁc scale (ξ ∼ 1 fm); a reader who rejects the lattice may still

treat these as an eﬀective two-parameter law and test the cosmological consequences, which is how

we recommend the observational sections be read.

We use natural units ℏ = c = 1 with lengths in Planck units, G = ℓ

, restoring ℏ, c in

numerical results. The FCC bond length is L

√

4 ln 2 ℓ

≃ 1.665 ℓ

and the lattice constant

a =

√

2 L

≃ 2.355 ℓ

, ﬁxed in

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

= a/

√

2 [27]. The space

between four mutually touching FCC sites (a tetrahedral void) has inscribed-sphere radius

min

√

. (1)

This is the minimum internode distance compatible with FCC topology: below r

min

, the tetrahedral

voids collapse and the lattice cannot maintain its K = 12 connectivity. The matter paper [27]

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 The K = 0 topological vacancy

Deﬁnition 1 (Topological vacancy). A black hole in the SSM framework is a connected region

V ⊂ R

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. We

assign each bond the characteristic energy

bond

ℏc

, (2)

which follows from the SSM structure-tensor result c = 4v

lat

= 4L

/τ [28] (one quantum trans-

mitted per bond per lattice timestep τ = 4L

/c): the natural energy quantum per bond is

bond

= ℏ/τ = ℏc/(4L

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 =

, G =

8 ln 2

. (3)

Proof. Each removed octahedral X-stabilizer of the CSS code [29] on the horizon contributes ∆S =

ln 2 to the entanglement entropy between interior and exterior, and we assign it horizon area

plaq

= L

= a

/2 (one severed stabilizer per plaquette of the 2D boundary sheet; this is the ﬂat-

plaquette area convention adopted in the Introduction). Bekenstein–Hawking entropy matching at

the level of a single stabilizer then gives G = A

plaq

/(4 ln 2) = a

/(8 ln 2). For a horizon of area

A comprising N = A/L

severed stabilizers, S = N ln 2 = A ln 2/L

= A/(4G) via the algebraic

identity L

= 4G ln 2. This is the one black-hole relation the model derives internally, and it is

what ﬁxes the numerical value of L

used throughout.

Proposition 3 (Consistency with the Schwarzschild radius). The boundary of the topological va-

cancy can be consistently identiﬁed with the Schwarzschild radius R

= 2GM/c

Proof. This is a consistency requirement, not an independent derivation: we do not derive the

Einstein equations from the lattice, but impose the standard general-relativistic trapped-surface

condition and check that the vacancy boundary is placed consistently with it. The vacancy nucleates

where the energy density of gravitational self-compression ﬁrst reaches the metric-wall density, and

extends outward through every shell whose enclosed mass M (r) satisﬁes 2GM (r)/(rc

) ≥ 1. The

outermost such shell is r = 2GM/c

= R

. The model thus places the K = 12/K = 0 boundary

at the horizon by construction; it does not predict the Schwarzschild relation from the lattice

dynamics.

Proposition 4 (Consistency with the Hawking temperature). If the dangling-bond emission from

the horizon is in local thermal equilibrium with the exterior, its temperature is the standard Hawking

value T

= ℏc

/(8πGMk

Proof. This proposition imports the standard general-relativistic surface-gravity result rather than

deriving it from the lattice. Assuming the dangling Bell-pair halves on ∂V are in detailed balance

with the intact K = 12 exterior at the local Tolman-rescaled temperature, the Unruh relation T =

ℏκ/(2πk

c) with Schwarzschild surface gravity κ = c

/(4GM) yields the stated T

. The content

of the proposition is therefore that the topological-vacancy picture is compatible with the Hawking

temperature, not that it derives it; the surface-gravity input is taken from GR.

The topological-vacancy model is therefore consistent with the three standard thermodynamic

black-hole relations, though the three are not on equal footing. The entropy S = A/(4G) (Proposi-

tion 2) follows from the bond-counting structure of the CSS code and is the one relation the model

genuinely derives; the Schwarzschild radius (Proposition 3) and the Hawking temperature (Propo-

sition 4) are imposed from general relativity and checked for compatibility. The single CSS-code

parameter G = a

/(8 ln 2) = ℓ

ﬁxes the entropy coeﬃcient; the other two relations carry their

standard GR coeﬃcients.

3 Geometric Evaporation Dynamics

3.1 Horizon surface tension

Theorem 5 (Discrete surface tension). The surface tension of the K = 12/K = 0 phase boundary

σ =

ℏc

. (4)

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 matching

(Proposition 2), the entropy per unit horizon area is ln 2/L

= 1/(4G), corresponding to one

removed CSS stabilizer per area L

(Section 2.3). Each removed unit dissolves into Bell-pair halves

of characteristic energy E

bond

= ℏc/(4L

) (Eq. (2)). The total surface energy per unit horizon area

is therefore

σ =

bond

ℏc

, (5)

independent of the microscopic dissolution sequence.

3.2 Boundary re-stitching dynamics

The K = 0 vacancy is unstable: the surrounding intact lattice exerts an inward Laplace pressure

P = 2σ/R

on the boundary, driving re-stitching of K = 12 lattice across ∂V. We model the

resulting boundary motion as an overdamped (mobility-limited) interface,

R = −µ

int

P , where µ

int

is the interface mobility and P the driving pressure. This is the Allen–Cahn form of curvature-

driven boundary motion [15]; what the lattice supplies is the value of µ

int

Theorem 6 (Recession velocity). Taking the interface mobility to be the unique combination

int

= L

/ℏ that can be built from the bond length L

and ℏ with the dimensions of a mobility,

the overdamped boundary law

R = −µ

int

P with Laplace pressure P = 2σ/R

and σ from Theo-

rem 5 gives

R = −

. (6)

Proof. A mobility relating an interface velocity (units m s

−1

) to a pressure (units J m

−3

) has di-

mensions m

(J s)

−1

. The only such combination available from the microscopic data {L

, ℏ} is

int

= β L

/ℏ with β a dimensionless O(1) constant; we set β = 1, and ﬂag this choice as the single

undetermined coeﬃcient of the dynamics (it rescales τ

Geo

but not its M

scaling). Then

R = −µ

int

2σ

= −

ℏ

ℏc

= −

, (7)

with the minus sign indicating inward motion. The result is subluminal for R

> L

/2, i.e. for

every horizon larger than half a bond length, which includes every black hole.

Two features of this derivation deserve separation. The M

scaling of the lifetime (below) is

ﬁxed by the curvature-ﬂow structure

R ∝ 1/R

together with R

∝ M, and does not depend

on β. The overall coeﬃcient 2.40 in Eq. (10), in contrast, is ﬁxed only up to the O(1) mobility

constant β and should be read as an order-of-magnitude estimate rather than a precise prediction.

We make no appeal to transition-state (absolute-rate) theory: that framework carries an attempt

frequency and an activation exponential that play no role here, and the content of the step is the

dimensional mobility ansatz above.

3.3 Mass-loss rate and lifetime

Combining the Schwarzschild relation R

= 2GM/c

(Proposition 3) with the recession velocity (6)

of Theorem 6:

M =

R = −

. (8)

Separating and integrating from initial mass M to M = 0:

′

Geo

=⇒ τ

Geo

. (9)

Using the algebraic identity L

√

4G ln 2 from G = a

/(8 ln 2), and expressing in Planck units

G = ℏc/m

, ℓ

= ℏ/(m

c), t

= ℓ

/c, with L

/ℓ

√

4 ln 2:

Geo

4ℓ





√

ln 2





≈ 2.40 t





. (10)

This τ ∝ M

scaling is parametrically faster than Hawking’s τ ∝ M

For a PBH at the historically discussed scale M = 10

g = 10

kg, with M/m

≈ 4.6 ×10

Geo

(10

g) = 2.40 × (5.4 × 10

−44

s) × (4.6 × 10

)

≈ 0.27 ms, (11)

up to the O(1) mobility constant β of Theorem 6. The standard Hawking lifetime for the same mass

is τ

Hawk

≈ 2700 Gyr [5]—about 200 times the age of the universe, which is the familiar statement

that ∼ 10

g is near the mass that completes Hawking evaporation around the present epoch.

The geometric channel, where it operates unsuppressed, is faster by a factor ∼ 10

. This raw

0.27 ms ﬁgure applies only to the unsuppressed channel; the physically relevant lifetime includes

the code-suppression factor of Section 4, which is already non-negligible at this mass (R

≈ 1.5 fm)

and which shifts the survival picture substantially. The properly code-suppressed survival cutoﬀ is

derived in Section 4.3.

4 Boundary locking from the code-distance threshold

Equation (10) 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

because the advance of the horizon boundary is a logical operation on the surrounding stabilizer

code, and is suppressed by the code-distance threshold of that code. The resulting sharp suppression

of geometric evaporation above R

∼ ξ is visible in Fig. 1 as the rapid upturn of the solid orange

curve. Because the suppression follows from the same stabilizer-code structure that ﬁxes the entropy

(

2), it is not an independent assumption added to stabilize macroscopic holes; it is a property of

the code the model already posits.

PBH mass

(g)

Lifetime (s)

asteroid-mass PBH

dark-matter window

Age of universe

Planck time

cut

16.5

Hawking:

Geometric (unsuppressed):

Geometric + code threshold

Figure 1: Black-hole lifetime as a function of mass under three scenarios. Blue: standard Hawking

radiation, τ

Hawk

∝ M

. Orange dashed: unobstructed geometric channel, τ

Geo

= 4G

/(c

parametrically faster (overall coeﬃcient uncertain at the O(1) level). Orange solid: geometric

channel with code-distance suppression exp(R

/ξ), active above the code-coherence length ξ ∼

1 fm. The code-suppressed geometric lifetime equals the age of the universe near M

cut

∼ 10

16.5

∼ 40 fm): below this mass PBHs evaporate geometrically in well under a second; above

it (including the asteroid-mass dark-matter window) the suppression factor locks the geometric

channel and only standard Hawking decay survives. The qualitative content is the change in

scaling and the sharp lower cutoﬀ, not the precise coeﬃcient.

4.1 Boundary advance as logical error propagation

In the SSM the intact vacuum is the codespace of the [[192, 130, 3]] CSS code [29], and the black-

hole interior is a region where that code has been destroyed (K =0). For the boundary ∂V to recede

by one step, the disordered (logical-error) conﬁguration of the interior must propagate one code

block into the surrounding intact code—a logical operation on the protected subspace. Quantum

error-correcting codes resist exactly this. Below a threshold ambient error rate p < p

, a code of

distance d suppresses the logical-error probability as

∼





d/2

= exp



−



, (12)

the hallmark exponential protection of the threshold theorem [17]. The eﬀective distance protecting

the receding boundary follows from what code distance means. In a topological (surface-type) sta-

bilizer code the distance is the length of the shortest logical operator—the shortest non-contractible

error chain that maps one codeword to another—and for a code patch of linear extent L this short-

est chain scales linearly, d ∼ L/ξ

, where ξ

is the code-block size. Advancing the horizon requires

precisely such a logical operation: an error chain must thread the protected region surrounding the

boundary. The shortest chain that does so spans the boundary’s linear extent, not its area, so

eﬀ

≃

, (13)

with ξ the code-coherence length (ﬁxed below). This linear scaling is selected on two independent

grounds. First, it is the surface-code-correct one: the minimal logical operator is a path, whose

length grows with the linear size of the region it must cross, not with the enclosed area. Second, it is

the only scaling consistent with the vacuum being a stable code. The area alternative d

eﬀ

∼ (R

/ξ)

would, to reproduce the observed cutoﬀ (

4.3), require p/p

≃ 0.95—the vacuum sitting essentially

at its error threshold, the marginal regime in which a code does not protect its logical state—

whereas the linear scaling requires p/p

≃ 0.13, an order of magnitude below threshold, the regime

of a functioning code. A stable vacuum code therefore naturally selects the linear law over the

area alternative. Combining Eqs. (12) and (13), the boundary mobility inherits the exponential

suppression, µ

eﬀ

= µ

exp(−R

/ξ), and the geometric lifetime is correspondingly lengthened,

eﬀ

Geo

= τ

Geo

exp





. (14)

The exponential-in-R

form is therefore not imposed: it is the threshold theorem’s exponential-in-

distance protection (Eq. 12), evaluated at the distance the boundary actually presents, d

eﬀ

≃ R

/ξ

(Eq. 13). Two ingredients remain inputs, and we ﬂag them as such: the coherence length ξ (an

external scale, ﬁxed below) and the O(1) proportionality in Eq. (13) together with the sub-threshold

ratio p/p

, which we do not compute from the code’s detailed fault-tolerance properties. What

is not an input is the functional form or the linear-in-R

exponent, both of which follow from

the code structure. The condensed-matter Peierls–Nabarro picture of a domain wall pinned in a

periodic potential [16] is the classical analog of the same statement and gives an identical functional

form.

4.2 The coherence length ξ ∼ 1 fm

The coherence length ξ is not the Planck-scale lattice spacing a: with d

eﬀ

= R

/a the suppression

would be astronomically strong already at R

∼ a, locking even the lightest holes. We therefore

identify ξ with the length over which the stabilizer code would plausibly stay coherent before gauge-

ﬁeld ﬂuctuations introduce uncorrectable errors, which we take to be of order the QCD conﬁnement

scale, ξ ∼ 1 fm [27]. Below this length the code would correct faster than errors accumulate and

the logical state (hence the boundary) is protected; above it the code decoheres and the protection

is lost. This identiﬁcation is a physical input, not a result derived from the code: the SSM does

not independently ﬁx a conﬁnement scale, and we do not claim to derive ξ. We adopt it as the

one order-of-magnitude scale between the Planck length and the macroscopic regime at which a

discrete error-correcting vacuum would plausibly lose coherence, and treat the cutoﬀ location it

produces as a consequence of that input rather than a prediction of the framework. A reader who

rejects the identiﬁcation can instead regard ξ as the free length scale that determines the cutoﬀ.

With this identiﬁcation the numbers are sensible rather than tuned. Reproducing the survival

cutoﬀ at M

cut

≃ 10

16.5

g (R

≃ 42 fm) requires R

/ξ ≃ 42, i.e. d

eﬀ

≃ 42 coherence lengths

and, from Eq. (12), p/p

≃ 0.13—the vacuum operating about an order of magnitude below its

own error threshold, the expected regime for a stable code, and (as shown above) the regime that

selects the linear d

eﬀ

∝ R

law over the area alternative. For the 10

g PBH, R

≈ 1.5 fm, so

/ξ ≈ 1.5 and the factor is mild, e

−1.5

≈ 0.22; the corrected lifetime τ

eﬀ

Geo

≈ 1.2 ms remains far

shorter than the age of the universe. For a stellar-mass black hole (M = 10 M

⊙

, R

≈ 30 km),

/ξ ≈ 3 × 10

and the suppression is exp(−3 × 10

), indistinguishable from zero; such objects,

and supermassive ones, are locked and decay only through the standard Hawking channel.

4.3 The survival cutoﬀ and the surviving PBH spectrum

The physically meaningful prediction is the boundary in mass between PBHs that the geometric

channel removes before the present epoch and those that survive. Setting the code-suppressed geo-

metric lifetime equal to the age of the universe, τ

Geo

exp(R

/ξ) = t

univ

, with τ

Geo

= (2/

√

ln 2) t

(M/m

)

and R

= 2GM/c

, gives a sharp survival cutoﬀ at

cut

≈ 10

16.5

g, R

cut

) ≈ 40 fm. (15)

PBHs below M

cut

have R

small enough that even the code-suppressed geometric channel evap-

orates them within a fraction of a second of formation; PBHs above M

cut

are locked and follow

the standard Hawking τ ∝ M

evolution. The transition is sharp because the suppression factor

exp(R

/ξ) varies by many orders of magnitude across a narrow mass range. Table 1 summarizes

the geometric fate across the full black-hole mass spectrum.

Relation to the asteroid-mass dark-matter window. This cutoﬀ sits below the asteroid-

mass band M ∼ 10

–10

g in which PBHs remain a viable dark-matter candidate [5]. The

geometric channel therefore does not destroy the asteroid-mass dark-matter scenario; rather, it

predicts a hard lower edge to the surviving PBH mass function near 10

16.5

g, sharper than the

gradual Hawking roll-oﬀ near ∼ 10

–10

g, and leaves the dark-matter window above it intact.

We regard the location of this lower edge as the principal quantitative prediction of the geometric

channel and note that it depends on the (here order-of-magnitude) coherence length ξ and the

mobility constant β. Because ξ enters the exponent of the suppression while β enters only the

power-law prefactor, the cutoﬀ is far more sensitive to ξ than to β, and only logarithmically to

either. Table 2 makes this explicit: over a factor-of-four range in ξ and a factor-of-ten range in

β, log

cut

/g) spans just 16.2–16.8, i.e. the cutoﬀ stays within 0.6 of a decade of 10

16.5

g. A

four-fold change in ξ moves it by 0.57 decade; a ten-fold change in β moves it by 0.02 decade. The

cutoﬀ scale is therefore well-constrained at the order-of-magnitude level despite the O(1) freedom

in the inputs.

Black hole M (g) R

Geometric fate (code-suppressed)

PBH (Planck mass) 2.2 × 10

−5

3.2 × 10

−33

cm ∼ 2.4 t

(unsuppressed)

PBH (below cutoﬀ) 10

1.5 fm τ

eﬀ

Geo

≈ 1.2 ms; evaporates

PBH (survival cutoﬀ) ∼ 10

16.5

∼ 40 fm τ

eﬀ

Geo

∼ t

univ

PBH (DM window) 10

–10

0.1–10

pm code-locked; survives

Stellar (e.g. Cyg X-1) 4 × 10

62 km code-locked

SMBH (e.g. Sgr A*) 8 × 10

1.2 × 10

km code-locked

Table 1: Geometric-channel fate across the BH mass spectrum, including the code-distance sup-

pression. PBHs below the survival cutoﬀ M

cut

∼ 10

16.5

g evaporate geometrically in well under

a second; PBHs above it—including the asteroid-mass dark-matter window—are code-locked and

evolve only through the standard Hawking channel.

ξ = 0.5 fm ξ = 1 fm ξ = 2 fm

β = 0.3 16.18 16.46 16.75

β = 1 16.17 16.45 16.74

β = 3 16.15 16.44 16.73

Table 2: Sensitivity of the survival cutoﬀ to representative variations in the two uncertain inputs,

given as log

cut

/g). The mobility constant β rescales the lifetime prefactor; the coherence

length ξ sets the exponential suppression scale. The cutoﬀ varies by < 0.6 decade across the full

box, and is essentially insensitive to β.

5 Evaporation history of the sub-cutoﬀ population

The phenomenological core of this paper is that the geometric channel does not merely change

how fast sub-cutoﬀ PBHs evaporate but when, moving their entire energy release from the present

epoch (where the standard γ-ray bounds operate) into the early Universe. This section establishes

the mass-to-epoch mapping;

6 turns it into quantitative abundance limits.

5.1 Formation and evaporation times

A PBH forms with a mass of order the horizon mass at formation, M ∼ M

hor

∼ c

form

/G, so

form

∼ GM/c

. For the masses of interest (10

–10

16.5

g) this is ≲ 10

−23

s, utterly negligible beside

the geometric lifetime. A sub-cutoﬀ PBH therefore evaporates, to excellent approximation, at

cosmic time

evap

≃ τ

eﬀ

Geo

(M) =

√

ln 2





exp





. (16)

In the radiation era, cosmic time maps to photon temperature through T ≃ 1 MeV (t/s)

−1/2

(taking

⋆

∼ 10; the weak g

⋆

dependence shifts T by an O(1) factor and does not move the epoch assign-

ments), and to redshift through 1 + z = T/T

with T

= 2.35 × 10

−4

eV. This ﬁxes the redshift of

evaporation z

evap

(M) shown in Fig. 2, a steeply decreasing function: a 10

g hole evaporates at

z ∼ 10

(well before BBN), a 10

g hole at z ∼ 10

, and a 2 ×10

g hole only at z ∼ 5 ×10

, just

before recombination.

5.2 Six evaporation bands

Sweeping M from the formation ﬂoor to the survival cutoﬀ, the evaporation redshift crosses every

major cosmological epoch in turn, dividing the sub-cutoﬀ population into the six bands of Table 3.

The boundaries follow from setting t

evap

(M) equal to the time of each epoch: BBN onset (t = 1 s)

at M ≃ 3.8×10

g, BBN end (t = 300 s) at 6.8×10

g, the photodissociation threshold (t ≃ 10

at 8.8 × 10

g, the double-Compton thermalization redshift z

≃ 2 ×10

at 1.3 × 10

g, the µ/y

boundary (z ≃ 5 × 10

) at 1.75 ×10

g, and recombination (z ≃ 1100) at 2.2 ×10

g. Each band

is probed by a diﬀerent observable, so the model predicts not a single constraint but a structured,

mass-resolved set.

5.3 Inversion relative to Hawking

The qualitative content of Table 3 is an inversion of the standard timeline. Under Hawking evapo-

ration, lifetime grows as M

, so a 10

g hole has τ

Hawk

∼ 10

s—some twenty orders of magnitude

beyond the age of the Universe—and is, for all observational purposes, stable: it radiates a faint

Mass range (g) z

evap

Dominant cosmological probe

M ≲ 4 × 10

≳ 10

none (thermalizes before BBN)

4–7 × 10

∼ 10

–10

BBN light-element abundances

0.9–1.3 × 10

∼ 10

6.5

–10

7.5

D/He photodissociation

1.3–1.8 × 10

∼ 10

–10

6.3

CMB µ-distortion (FIRAS)

1.8–2.2 × 10

∼ 10

–10

4.7

CMB y-distortion (FIRAS)

2.2 × 10

–M

cut

≲ 10

recombination / present γ-rays

Table 3: Geometric-channel evaporation epoch as a function of sub-cutoﬀ PBH mass, and the

observable that constrains each band. Boundaries are accurate to the O(1) normalization of Eq. (10)

with the code-suppression factor of

PBH mass

(g)

Redshift of evaporation

evap

thermalized (no distortion)

-distortion

post-recomb / present

BBN (

)

Figure 2: Redshift at which a sub-cutoﬀ PBH evaporates through the geometric channel. 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.

present-day ﬂux that the extragalactic and Galactic γ-ray backgrounds constrain. Under the ge-

ometric channel the same hole evaporated at z ∼ 10

and is long gone; its energy was deposited

into the pre-recombination plasma. The observable consequence therefore shifts in kind, from a

present-day photon ﬂux to an early-Universe energy injection, and the constraints that apply shift

correspondingly, from γ-ray telescopes to BBN and the CMB spectrum. Quantifying that shift is

the subject of the next section.

6 Cosmological constraints

6.1 Energy-injection formalism

A PBH component behaving as non-relativistic matter has energy density ρ

PBH

(z) = f

PBH

DM,0

(1+

, where f

PBH

is the fraction of the DM density the holes would constitute today absent evap-

oration. When the holes evaporate at z

evap

they release a fraction f

of this rest energy into

electromagnetically interacting species (the remainder escaping as neutrinos/gravitons). The frac-

tional energy injected into the photon bath is

∆E

= f

PBH

Ω

1 + z

evap

≃ f

PBH

4.85 × 10

1 + z

evap

, (17)

using Ω

/Ω

≃ 4.85 ×10

. We adopt f

= 0.5 as a ﬁducial value and treat it, with the mobility

normalization, as an overall O(1) uncertainty. Equation (17) shows that later (heavier) evaporation

injects more energy relative to the bath, so constraints tighten toward the cutoﬀ.

6.2 Pre-BBN: the safely cleared population

For M ≲ 4 × 10

g the holes evaporate at t < 1 s (T > 1 MeV), when the plasma is in full

thermal equilibrium. The injected energy thermalizes completely and leaves no spectral or nu-

cleosynthetic trace; the only residual eﬀects are a possible contribution to N

eﬀ

if a fraction of

the dangling-bond emission is relativistic and decoupled, and a negligible entropy dilution. This

sub-population—potentially the bulk of a bottom-heavy PBH mass function—is therefore erased

without observational consequence, the cleanest distinction from the Hawking case, in which the

same masses would persist and radiate today.

6.3 BBN and photodissociation

Holes in the band 4 × 10

–2.2 × 10

g evaporate during (1–300 s) or after BBN. We treat the

electromagnetic injection following the standard non-thermal-nucleosynthesis framework [18–20].

The natural variable is the released electromagnetic energy per background photon. Writing the

energy released per evaporation as f

and the comoving number density as f

PBH

DM,0

/M ,

the product is independent of M —heavier holes are fewer but each releases more—so

≡ E

inj

PBH

= f

PBH

DM,0

γ,0

≃ 3.1 × 10

−9

PBH

GeV, (18)

using ρ

DM,0

γ,0

≃ 3.1 ×10

−9

GeV. The PBH mass enters the constraint only through the evap-

oration time t

evap

(M), which selects which published limit on ζ

applies. Two features of the

cascade physics matter. First, photodissociation is inactive for t

evap

≲ 10

s: at higher tempera-

tures the injected energy exceeds the pair-production threshold E

≃ m

/(22T ) on the CMB, and

is degraded below the nuclear thresholds before it can dissociate anything [19, 20]. Second, deu-

terium destruction dominates for 10

≲ t

evap

/s ≲ 10

, giving way to

He photodissociation (which

overproduces D and

He) at later times, the regime where the bound is strongest.

We adopt the published ζ

evap

) exclusion—the deuterium branch of Refs. [18, 19] and the

He branch anchored to ζ

< 2.1 × 10

−10

GeV (t

evap

/10

1/4

for t

evap

> 10

s [19]—and invert

Eq. (18) to obtain f

PBH

(M). The result is the dashed curve in Fig. 3, with the shaded band giving

the spread under f

∈ [0.3, 0.7] and a factor-of-two mobility uncertainty in t

evap

. The bound

runs from f

PBH

≲ 1 at the onset (∼ 9 × 10

g, where E

ﬁrst clears the deuterium threshold)

to f

PBH

≲ 10

−2

near 2 × 10

g, and switches oﬀ above 2.2 × 10

g, where evaporation falls after

recombination and the constraint passes to the CMB. Unlike the earlier version of this analysis, this

band is now computed from the time-resolved injection rather than adopted as a ﬂat threshold. The

one regime we do not compute from ﬁrst principles is the hadronic injection during nucleosynthesis

evap

< 300 s, M ≲ 7 × 10

g), which alters light-element yields through meson-driven p ↔ n

conversion and requires a reaction-network treatment; we ﬂag that sliver explicitly and do not rest

any conclusion on it.

6.4 CMB spectral distortions

Energy injected between z ≃ 5 ×10

and the double-Compton thermalization redshift z

≃ 2×10

cannot be fully thermalized and survives as a µ-type spectral distortion; injection at z ≲ 5 × 10

produces a y-type distortion [21]. For a localized injection at z

evap

we use the standard estimates

µ ≃ 1.4

∆E

exp

−



evap



5/2

, y ≃ 0.25

∆E

, (19)

the exponential being the thermalization visibility. Confronting these with the COBE/FIRAS limits

|µ| < 9 × 10

−5

and |y| < 1.5 × 10

−5

[22] and solving for the abundance at which each distortion

saturates its limit gives the bounds plotted in Fig. 3. Three regimes appear, all following from

Eqs. (17)–(19):

(i) Thermalization-erased (M ≲ 1.3×10

g). These holes evaporate at z

evap

> z

; the visibility

exponential suppresses µ by factors such as e

−(7.6)

5/2

at 10

g, so no distortion survives and FIRAS

places no bound. The energy thermalizes, exactly as in the pre-BBN case but at lower redshift.

(ii) µ-band (1.3–1.8 × 10

g). Here z

evap

runs from ∼ 10

down to ∼ 2 × 10

. Even with

the entire band assumed to be all the dark matter (f

PBH

= 1), the distortion is µ ≃ 3 × 10

−3

1.5 ×10

g, only thirty times the FIRAS limit, so the bound is loose: f

PBH

≲ 3 ×10

−2

at the light

end of the band, tightening to f

PBH

≲ 10

−4

near 1.8 × 10

(iii) y-band and approach to the cutoﬀ (1.8–3 × 10

g). As z

evap

falls, the injected fraction

∆E/E

∝ (1 + z

evap

)

−1

grows, so the bound tightens steadily: f

PBH

≲ 2 × 10

−6

at 2 × 10

g and

PBH

≲ 10

−7

for the heaviest sub-cutoﬀ holes, which evaporate near recombination where energy

injection is most damaging. This is the one band where the geometric channel is comparably

constrained to the standard picture.

6.5 A shifted constraint landscape

The standard evaporation bounds on 10

–10

g PBHs, f

PBH

≲ 10

−8

from the extragalactic and

Galactic γ-ray backgrounds and from Voyager e

[8–10], assume the holes survive and radiate today.

In the geometric scenario they do not: every sub-cutoﬀ hole evaporated in the early Universe, so the

present-day γ-ray bound does not apply. We are careful about what this does and does not mean. It

is not that we weaken an applicable constraint—one cannot relax a bound that does not operate—

but that the constraints on a given primordial abundance change in kind: the present-day photon

limits are replaced by the early-Universe limits computed above. The substantive, non-tautological

content is the comparison at ﬁxed primordial abundance. A population that standard Hawking

evaporation would exclude at the f

PBH

∼ 10

−8

level (because it would radiate a present-day γ-ray

ﬂux) is, in the geometric scenario, constrained only at f

PBH

∼ 10

−2

–10

−6

by its early-Universe

imprint. The geometric channel therefore admits intermediate-mass primordial abundances that

standard evaporation forbids—by several orders of magnitude over most of the band, narrowing to

none at the cutoﬀ, where near-recombination injection drives the y-distortion bound back down to

the γ-ray level. We plot the γ-ray line in Fig. 3 only as this ﬁxed-abundance reference, not as a

bound that applies in the scenario.

6.6 Combined exclusion

Figure 3 collects the computed bounds and their combined exclusion envelope, with the shaded

bands propagating the O(1) uncertainties in f

and the mobility normalization through every

channel. The envelope runs from f

PBH

∼ 1 at the light (thermalization-erased) end, through

PBH mass

(g)

PBH

(would-be DM fraction)

survival

cutoff

BBN/photodiss. band

combined exclusion band

BBN / photodissociation

-distortion (FIRAS)

combined exclusion

Hawking -ray (does not apply)

Figure 3: Maximum allowed PBH abundance f

PBH

versus mass for the geometric channel, all

computed in this work. Solid orange: µ-distortion; solid purple: y-distortion (both against

COBE/FIRAS). Dashed red with shaded band: BBN/photodissociation, computed from the time-

resolved electromagnetic injection [Eq. (18)], the band spanning f

∈ [0.3, 0.7] and a factor-of-

two mobility uncertainty. Solid black: the combined exclusion (strongest applicable bound at each

mass). Dotted blue: the standard present-day Hawking γ-ray bound, which does not apply in this

scenario and is shown only for comparison. The geometric-channel bounds sit well above it across

most of the band: at ﬁxed primordial abundance the early-Universe limits are weaker than the

γ-ray reference, converging to it only at the survival cutoﬀ.

PBH

≲ 10

−2

–10

−4

across the µ-band, to f

PBH

≲ 10

−6

near the cutoﬀ. The mass-resolved structure

is the observable content of the scenario: it predicts that any PBH signature in this band must be

an early-Universe one—a BBN light-element shift or a sub-FIRAS-level distortion, ordered in mass

as in Table 3—rather than a present-day γ-ray excess. A detection of a present-day γ-ray signal

from 10

–10

g PBHs would favor standard Hawking evaporation over this scenario; a future

PIXIE-class measurement of a small µ or y distortion, correlated with a BBN anomaly along the

mass–epoch map, would favor this one. Neither test requires committing to the SSM microphysics

(

1.1).

7 Discussion

A falsiﬁable mass–probe map. The central observational content of the scenario is Table 3: a

given sub-cutoﬀ PBH mass is tied to a speciﬁc cosmological epoch, and hence a speciﬁc probe. A

detected FIRAS-level µ- or y-distortion (for instance by a future PIXIE-class mission), combined

with a BBN light-element anomaly, would test the predicted ordering directly; conversely, the

absence of any surviving PBH below M

cut

with a present-day γ-ray signature is a generic prediction.

This testable content reduces to an eﬀective two-parameter law—the M

scaling and the cutoﬀ

scale—and does not require the lattice hypothesis. We do not present that independence as a

strength of the microphysics: on the contrary, it means the cosmological tests constrain the eﬀective

law, not the SSM speciﬁcally. A reader who ﬁnds the lattice unpersuasive can read

§§

5–6 as the

phenomenology of an accelerated, sharply-cut-oﬀ evaporation channel; the microphysics of

§§

2–4 is

what motivates that particular law and scale, no more.

Macroscopic black holes. For stellar-mass and supermassive holes the horizon radius exceeds ξ

by twenty orders of magnitude or more, so the geometric channel is frozen and only Hawking emis-

sion operates. The model therefore reproduces standard astrophysical black-hole phenomenology

exactly where that phenomenology is tested, and departs from it only in the primordial regime.

Relation to memory burden. The modiﬁcation here has the opposite sign to the memory-

burden program [11–13]: there, backreaction slows evaporation and can open a low-mass DM

window; here, the geometric channel accelerates evaporation, clearing the low-mass population

early and admitting intermediate-mass abundances that the present-day γ-ray bounds would forbid.

Both face the same arbiter—the cosmological energy budget across BBN and recombination [14]—

and the geometric channel passes it for any sub-dominant abundance, because acceleration pushes

injection to high redshift where the radiation bath is overwhelming.

Information. Read as information bookkeeping, the same lattice picture suggests the radiation

entropy rises monotonically, with no Page turnover. We regard this as considerably more speculative

than the energy-injection phenomenology and develop it, with its caveats, only in Appendix A; it is

a model-level interpretation stated as an assumption, not a result, and nothing in the observational

analysis depends on it.

Limitations. The scenario inherits the unveriﬁed SSM assumption; we have treated it as an

assumption and tested its consequences, not validated it. Within the channel, the lifetime nor-

malization (β), the electromagnetic fraction (f

), and the correlation length (ξ) are O(1) or

order-of-magnitude inputs; the M

scaling, the epoch ordering, and the qualitative shift in con-

straint kind are insensitive to them, and we propagate the f

and mobility uncertainties into the

quoted abundance bounds as the shaded band of Fig. 3 rather than leaving them implicit. The

BBN/photodissociation bound is computed from the time-resolved injection using the standard

non-thermal-nucleosynthesis limits on ζ

evap

); the one part we do not treat from ﬁrst principles

is the hadronic injection during nucleosynthesis (t

evap

< 300 s), which needs a reaction-network

code and on which no conclusion rests. The distortion estimates use the standard visibility ap-

proximation rather than a full Green’s-function treatment; we expect factors of a few, not orders

of magnitude, from a reﬁned analysis.

8 Conclusions

If the vacuum is a discrete close-packed lattice, PBH evaporation acquires a geometric channel

with τ ∝ M

, frozen by the code-distance threshold above a sharp cutoﬀ M

cut

≃ 10

16.5

g. Trac-

ing the evaporation history of the sub-cutoﬀ population yields a deﬁnite mapping of PBH mass to

cosmological epoch: the lightest holes vanish before BBN without trace, while a band up to the cut-

oﬀ injects energy across nucleosynthesis, photodissociation, and the CMB spectral-distortion eras.

Confronting this with BBN and COBE/FIRAS, the intermediate-mass abundance is bounded at

PBH

≲ 10

−2

–10

−6

. These early-Universe limits replace the present-day γ-ray bounds that ap-

ply under standard evaporation (and that do not apply here, since the holes are gone), so the

scenario admits intermediate-mass primordial abundances that standard evaporation excludes—

though, since the beneﬁting masses lie below the survival cutoﬀ, as a transient early-Universe

population rather than as surviving dark matter. The scenario is compatible with all current data

for sub-dominant abundances and makes a falsiﬁable, mass-resolved prediction for spectral distor-

tions and BBN that a future PIXIE-class experiment could test. A more speculative information-

theoretic corollary—a monotonic radiation entropy with no Page turnover—is set out separately in

Appendix A. A dedicated BBN-cascade computation and a Green’s-function distortion treatment

are the natural reﬁnements.

Data availability. The computations and ﬁgures are reproducible with the script compute pbh.py,

available at https://github.com/raghu91302/ssmtheory/blob/main/compute_pbh.py.

Acknowledgements. This work builds on the FCC lattice studies of Refs. [27–29].

A Information content of the geometric-channel radiation

This appendix develops a speculative corollary of the geometric channel that is logically separable

from the paper’s observational results; none of the constraints in the main text depend on it, and

a reader interested only in the phenomenology may skip it. Because the geometric channel makes

the radiation an explicit re-emission of severed lattice bonds, the framework aﬀords a concrete

bookkeeping of what becomes of the information that fell in. We develop that here. The discussion

is interpretive rather than observational, and we keep its conclusions ﬂagged as model-level: the

proposal is that information is conserved in its lowest form, the bit count of severed bonds, while

the hierarchical structure of that information is not.

A.1 Four levels of information

The SSM vacuum is a tensor network: a decoration of the FCC lattice with Bell-pair bonds on

each of the K = 12 edges per site, organized into the [[192, 130, 3]] CSS stabilizer code [29]. Its

information content is hierarchical:



L1 (bond): the existence of an individual Bell pair on a given edge, carrying ln 2 of entanglement

entropy. Bell pairs are intrinsically indistinguishable, so L1 is a pure counting statistic—the

cardinality of the set of present bonds.



L2 (plaquette): the eigenvalue of a weight-4 CSS stabilizer A

= X

, one bit distin-

guishing conﬁgurations with the same L1 count but diﬀerent local correlations.



L3 (cell): the conﬁguration of the twelve bonds in a cuboctahedral K = 12 neighborhood—the

natural unit of geometric information, the discrete analog of a local curvature.



L4 (global): the joint stabilizer-eigenvalue pattern ﬁxing the logical state of the code, up to 2

130

states from the n −k = 62 stabilizers of the [[192, 130, 3]] code.

Each level is built from the previous but is not reducible to a count of its constituents: it lives in

their relational structure, as ink, letters, words, and catalog do in a library.

A.2 L1 is preserved as the thermal radiation; L2–L4 are not

When the lattice shatters at the metric wall, every Bell pair in the vacancy region is severed, and

each severed bond becomes a dangling half that radiates one thermal quantum (Proposition 4).

The number of quanta emitted over the hole’s life equals the number of bonds severed, so

quanta

= N

severed

ln 2

4G ln 2

, S

= N

severed

ln 2 =

. (20)

A thermal spectrum carries maximal entropy per energy—one bit, the quantum’s existence, per

quantum—so Hawking’s leading semiclassical thermal result [7] is, in this language, the statement

that only L1 information leaves the horizon, and S

is the L1 bit count of the horizon.

The higher levels do not cross the K = 12 → K = 0 boundary as structure. A stabilizer

eigenvalue is a joint property of four edges; once all four are severed the operator acts on a Hilbert

space of changed dimension and the pre-severance value is unrecoverable from the dangling-bond

ensemble (L2). The cuboctahedral cell requires its central site as a coordination center; once that

site is shattered its twelve bonds scatter into the boundary ensemble (L3). The global logical state is

ﬁxed by all 62 stabilizers; severing a macroscopic fraction leaves it undeﬁned, and any re-crystallized

K = 12 region carries a fresh, statistically independent L4 state rather than a unitary image of

the original (L4). The common cause is that indistinguishable constituents cannot carry relational

structure once their relations are broken: the count survives and is radiated, the organization does

not—as burning an ordered library preserves the atoms while losing the hierarchy of letters, words,

and catalog.

A.3 Unitarity, level by level

This reﬁnes the meaning of unitarity. L1 unitarity is fundamental and globally preserved: every

severed bond emerges as one quantum, with no L1 bits created or destroyed, even at the phase

boundary. By the mass–energy–information equivalence of the SSM framework [28], conservation

of the L1 bit count is identical to conservation of the radiated energy. L2–L4 unitarity is emergent

within the K = 12 phase, where stabilizer-code dynamics is a norm-preserving map on a ﬁxed

logical Hilbert space [29]: all standard physics lives here and sees full unitarity, and the strict

unitarity of ordinary quantum mechanics is, in this language, the L4 unitarity of the K = 12 phase.

But L2–L4 unitarity, deﬁned on a ﬁxed Hilbert space, simply does not extend across a transition in

which that space changes dimension. The standard information paradox, in this reading, arises only

from assuming strict L4 unitarity in a regime where the coordination-changing boundary makes

the assumption inapplicable.

A.4 Phase-boundary processes: black holes and the Big Bang

Two physical processes reduce the hierarchy to its lowest level. In black-hole formation and evap-

oration (K = 12 → K = 0 → K = 12), collapse reduces interior L2–L4 information to the L1

bit count of severed bonds; geometric evaporation (

3) then re-stitches K = 12 lattice across the

former vacancy, the radiation carrying away the L1 count while a fresh, statistically independent

L4 sector crystallizes in the re-stitched vacuum. In Big Bang crystallization, the matter paper [27]

establishes that the K = 12 vacuum emerges not in one step but through an ordered sequence

K = 0 → K = 6 → K = 4 → K = 12, as two-dimensional triangular sheets form and register into

FCC stacking. The four information levels switch on in step: L1 with the ﬁrst sheet (K = 6), L2

within sheets, L3 once cuboctahedral coordination exists, and L4 only after full registration. The

present vacuum’s L4 state was selected stochastically at the ﬁnal K = 4 → K = 12 transition and

is not unitarily related to any pre-Big-Bang state. Both processes are non-equilibrium transitions

in the coordination structure, outside the regime of ﬁxed-Hilbert-space L4 unitarity.

A.5 Relation to standard arguments

Banks–Susskind–Peskin [26]. Local non-unitary evolution is argued to violate energy conser-

vation or locality. Here L1 unitarity is preserved everywhere and the bit-count bookkeeping is local

at every step; the loss of L2–L4 structure is a Hilbert-space topology change with exact L1 (hence

energy) conservation, not local non-unitarity of the BSP type.

Page curve [23]. If evaporation is unitary at all levels, the radiation entropy must rise and then

fall, as replica-wormhole/island derivations reproduce within semiclassical gravity [24,25]. Here L4

unitarity is assumed not to survive the K = 12 → K = 0 reduction, so the radiation entropy rises

monotonically and no Page turnover occurs. This is a model assumption, not a result. The island

derivations rely on a smooth Euclidean path integral with a saddle across the horizon, which the

discrete SSM does not provide; the two address diﬀerent regimes and we do not claim the island

result is wrong. They do make contradictory predictions for the radiation entropy, and that is the

sharpest point of falsiﬁability: a ﬁrm demonstration of Page-curve recovery in a setting the SSM

claims to describe would rule out this picture.

AdS/CFT. A black hole dual to a unitary boundary CFT has L4 unitarity by construction.

The SSM vacuum is asymptotically ﬂat with a discrete K = 12 FCC structure and no evident

holographic dual; the framework lies outside the domain of the duality rather than in conﬂict with

it.

A.6 Summary

Level Information content Carrier Survives?

L1 (bond) Existence of one Bell

pair; ln 2 per bond

Severed-bond thermal

quantum

Yes (as radiation)

(plaquette)

CSS stabilizer eigenvalue;

weight-4 correlation

Stabilizer plaquette A

No (reduces to L1)

L3 (cell) Cuboctahedral K = 12

neighborhood

Local 12-bond pattern No (reduces to L1)

L4 (global) [[192, 130, 3]] logical-qubit

state

Joint stabilizer pattern No (reduces to L1)

Table 4: The four information levels and their fate across the K = 12 → K = 0 boundary. L1

is preserved and emerges as thermal radiation with total bit count A/(4G ln 2), recovering S

;

L2–L4 reduce to that ﬂux. The bit count is conserved; the hierarchy is not.

The picture is conditional and explicitly assumption-dependent: if information is conserved

only at the lowest level while the hierarchy is not, then the radiation is exactly thermal (the L1

channel), the Bekenstein–Hawking entropy is the L1 bit count, and the radiation entropy rises

monotonically with no Page turnover. The last is a consequence of the central assumption, not

an independent success, and is where the model departs from the unitary-at-all-levels picture and

where it is falsiﬁable. We carry it here as an interpretation of the geometric-channel radiation, not

as an observational result.

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