Horizon Area Quantization from Bond-Counted Entropy:
a Gravitational-Wave Test of the Selection–Stitch Model
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA raghu@idrive.com
August 10, 2026
Abstract
In the Selection–Stitch Model a black-hole horizon carries entropy by bond counting: S =
Nk ln 2 with N the integer number of boundary bonds, calibrated to A/4ℓ
2
P
in a published
companion. One line of algebra then quantizes the horizon area, A = (4 ln 2) N ℓ
2
P
, with area
quantum ∆A = 4 ln 2 ℓ
2
P
. This is exactly the Bekenstein–Mukhanov spectrum, proposed in 1995
on information-theoretic grounds; here it arrives from a mechanism, with no free parameter. In
the language of the Foit–Kleban gravitational-wave test, which treats the quantization constant
α in ∆A = αℓ
2
P
as free, the model fixes α = 4 ln 2 ≃ 2.77. The predicted transition-line spacing
is f
1
= αc
3
/(64π
2
GM): 89 Hz for a 10 M
⊙
remnant, 30 Hz at 30 M
⊙
— a universal 7.4%
of the fundamental ringdown frequency, independent of mass. The prediction is falsifiable on
both sides: a ringdown measurement consistent with an unquantized continuum at the relevant
sensitivity rules the model out, and a measured α different from 4 ln 2 — for instance Hod’s
4 ln 3 — rules it out just as cleanly. The derivation is conditional on the companion’s imported
normalization and is stated for static horizons; the rotating case is open.
1 Introduction
Bekenstein proposed that the horizon area of a black hole is an adiabatic invariant and therefore
quantized [1]. Bekenstein and Mukhanov sharpened the proposal: if each area quantum carries one
bit of information, the spectrum is A = (4 ln 2) Nℓ
2
P
with integer N [2]. The proposal has rivals.
Hod derived ∆A = 4 ln 3 ℓ
2
P
from the asymptotic quasinormal spectrum [3]; Maggiore’s reanalysis
returns 8πℓ
2
P
[4]; Bekenstein’s original argument gave 8πℓ
2
P
as well. Foit and Kleban showed that
the question is empirical: writing ∆A = αℓ
2
P
, near-future ringdown observations can measure or
exclude α [5], and with Cardoso they computed the post-merger echo structure that quantization
imprints [6]; see also Ref. [7]. In that program α is a free parameter.
This paper supplies a value. In the Selection–Stitch Model — a framework in which the vacuum
is a face-centered-cubic lattice running a stabilizer code [12] — a recently published companion
derived the Bekenstein–Hawking entropy by counting horizon bonds [13]. The count is an integer.
One line of algebra turns that integer into an area spectrum, and the spectrum is Bekenstein and
Mukhanov’s, with α = 4 ln 2 fixed by the published calibration. The model therefore enters the
Foit–Kleban test with no free parameter, and the test cuts both ways.
1
2 The derivation
The companion paper’s horizon entropy is bond counting [13]: a horizon of area A is crossed by N
lattice bonds, each contributing one bit,
S = N k ln 2 =
A
4ℓ
2
P
k, (1)
where the second equality is the paper’s calibration of the bond length against the Bekenstein–
Hawking coefficient. N is an integer. Solving Eq. (1) for the area,
A = (4 ln 2) N ℓ
2
P
, ∆A = 4 ln 2 ℓ
2
P
≃ 2.7726 ℓ
2
P
. (2)
The horizon area is quantized, and the quantum is the Bekenstein–Mukhanov value: in the con-
vention ∆A = αℓ
2
P
,
α = 4 ln 2. (3)
[derived, given the companion’s imported normalization]. Nothing was fit: the 4 is the Bekenstein–
Hawking coefficient, the ln 2 is one bit per bond, and both are in print [13].
For a Schwarzschild horizon, A = 16πG
2
M
2
/c
4
, so an area quantum corresponds to a mass
quantum ∆M = c
4
∆A/(32πG
2
M), and radiation exchanged with the hole comes in lines at mul-
tiples of
ω
1
=
∆Mc
2
ℏ
=
α
32π
c
3
GM
=
ln 2
8π
c
3
GM
, f
1
=
α
64π
2
c
3
GM
, (4)
the same ladder Refs. [5, 6] analyze for general α. Numerically: f
1
= 89.1 Hz for M = 10 M
⊙
,
29.7 Hz for 30 M
⊙
, 13.7 Hz for 65 M
⊙
— inside the LIGO–Virgo band for stellar-mass remnants
(Fig. 1). Dividing by the fundamental ℓ = 2 ringdown frequency, f
220
≃ 0.3737 c
3
/(2πGM) for a
nonrotating hole, gives a mass-independent ratio,
f
1
f
220
=
4 ln 2
64π
2
× 0.3737/(2π)
≃ 7.4%. (5)
The line spacing is a fixed fraction of the ringdown frequency for every mass. [derived]
3 Where the value sits
The classical prediction is the null hypothesis. In general relativity the horizon area is a smooth
variable: it can change by any amount, the hole absorbs and emits at any frequency, and the
ringdown is a continuous quasinormal spectrum with no internal line structure. Every quantization
proposal replaces that continuum with a frequency comb, and the proposals differ only in the comb’s
spacing. Table 1 sets the alternatives side by side.
Equation (3) lands on the oldest proposal in the list. Bekenstein and Mukhanov chose 4 ln 2
because a two-state degree of freedom per quantum is the minimal information-theoretic assign-
ment [2]; they had the value and lacked a mechanism. The bond-counted horizon is a mechanism:
each boundary bond is a two-state system because the code’s checks are binary, and the calibration
that fixes the bond length is the same one that reproduces Hawking’s temperature in the compan-
ion [13]. Figure 2 shows the three ladders against a single ringdown resonance: the combs differ
enough that the Foit–Kleban analysis distinguishes them [5]. Loop quantum gravity sits outside
the comb family altogether: its area spectrum is unevenly spaced [10] and effectively continuous at
stellar masses, so in this channel it predicts what classical relativity predicts. The test therefore
discriminates at two levels: a comb against no comb, and then the spacing among the equispaced
proposals.
2
10
1
10
2
10
3
remnant mass
M
[
M
]
10
0
10
1
10
2
10
3
10
4
frequency [Hz]
line spacing
f
1
=
c
3
/(64
2
GM
), = 4ln 2
fundamental ringdown
f
220
(spin 0)
LIGO-Virgo band
Einstein Telescope extension
Figure 1: Predicted transition-line spacing f
1
(solid) against remnant mass, with the fundamental
ringdown frequency f
220
(dashed) for scale. For stellar-mass remnants the spacing sits inside the
LIGO–Virgo band; the Einstein Telescope extends the reach to intermediate masses. The two
curves keep the fixed ratio of Eq. (5).
classical GR SSM (this work)
horizon area continuous quantized, ∆A = 4 ln 2 ℓ
2
P
emission/absorption continuous spectrum line comb, Eq. (4)
line spacing, 10 M
⊙
none 89.1 Hz
line spacing, 30 M
⊙
none 29.7 Hz
spacing / f
220
— 7.4%, mass-independent
rival combs, 10 M
⊙
: Hod 4 ln 3: 141 Hz Bekenstein–Maggiore 8π: 808 Hz
loop quantum gravity unevenly spaced [10], effectively continuous at stellar mass: no comb
Table 1: The classical continuum against the quantized alternatives. The SSM value is derived;
the rivals [3, 1, 4] are shown at the same remnant mass for scale.
4 Signatures and present reach
Three observational channels exist, all developed in the literature for general α and inherited here
with α fixed.
Ringdown filtering. A quantized horizon absorbs only at the lines, so the ringdown of a newly
formed hole is filtered relative to the classical prediction. Foit and Kleban show that a single
measurement consistent with classical general relativity constrains most proposed values of α, with
the discriminating power depending on the remnant’s spin [5]. Ringdown spectroscopy is an active
program: the fundamental mode and one overtone have been measured in GW150914 [8].
Echoes. Cardoso, Foit, and Kleban compute distorted post-merger echoes whose frequency
content encodes α [6]; Coates, V¨olkel, and Kokkotas refine the modeling [7]. Echoes alone are not
unique to quantization — exotic compact objects produce them too [11] — so the discriminator is
3
0 200 400 600 800
frequency [Hz] (remnant
M
= 30
M
, spin 0)
0.0
0.2
0.4
0.6
0.8
1.0
arbitrary amplitude
classical = 2 ringdown resonance (
Q
2.2)
= 4ln 2 (this work, derived)
= 4ln 3 (Hod)
= 8 (Bekenstein 1974)
Figure 2: Transition-line ladders for three proposed area quanta, drawn against the classical ℓ = 2
ringdown resonance of a 30 M
⊙
nonrotating remnant. The model’s derived α = 4 ln 2 (tallest, blue)
gives the densest ladder; Hod’s 4 ln 3 (red) and the Bekenstein–Maggiore 8π (green) are shown for
comparison. Line heights are for visual separation only.
the comb’s frequency content, not the existence of echoes. With α = 4 ln 2 the echo template is
fully specified up to the astrophysical parameters of the event.
Evaporation lines. The companion’s geometric evaporation law releases boundary bonds one
at a time [13]; the emission spectrum of a late-stage primordial black hole is then a line spectrum
with the spacing of Eq. (4). For the microsecond-scale terminal bursts predicted there, the lines
sit at gamma-ray energies; resolving them is beyond present instruments, and this channel is listed
for completeness. [conditional on the companion’s evaporation law]
Current ringdown precision on stellar-mass events is at the several-percent level in frequency;
the maximal line-snapping offset implied by Eq. (5) is 3.7%. The test is therefore marginal today
and decisive with the Einstein Telescope’s precision spectroscopy [9].
5 Falsifiers and limitations
The observational verdict is two-sided. A confirmed continuum refines nothing — it is the classical
expectation — but at sufficient precision it excludes the comb; a detected comb would establish
horizon quantization and, by its spacing, select among the proposals of Table 1. Two falsifiers,
both sharp. (F1) A ringdown measurement of a low-spin remnant consistent with an unquantized
continuum, at the sensitivity Ref. [5] computes for α ≃ 2.8, falsifies the bond-counted horizon and
with it the published entropy mechanism. (F2) A measured area quantum different from 4 ln 2 ℓ
2
P
— for instance Hod’s 4 ln 3 — falsifies it just as directly; the model has no freedom to move α.
Three limitations, stated as in the companion series. (L1) The derivation inherits the compan-
ion’s imported normalization: the 4 in Eq. (1) is calibrated to the Bekenstein–Hawking coefficient,
not derived from the code. (L2) The construction is static and spherically symmetric; the rotating
case requires a lattice treatment of angular momentum that does not yet exist, so the prediction
4
is stated for low-spin remnants and the extension is open. (L3) The line spacing is derived; the
line widths and transition rates are not, and they control detectability in detail. The companion’s
division of labor applies here unchanged: the count is the code’s, the geometry is the lattice’s, and
the two meet only through the calibration.
Data Availability Statement
All numbers and both figures are produced by one script, gw area quant.py, archived with the
series’ verification suites at github.com/raghu91302/ssmtheory.
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