The Quantum Zeno Effect Links an Error-Correcting
Code to Confined Fields:
The Selection–Stitch Lattice
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
Abstract
In the Selection–Stitch Model (SSM), the vacuum is a quantum code, and the mass of a defect is
the cost of verifying it. We ask how a verification cost becomes an energy. We let an excitation
that fails checks carry a local mass equal to their cost, and define a state’s verification cost as
the rise in its energy when this mass is switched on. If the checks fail everywhere outside a
region, the strong-coupling form of the quantum Zeno effect makes this cost equal to the lowest
mode energy of the field in that region. This energy has the form
xℏc/L
. We read it as an
instance of the SSM rule that a defect’s verification rate sets its Compton frequency. Landauer’s
principle, which prices erasure, does not apply to such standing information. For a massless
quark, only a mass gives the bag wall. A projector penalty does not, and neither does an electric
potential. A wall of finite strength
K
acts as a hard wall moved outward by about
ℏ/p
K
, so walls
for massless particles converge faster as
K
grows. On the lattice, a color singlet needs a whole
tetrahedral cell. With a bond-verification term
h
added to the code’s checks
λ
, the code maps
onto a transverse-field Ising model, and an exact spectral check confirms the mapping. Quantum
Monte Carlo on up to 2048 nodes finds free colors below
h/λ ≈
0
.
09 and a linear potential above
0
.
11, with a string tension that holds as the box grows. For the proton, the measured charge
radius fixes the size of its 13-site verification region. Three confined quarks then give 865 MeV,
8% below the proton mass and nearly independent of the cavity’s shape. The SSM’s count of
the same region’s checks gives 938 MeV.
Keywords: quantum Zeno effect; verification; confinement; string tension; quantum Monte Carlo;
bag model; proton mass; quantum error correction
1 Introduction
In the Selection–Stitch Model (SSM), the vacuum is a lattice of entangled bonds, and the lattice is
a quantum error-correcting code [
1
]. A state that passes every check of the code is a vacuum state.
A defect fails some checks, and its mass is the cost of checking it [
2
]. So far this cost has been a
count. It is the number of qubits a defect touches times the number of checks that can see it.
A count is about records. A mass is about energy and motion. This paper asks what links the
two, and what verification does to the field that carries a defect.
We make one assumption. An excitation that fails checks carries a local mass equal to their
cost. This is the SSM’s rule that mass is verification cost, applied at each point. We then define the
verification cost of a state as an energy. It is the rise in the ground-state energy when this mass is
switched on. A system that is checked often enough does not change. This is the quantum Zeno
effect [3]. Its strong-coupling form [4] gives the result we use:
1
An excitation held in a region by strong checks outside it has a verification cost equal to
the lowest mode energy of the field in that region.
One side of this statement counts failed checks. The other side solves a wave equation.
Five results follow.
1.
The cost must be a mass. A massless quark is not held by checks of its position, by a projector
penalty, or by an electric potential. It is held by a verification mass outside the region, which
gives the bag condition of hadron physics (Section 5).
2.
The energy of standing information. Landauer’s principle prices erasure and cannot give a
mass. The SSM’s rule for standing information applies instead [
5
,
6
]: a mass is a count times
a fixed energy per bit, and a defect’s verification rate sets its Compton frequency. We read
the cavity energy, xℏ times the crossing rate c/L, as an instance of this rule (Section 4).
3.
A Zeno law for walls. A verification mass
K
makes a region of size
L
act as one of size
L
+
ℏc/K
. A wall for a massless particle approaches the hard wall as 1
/K
, against 1
/
√
K
for
a slow particle. Only a penalty whose mass part outweighs its potential part holds both a
particle and its antiparticle. At a fixed charge radius, the wall’s strength moves the proton
mass by less than 1% (Section 6).
4.
A confined phase of the SSM code. A lone color is a fraction of a tetrahedral cell, and it fails
two checks at any distance. With a bond-verification term
h
added to the code’s checks
λ
, the
code maps onto a transverse-field Ising model. An exact spectral comparison on a fragment
of the code confirms this mapping. Quantum Monte Carlo on up to 2048 nodes finds free
colors below
h/λ ≈
0
.
09 and a linear potential, with a string tension that holds as the box
grows, above 0
.
11. In the confined phase, fractions are confined and complete patterns, such
as hadrons and leptons, are not (Section 7).
5.
The proton. Applied to the 13-site cluster of the SSM’s confinement code, with its size fixed by
the measured charge radius, the duality gives 865 MeV for three quarks. This is 8% below the
proton mass. The SSM’s count of the same region’s checks gives 938 MeV, a second estimate
of the same verification cost (Section 8).
2 The SSM framework
This section summarizes the parts of the SSM that this paper uses. Each rests on a published or
archived paper, except the use of one code at two lengths, which this paper takes as a premise. This
paper does not test them.
The vacuum is a code. The basic object of the SSM is a bond. A bond is a Bell pair that joins
two nodes at a fixed distance [
7
]. The bonds assemble into the face-centered-cubic (FCC) lattice, in
which each node has twelve neighbors. Ref. [
1
] puts a qubit on each edge of this lattice and defines
two kinds of check (Fig. 1). A
Z
-check acts on the twelve bonds of a node. An
X
-check acts on the
twelve edges of an octahedral void. The two kinds commute, so they define a stabilizer code. The
vacuum is its code space. In the SSM, a bond persists because the code keeps verifying it.
Mass is the cost of verification. Ref. [
2
] proposes that a defect’s mass is the cost of checking
it. The cost is a count of qubits and checks, turned into an energy. Mass ratios are then ratios of
integers. Ref. [
5
] states the step from a count to an energy as a postulate for standing information
2
(a)
Z
-check on a node (b)
X
-check on an octahedral void
(c) the 13-node cluster
Figure 1: The SSM code [
1
,
2
]. (a) A
Z
-check: the twelve bonds of one node. (b) An
X
-check: the
twelve edges of one octahedral void. (c) The 13-node cluster of a node and its neighbors. In the
confinement code this cluster is the proton’s verification region.
(Section 4). Ref. [
6
] showed that a verification cost couples to geometry as a rest mass, if verification
ticks measure proper time.
One geometry, two scales. We take the same FCC code to run at two lengths. The fine code
has a spacing near the Planck length. It carries spacetime and gravity, the photon, the leptons, the
neutrinos, the Higgs condensate, and the
W
and
Z
bosons. The confinement code has a spacing
L
c
near 1 fm. It carries color and the hadrons. The fine code does not carry color. The proton is
a defect, a node trapped in a tetrahedral void [
7
]. Its mass lives in the 13-site FCC cluster that
verifies it, a node and its twelve neighbors, with 36 bonds (Fig. 1c). We call this cluster the proton’s
verification region. In this paper, the verification region is also the cavity. Counting the checks of
this cluster gives m
p
/m
e
= 1836 [2].
The ocean. The vacuum’s fluctuations persist around the finished lattice. We call them the
ocean [
6
]. The bonds and the defects were stitched from the ocean. The ocean itself is unverified,
and no check acts on it. This paper uses one property of the ocean. Inside a verified region, its
modes are the modes of a cavity.
3 Verification as an energy
We need a verification cost that is an energy. In the SSM the mass of a defect is its verification
cost [
2
]. We apply this point by point. Let a field excitation at a point x fail
n
(x) checks of the
code, each with cost
λ
. Then the excitation carries a verification mass
m
v
(x) =
λ n
(x) there. For a
fermion field ψ, a mass enters the Hamiltonian as a Lorentz scalar:
H = H
field
+
Z
m
v
(x)
¯
ψψ d
3
x. (1)
We define the verification cost of a state as the rise in its energy when this term is switched on:
∆E = E
0
(H) − E
0
(H
field
). (2)
3
10
−
4
10
−
3
interval between checks,
τ
9.1
9.2
9.3
9.4
9.5
9.6
9.7
lowest energy (
ħ
2
/2
mL
2
units)
hard wall (same grid)
(a) particle checked for being inside
10
1
10
2
check strength
KL
/
ħ
c
1.425
1.450
1.475
1.500
1.525
1.550
1.575
lowest energy (
ħ
c
/
L
)
bag wall,
π
/2
(b) massless quark, checked outside
check on scalar density
Figure 2: Walls from checks, in one dimension. (a) A particle checked at intervals
τ
for being inside
an interval. Its lowest energy approaches the hard-wall value on the same grid. (b) A massless
quark with the verification mass
K
of Eq.
(1)
outside the interval. Its lowest energy approaches the
bag value πℏc/2L, and the curve follows πℏc/2(L + ℏc/K).
No measuring apparatus enters this definition. Equation
(1)
is the only assumption. It says that a
verification cost acts on a field as a local mass.
Now let the excitation fail no checks inside a region Ω, and fail checks everywhere outside it.
Then
m
v
= 0 inside and
m
v
≥ λ
outside. As
λ
grows, the excitation is pushed out of the outside
region. This is the strong-coupling form of the quantum Zeno effect. A large term in the Hamiltonian
splits the dynamics into that term’s eigenspaces [
4
]. Here the eigenspaces are the inside and the
outside. In the limit, the excitation lives in Ω, with a boundary condition set by the form of the
term. A free massless excitation in open space has energies down to zero. The confined one has a
lowest energy, the lowest mode of the cavity Ω. So
in the limit of strong checks, the verification cost of an excitation held in a region by the
checks equals the lowest mode energy of the field in that region.
This is the duality of Section 1. It follows from Eqs. (1) and (2). Section 5 shows that it holds for
quarks. It also shows that the term must be a mass.
Pulsed checks. A simpler setting shows the same wall. Let a nonrelativistic particle,
H
=
p
2
/
2
m
,
be checked at intervals
τ
by a projector
P
onto a region Ω. After
N
checks over a time
t
=
Nτ
, it
has evolved by (
P e
−iHτ
P
)
N
. As
τ →
0 at fixed
t
, this tends to evolution by the kinetic energy in
Ω with the wave function set to zero on the boundary [
3
,
8
,
9
]. Figure 2a shows this numerically.
Pulsed projections restrict the states the particle can reach. They do not by themselves fix what
energy is paid. That is why we define the cost through Eq. (2).
4 The energy of standing information
Why Landauer’s principle does not apply. Ref. [
2
] used Landauer’s principle to turn counts
into energies. Landauer’s principle says that erasing one bit at temperature
T
costs at least
k
B
T ln
2 [
10
]. It describes erasure, an irreversible step that gives off heat. It assigns no cost to
information that is kept and never erased, and its bound vanishes as
T →
0. A rest mass is the
4
Table 1: The factor
x
in
E
=
x ℏν
v
for fields held by strong checks.
L
is the length of the interval
in one dimension, the radius of the sphere, and the circumradius of the cluster.
field region x
massless quark (bag wall) interval π/2 = 1.571
massless scalar (hard wall) interval π = 3.142
massless quark (bag wall) sphere 2.043
massless quark (bag wall) 13-site cluster 2.60
gluon (bag wall) sphere 2.744
opposite case. It persists, it exists at zero temperature, and it does not depend on
T
. So Landauer’s
principle cannot turn a verification count into a mass.
The rule for standing information. Ref. [
5
] replaces Landauer’s principle with a postulate (its
Postulate 1, Section 4.1). In a code with no independent substrate, standing syndrome information
that cannot be exported is localized energy at the site that requires it, at a fixed amount
ε
b
per bit:
E = C ε
b
, Mc
2
= C ε
b
, (3)
where
C
is the count. Only ratios are used,
M
x
/M
y
=
C
x
/C
y
, so
ε
b
cancels and no temperature
enters. Nothing is erased, because the defect is never repaired and its syndrome is constant. There
is no environment to take up heat. So Landauer’s bound on the cost of erasure does not apply. It
supplies only the precedent that information and energy convert [
5
]. Ref. [
5
] states the rule as a
postulate, not as a theorem. Ref. [
6
] adds the dynamics. If each verification adds a fixed phase
φ
0
to
the state, and verification ticks measure proper time with a tick duration
τ
0
, then the cost couples to
geometry as a rest mass, with
ε
b
=
ℏφ
0
/τ
0
. The phase then turns at the angular frequency
Mc
2
/ℏ
:
the verification rate sets the Compton frequency. A particle’s mass is known to set a clock at this
frequency [
11
]. The rule needs no change of state. A stationary state turns its phase at the rate
E/ℏ, and a check that confirms the state without changing it fits the rule.
The cavity form. The duality of Section 3 gives the energy of a confined mode as
E
=
x ℏc/L
.
Write
ν
v
=
c
L
, (4)
the rate at which a signal at the speed of light crosses the region. Then
E
=
x ℏν
v
. This is a
statement about the cavity spectrum, and it holds for any confined mode. In the SSM we read
it as Eq.
(3)
for one confined mode: the mode’s phase turns at
x ν
v
, and this rate plays the role
of
Cφ
0
/τ
0
in Ref. [
6
]. That reading is an interpretation. It assumes that the region is verified at
the rate
ν
v
, which is the fastest rate a signal allows, not a rate the code is shown to have. Table 1
gives
x
for the fields of this paper. For a sphere,
L
is the radius. For the 13-site cluster,
L
is its
circumradius L
c
, as in Section 8.
A consistency check. If each verification changed the state into an orthogonal one in the time
L/c
, the quantum speed limit would require
E ≥
(
π/
2)
ℏc/L
[
12
]. All the values of Table 1 meet
this bound, and the one-dimensional bag quark meets it exactly. But a verification need not change
the state, so the bound is a check on the values, not a derivation of them.
5
Two rules for information. Landauer’s principle and Eq.
(3)
answer different questions. Lan-
dauer’s principle prices an erasure. Its scale is
k
B
T
, and the energy leaves as heat. Equation
(3)
prices standing information. Its scale is a fixed energy per bit,
ℏφ
0
/τ
0
in Ref. [
6
]. It does not depend
on temperature, and the energy stays as rest energy. A mass is standing information, so Eq.
(3)
is
the rule that applies to it.
5 Which term holds a quark
Quarks are massless Dirac particles, with
H
field
=
c α ·
p. We tested Eq.
(1)
against other ways to
charge a quark for being outside a region. We worked in one dimension, with
H
field
=
c σ
x
p
and
β = σ
z
, on an interval of length L. The bag condition of hadron physics [13] gives a lowest energy
of πℏc/2L.
Pulsed position checks. A massless particle moves at the speed of light from the first instant.
Its probability to leave the region in a time
τ
is linear in
τ
, not quadratic. Each check removes
the part that has crossed, and nothing slows the rest. So pulsed checks of position do not hold a
massless quark.
Three penalties. Table 2 compares three terms of strength
λ
outside the interval. The first
is the verification mass of Eq.
(1)
,
λβ
. The second is a stabilizer-style penalty,
λ
(1
− S
)
/
2 with
S
=
−β
, which charges the quark for having its upper component outside. The third charges any
quark density outside,
λ
, like an electric potential. Only the mass gives the bag wall. The projector
holds quarks, but its wall is different. In one dimension its lowest energy tends to
πℏc/L
. This
is the zigzag condition, in which one spinor component vanishes on the wall [
14
]. It also holds
no antiquark, so it treats matter and antimatter differently. The electric potential holds nothing,
because a massless quark crosses it (the Klein effect [
15
]). So a penalty on a quark’s presence is not
enough. The penalty must be a mass, as Eq.
(1)
states. In three dimensions, a large mass outside a
smooth region gives the bag condition in the limit [16].
The wall is the vacuum outside. In the SSM, the verification mass outside the cluster says
that carrying a quark there fails checks. Inside the cluster a quark’s color is part of a complete
pattern with the other two colors, and fails none. Outside, it is separated from them. Section 7
shows that this fails at least two checks, so
m
v
≥
2
λ
there. Confinement is the limit of large
λ
. So
the wall is not a check on the quark inside. It is the verified vacuum outside, which does not carry
a lone color.
6
Table 2: Three penalties of strength
λ
on a massless quark outside an interval of length
L
, in one
dimension. Energies in units of ℏc/L. The bag value is π/2 = 1.571.
penalty outside λ = 20 50 100 200 antiquark held
verification mass, λβ 1.496 1.540 1.556 1.572 yes
projector, λ(1 + β)/2 2.43 2.68 2.81 2.90 no
any density, λ none none none none no
6 The strength of the wall
A finite verification mass
K
makes a soft wall. The quark leaks a short distance into the outside
region, and its energy falls. In one dimension the numerical energies follow
E
1
=
πℏc
2 (L + ℏc/K)
(5)
to within 0
.
3% for
KL/ℏc ≥
10 (Fig. 2b). The wall acts as if the region were longer by
ℏc/
2
K
on
each side. The energy is 9% low at KL/ℏc = 10 and 3% low at 30.
How large is
K
in the SSM? Outside the cluster a quark fails at least two checks, so
K ≥
2
λ
. In
the confined phase of Section 7, its cost also grows with distance, as a string tension
σ
. Leaving
across one bond length then costs at least
σL
c
, so
KL
c
/ℏc ≳ σL
2
c
/ℏc
. With
σ ≈
1 GeV/fm and
L
c
= 1.45 fm, this is about 10. So the wall is not hard. It lies near the softest case tested below.
In three dimensions we repeated the cavity calculation of Section 8 at three strengths,
KL
c
/ℏc
=
12, 24, and 48, on the same grid (Table 3). A softer wall lowers the mode energy
x
, as in one
dimension. It also spreads the quark density outward, so its rms radius
ρ
grows. Section 8 shows
that the mass at a given charge radius depends on the product
xρ
. The two changes nearly cancel
in that product. From
KL
c
/ℏc
= 48 to 12,
x
falls by 3
.
6% for the cluster and
ρ
rises by 4
.
7%. The
product changes by 0
.
9% for the cluster and 0
.
7% for the sphere. So the strength of the wall moves
the predicted mass by less than 1%.
A Zeno law for walls. The results so far combine into one law for walls made by checks. It has
four parts.
1.
The kind of check. For a slow particle, frequent checks of position make a wall [
8
,
9
]. For a
massless Dirac particle in one dimension they do not. Its two chiral parts move at
±c
, and a
check of position does not mix them, so after a time
L/c
the particle has left, however often it
is checked. We expect the same in three dimensions but have not shown it. A strong penalty
works if its mass part, the part that reverses the velocity normal to the wall, outweighs its
potential part. Table 4 shows the cases in one dimension. A pure mass, scalar or pseudoscalar,
gives the bag wall and treats the particle and its antiparticle alike. A mixture with a smaller
potential part still holds both, but it shifts their levels in opposite directions. In each case
the two levels add to about
πℏc/L
, as for the generalized bag conditions of Ref. [
14
], which
arise from scalar and electrostatic potentials. Equal parts, the projector of Section 5, hold the
particle only, which is the zigzag limit of the same family [
14
]. A larger potential part, or a
pure potential, holds neither (the Klein effect [15]).
2.
The approach to the hard wall. A wall of finite strength
K
acts as a hard wall moved outward
on each side by a penetration length
ℓ
Z
, as long as
K
is much larger than the confined energy.
For a slow particle
ℓ
Z
=
ℏ/
√
2mK
, and for a massless one
ℓ
Z
=
ℏc/
2
K
(Table 5). Both are of
7
Table 3: Soft walls in three dimensions: lowest quark mode
x
and rms radius
ρ
of its density, in
units of the circumradius, on a grid of 40
3
sites with circumradius 12 sites. The product
xρ
fixes
the mass at a given charge radius.
sphere 13-site cluster
KL
c
/ℏc x ρ xρ x ρ xρ
12 2.000 0.743 1.486 2.525 0.595 1.503
24 2.042 0.724 1.480 2.592 0.576 1.494
48 2.062 0.716 1.476 2.620 0.569 1.490
Table 4: Strong penalties on a massless Dirac particle outside an interval of length
L
, in one
dimension, at strength
λ
= 200
ℏc/L
. The penalty is
λ
(
β
+
a
), a mass
β
plus a potential of relative
size
a
, or
λσ
y
, a pseudoscalar mass. Energies of the lowest held particle and antiparticle, in units of
ℏc/L. The bag value is π/2 = 1.571. None means that no level stays finite as λ grows.
penalty particle antiparticle wall
mass, a = 0 1.572 1.572 bag
pseudoscalar mass 1.572 1.572 bag
a = 0.5 2.086 1.047 generalized bag
a = 0.9 2.659 0.446 generalized bag
a = 1 (projector) 2.97 none zigzag
a = 1.5 none none none
pure potential none none none
order
ℏ/p
K
, where
p
K
is the momentum that the energy
K
can buy. The three-dimensional
data of Table 3 agree: for the sphere, a fit of
x
=
x
∞
/
(1 +
a ℏc/KL
) gives
a
= 0
.
504, against
1
/
2 from
ℓ
Z
=
ℏc/
2
K
. So the wall for a massless particle approaches the hard wall as 1
/K
,
and the wall for a slow particle as 1
/
√
K
. Which wall is softer at a given
K
depends on how
K compares with mc
2
. The law concerns how fast each wall converges.
3.
The cost. The confined field’s energy is the lowest mode of the region,
x ℏc/L
for a massless
field. In the SSM this is read as ℏ times the verification rate (Section 4).
4.
Competing checks. In the SSM code the two kinds of check, the code’s checks and bond
verification, do not commute. The system switches between their two Zeno limits at a sharp
transition, the confinement transition at
h/λ ≈
0
.
10 (Section 7). Terms that do not commute
need not produce a sharp transition in general. In this code they do.
Each part rests on known physics: evanescent waves, the Klein effect [
15
], the infinite-mass and
generalized bag limits [
16
,
14
], and Wegner’s duality [
17
]. The law gathers them as statements
about walls made by checks.
8
Table 5: The approach of a Zeno wall of strength
K
to the hard wall, on an interval of length
L
in
one dimension, for
K
much larger than the confined energy.
E
∞
is the hard-wall energy, and
ℓ
Z
the
penetration length on each side. The agreement is with a direct numerical solution.
particle p
K
ℓ
Z
lowest energy at strength K agreement
slow, mass m
√
2mK ℏ/p
K
E
∞
L/(L + 2ℓ
Z
)
2
0.4% for K = 100–1600
massless (Dirac) K/c ℏ/2p
K
E
∞
L/(L + 2ℓ
Z
) 0.01% for K = 25–100
7 Which species are confined
The duality confines only what the checks act on. This section shows which species that is. The
answer follows from the cells of the FCC lattice and from the cost of verifying bonds.
The cells of the cluster. The FCC lattice fills space with two kinds of cell, tetrahedra and
octahedra. Eight tetrahedral cells and six octahedral cells meet at each node (Fig. 3a). Every
corner of the eight tetrahedral cells lies in the node’s 13-site cluster. Each of the six octahedral
cells has one corner outside the cluster, at the second-neighbor distance
√
2 L
c
from the center. So
the tetrahedral cells fit inside the cluster, and the octahedral cells reach out of it. Every face of a
tetrahedral cell is shared with an octahedral cell, never with another tetrahedral cell.
Color fits only in tetrahedral cells. The FCC lattice is four interpenetrating cubic lattices.
Every tetrahedral cell has exactly one site from each of the four. One of them anchors the cell, as
the trapped node’s anchor bond does in Ref. [
7
]. The other three are the cell’s three colors. Ref. [
7
]
takes the three colors from the tetrahedron’s three pairs of opposite edges. Each pair contains one
edge from the anchor, and that edge ends on one of the three color sites. So the two descriptions
agree. So a color singlet needs a whole tetrahedral cell. No octahedral cell can hold one, because
each octahedral cell lacks one of the four sublattices. A tetrahedral cell inside the cluster also shares
edges with tetrahedral cells outside it. Each of those is again a complete singlet, with its own anchor.
Passing through a shared edge moves a whole singlet, a hadron. It never separates a color.
A lone color fails two checks at any distance. In the SSM code each bond carries a qubit,
and each octahedral cell carries an
X
-check on its twelve edges [
1
]. Apply
Z
to the three bonds
that join an anchor to the three colors of one tetrahedral cell. This operator commutes with every
check, so it is a singlet. A
Z
on one of those bonds, a single color, fails the two octahedral checks
that contain that bond. These two failures can be pulled apart by flipping a chain of bonds that
links neighboring octahedral cells. The number of failed checks stays at two, whatever the distance.
The number of flipped bonds grows with it, about
√
2
per
L
c
along a cube axis (Fig. 3b). So the
checks alone charge a lone color a fixed 2λ. They do not charge it more for going farther.
Bond verification and the confined phase. A cost that grows with distance needs a term
that sees the chain’s bonds. In the SSM a bond persists because it is verified (Section 2). We write
this as a second term:
H
code
= λ
X
a
1 − S
a
2
+ h
X
e
1 − X
e
2
, (6)
where the
S
a
are the code’s checks and
X
e
detects a flip of bond
e
. Each
X
e
fails to commute
with the node checks that contain
e
. So the two terms compete. Systems of this kind have two
9
(a) cells of the 13-site cluster
1 2 3 4 5
distance between the violated checks (
L
c
)
1
2
3
4
5
6
7
8
count
(b) pulling one color away
flipped bonds
violated checks
Figure 3: (a) The 13-site cluster. The shell sites are colored by sublattice. The center (black) is
an anchor site. One tetrahedral cell (gold) lies inside the cluster and has one site of each color.
One octahedral cell (purple, dashed) reaches one site beyond the shell (square). (b) Pulling one
color away in the SSM code. The failed checks stay at two, while the flipped bonds grow with the
distance.
phases [
17
,
18
]. When the checks dominate, a lone color moves at the fixed cost 2
λ
: it is free. When
the bond term dominates, each flipped bond costs about
h
, and a lone color pulled a distance
r
from its partners costs about
√
2 h r/L
c
. That is a string tension,
σ ≈
√
2 h/L
c
, and the color is
confined. A singlet stays inside one tetrahedral cell and pays no such cost. For the toric code in two
dimensions, the transition lies near h/λ ≈ 0.33 [18]. We compute it for the FCC code next.
The confined phase of the FCC code. The octahedral checks and the bond terms all commute
with Eq.
(6)
. So the failed octahedral checks are conserved, and two of them form a pair of static
charges. In the sector with two failed checks, write each bond’s value as
x
e
=
J
e
s
u
s
w
, with
s
=
±
1
on the nodes and J
e
= −1 on a chain of bonds that links the two failed checks. Equation (6) then
becomes a transverse-field Ising model on the FCC nodes,
H
code
=
h
2
X
e
1 − J
e
s
u
s
w
+
λ
2
X
v
1 − σ
x
v
+ λ n
f
, (7)
where
n
f
is the number of failed octahedral checks, two for a pair and none for the vacuum. This
is the duality of Ref. [
17
] applied to this code. The node checks become the transverse field.
Appendix A derives Eq.
(7)
and checks it against the exact spectrum of the original Hamiltonian.
The sector with two failed checks splits into 2
k
blocks, one for each class of couplings
J
, where
k
is
the number of logical qubits of the code. Equation
(7)
is one block, restricted to the even sector of
the global spin flip. The model has no sign problem. We sampled it by path-integral quantum Monte
Carlo on periodic boxes of 256, 864, and 2048 nodes. The static potential
V
(
r
) =
E
pair
(
r
)
− E
vac
follows from switching the chain’s couplings from +1 to
−
1 and integrating the change in energy.
Figure 4 shows the result. Below
h/λ ≈
0
.
09,
V
stays within 0
.
02
λ
of 2
λ
out to
r
= 5
.
7
L
c
: the
colors are free. Above
h/λ ≈
0
.
11,
V
rises with
r
. Its slope is the effective string tension
σ
eff
.
The largest separation a box allows is half its length. Away from it,
σ
eff
is the same at different
separations and box sizes. At
h/λ
= 0
.
15 it is 0
.
125
±
0
.
001
λ/L
c
between 1
.
4 and 2
.
8
L
c
on 864
10
2 3 4 5
separation of the failed checks,
r
(
L
c
)
0.0
0.5
1.0
1.5
2.0
V
(
r
)
−
2
λ
(
λ
)
(a) potential: 864 nodes (solid), 2048 (open)
h
/
λ
= 0.09
h
/
λ
= 0.11
h
/
λ
= 0.15
h
/
λ
= 0.20
h
/
λ
= 0.30
0.00 0.05 0.10 0.15 0.20 0.25 0.30
h
/
λ
0.0
0.1
0.2
0.3
0.4
σ
eff
(
λ
/
L
c
)
(b) effective string tension
√
2
h
(bond term alone)
864 nodes,
r
=1.4
–
2.8
2048 nodes,
r
=2.8
–
4.2
256 nodes,
r
=1.4
–
2.8 (to half box)
864 nodes,
r
=2.8
–
4.2 (to half box)
2048 nodes,
r
=4.2
–
5.7 (to half box)
Figure 4: Static potential between two failed checks in the SSM code with bond verification, Eq.
(6)
,
from quantum Monte Carlo. (a)
V
(
r
)
−
2
λ
on 864 nodes (solid; mean and standard error of three
runs) and 2048 nodes (open). (b) The effective string tension against
h/λ
. Filled: steps away
from the largest separation the box allows. These agree between box sizes. Open: the step to that
separation, half the box, which is larger in every box. The dotted line is the bond-term value
√
2 h
.
The colors are free below h/λ ≈ 0.09 and confined above 0.11.
nodes and 0
.
124
λ/L
c
between 2
.
8 and 4
.
2
L
c
on 2048 nodes. At
h/λ
= 0
.
30 the two values are 0
.
379
and 0
.
378
λ/L
c
, close to the bond-term value
√
2 h
= 0
.
42
λ/L
c
. The step to the half-box separation
is larger in every box, by 4 to 30%, which we take to be a finite-size effect of the periodic box. The
transition lies at
h
c
/λ ≈
0
.
10. This is close to the mean-field value 1
/
12 for a lattice with twelve
neighbors. Repeated runs give statistical errors of 0
.
001 to 0
.
004
λ
in
V
. Halving the time step and
doubling the inverse temperature change V by at most 0.002 λ.
Two limits apply. First, each pair is held on one shortest chain, which fixes one block. Other
blocks give other states, and the pair’s ground state is the lowest of them. Three other chains
changed
V
by up to 0
.
016
λ
, so the computed
V
is an upper bound at about that level. Second,
the separations reach only 5
.
7
L
c
, and the steadiness of
σ
eff
rests on two box sizes away from the
half-box step. The tension is bounded above by
√
2 h
, the cost of the chain’s bonds with all other
bonds intact.
Two codes, two phases. The result suggests a picture of the SSM’s two codes. The confinement
code is in the confined phase. There, bond verification is at least about a tenth of the check cost,
h/λ ≳
0
.
1, and a lone color’s cost grows with distance. The fine code is in the free phase,
h/λ ≲
0
.
1.
There, its checks dominate, and its excitations move at fixed cost. We state this as a condition on
the two codes, not as a result.
Lemma (confinement by fraction). In the confined phase, a species is confined by the cluster
if and only if it is a fraction of a minimal verified pattern. A lone color is a third of a tetrahedral
cell, so quarks are confined. Gluons carry color without being singlets, so they are confined too.
Hadrons are complete tetrahedral patterns and move freely. Electric charge in the SSM comes from
the geometry of bonds, not from the code [7], so no check acts on it and no string forms.
11
Leptons. The leptons also come in threes, but the three has a different origin. In the SSM
the charged leptons are worldlines of the fine code at three depths of the
D
4
lattice’s triality
decomposition [
19
]. Triality relates one complete object to another. A lepton needs no partner, so
it is not a fraction and is not confined. We take this from the SSM’s description of the leptons. We
have not computed the fine-code pattern of a single lepton.
Gravity counts species differently. Every species fluctuates on the fine geometry, so every species
adds to the induced action of gravity [
6
]. A cavity counts only the fractions that its bonds tie
together.
8 The proton
The cavity. The proton’s verification region is the 13-site cluster of the confinement code. It is a
node and its twelve neighbors at distance
L
c
. Its outer surface is a cuboctahedron whose edge and
circumradius both equal
L
c
. A confined massless field has a lowest mode of energy
E
=
x ℏc/L
c
,
where
x
depends on the species, its wall, and the shape. We computed
x
numerically for the
cuboctahedron. For the quark we solved the Dirac equation with a large verification mass outside
the cluster, Eq.
(1)
of Section 3. We did the same for the sphere of radius
L
c
on the same grid,
and multiplied the exact sphere value, 2
.
043 [
13
], by the ratio. The quark ratio is 1
.
27, which gives
x
= 2
.
60. For the gluon we used the wall of the bag model, the dual of a conducting wall. The same
method gives
x
= 3
.
2 to 3
.
4. The cuboctahedron is smaller than the sphere through its corners, so
its modes are about 25% higher.
The size of the cluster. The measured proton radius,
r
p
= 0
.
841 fm [
20
], is a charge radius. It
is the root-mean-square radius of the charge. In the cavity the charge is carried by the quarks, so
r
p
is the rms radius of the quark density in its lowest mode. This density peaks inside the cluster and
falls toward the wall. For the sphere the rms radius is 0
.
729 times the radius, an exact result. For
the cuboctahedron we computed it on the same grid and corrected it by the sphere’s grid error of
2%. The result is ρ = 0.579 in units of L
c
. Setting ρL
c
= r
p
gives
L
c
=
r
p
ρ
= 1.45 fm. (8)
No mass enters this step. Table 6 gives the mode energies at this size.
Three quarks. The proton holds three quarks in their lowest mode. Their energies add to 3
E
.
Part of this is the motion of the cluster’s center of mass. Each quark has
⟨p
2
c
2
⟩
=
E
2
, so removing
the center-of-mass motion gives M =
p
(3E)
2
− 3E
2
=
√
6 E [21]. Then
M =
√
6 xρ
ℏc
r
p
= 3.68
ℏc
r
p
= 865 MeV. (9)
The proton’s mass is 938 MeV. The estimate is 8% low. It uses the measured radius and no fitted
length.
The shape nearly drops out. The product
xρ
fixes the mass at a given charge radius. Going
from the sphere to the cuboctahedron raises
x
by 27%. It also pulls the quark density inward,
and
ρ
falls by 21%. The two changes nearly cancel. For the sphere,
Mr
p
= 3
.
65
ℏc
. For the
cuboctahedron,
Mr
p
= 3
.
68
ℏc
. If the cavity were the tetrahedral void itself, the same matching
would give
Mr
p
≈
3
.
9
ℏc
, or about 910 MeV. But it would need a bond length of 4
.
1 fm, far from the
12
Table 6: Lowest cavity modes of the confined species in the 13-site cluster, a cuboctahedron of edge
and circumradius
L
c
= 1
.
45 fm. This size makes the rms radius of the quark density equal to the
proton charge radius.
species x E = x ℏc/L
c
(MeV)
quark 2.60 353
gluon 3.2–3.4 430–460
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
rms radius of the quark density (fm)
400
600
800
1000
1200
1400
three-quark mass
(
MeV
)
proton
r
p
cuboctahedron
sphere
Figure 5: Mass of three confined quarks, with the center-of-mass motion removed, against the rms
radius of their density. Solid: the 13-site cluster (cuboctahedron). Dashed: the sphere. The dotted
line is the proton’s mass, and the vertical line is the measured charge radius.
confinement scale. The 13-site region needs 1
.
45 fm. So the mass depends on the measured radius,
and on the cavity’s shape only to within 5% across the sphere, the cluster, and the tetrahedral void
(Fig. 5). The size of the cell does depend on the shape, and it singles out the 13-site region.
Two estimates of one cost. The SSM now has two estimates of the proton’s verification cost.
Ref. [
2
] counts the cluster’s qubits and checks and obtains 1836
m
e
= 938 MeV. This paper computes
the energy shift of Eq.
(2)
for the cluster and obtains 865 MeV from the charge radius. Both are
verification costs, but they are computed differently. One is a count, and the other is a ground-state
energy. This paper does not show that the count equals the energy shift. The two agree to 8%.
Read the other way, the counted mass and Eq.
(9)
predict a charge radius of 0
.
775 fm, against the
measured 0.841 fm.
What the estimate leaves out. Two things can move the 8%. First, the quarks share a color
field. Its energy depends on how the three modes overlap, and the cavity estimate does not include
it. Second, part of the measured charge radius may come from outside the three quark modes, so
r
p
equals the quark rms radius only at leading order. The wall’s finite strength is not among them,
since it moves the mass by less than 1% (Section 6). The zero-point energy of the confined fields is
small. For a conducting sphere it is +0.046 ℏc/R [22].
13
9 What remains open
The size of the cluster. Equation
(8)
gives
L
c
= 1
.
45 fm. This is larger than the charge radius,
because the charge sits well inside the wall. It is also larger than a simple stability estimate. If the
counted mass equals the energy that stretches one bond to breaking,
m
p
c
2
≈ σL
c
, then the string
tension of QCD,
σ ≈
0
.
98 GeV/fm, gives
L
c
≈
0
.
96 fm. The two conditions differ, and reconciling
them is open. In the confined phase, Section 7 gives the tension a source,
σ ≈
√
2 h/L
c
, so fixing
the bond term h would settle both.
The phases of the two codes. Section 7 locates the transition of the FCC code at
h
c
/λ ≈
0
.
10.
Whether the confinement code lies above it and the fine code below needs
h
and
λ
in terms of the
SSM’s counts. Three refinements of the calculation are also open. The first is a finite-size scaling of
h
c
. The second is larger boxes and separations, to confirm the tension beyond 5
.
7
L
c
. The third is
the lowest pair state over all blocks. The lemma also takes from the SSM that a lepton is a complete
pattern of the fine code. Computing that pattern would test it.
The count and the energy. Equation
(2)
defines a verification cost as a ground-state energy.
Equation
(3)
gives it as a count times the energy
ε
b
of one bit. The cavity form,
E
=
xℏν
v
, reads
the first as an instance of the second, but this paper does not derive one from the other. Expressing
ε
b
through λ and h would turn the 8% agreement of Section 8 into a test.
The form of the wall. Section 7 gives a cost that grows with distance, a linear wall. Sections 6
and 8 model it as a step of fixed strength at the cluster’s surface. Table 3 shows that the mass at a
fixed charge radius barely depends on the step’s strength, so a linear wall should change it little.
Solving the linear wall directly would check this. It would also test other hadron properties, which
may be more sensitive.
The gluon wall. We tested the verification mass for quarks. For gluons we used the gluon wall
of the bag model, the dual of a conducting wall. Deriving it from a check on the gluon field is open.
10 Conclusion
If a verification cost acts on a field as a local mass, then verification and confinement are two
descriptions of one thing. An excitation held in a region by strong checks has a verification cost
equal to the region’s lowest cavity mode. A mass is standing information. Its energy follows the rule
that the verification rate sets the Compton frequency, not Landauer’s principle. For quarks the cost
must be a mass, and its limit is the bag condition. These results combine into a Zeno law for walls:
the kind of check that works, the penetration length of a finite wall, the cost of confinement, and, in
this code, a sharp transition between competing checks. On the lattice, a color singlet needs a whole
tetrahedral cell, and a lone color’s cost grows with distance in the phase where bond verification
dominates. For the FCC code that phase begins near
h/λ ≈
0
.
10. Applied to the SSM’s proton,
with its size fixed by the charge radius, the duality gives 865 MeV. Counting the same cluster’s
checks gives 938 MeV.
14
A The duality of Eq. (7)
The mapping. Work in the eigenbasis of every
X
e
, with values
x
e
=
±
1. Three kinds of term
appear in Eq.
(6)
. The bond terms,
h
(1
−x
e
)
/
2, are diagonal. The octahedral checks,
X
o
=
Q
e∈o
X
e
,
are diagonal, with values
Q
e∈o
x
e
. They commute with every term, so their values label sectors. The
node checks,
Z
v
=
Q
e∋v
Z
e
, flip
x
e
on the twelve bonds of node
v
and act on nothing else. So the
Hamiltonian connects a configuration only to those reached from it by flipping node stars. Fix one
configuration
x
0
in a sector and set
J
e
=
x
0
e
. Flipping the stars of a set of nodes gives
x
e
=
J
e
s
u
s
w
,
with
s
v
=
−
1 on the flipped nodes and +1 elsewhere. The set of these configurations, the orbit of
x
0
, is closed under the Hamiltonian. On it, the bond terms become
h
2
(1
− J
e
s
u
s
w
), the node checks
become
σ
x
v
acting on
s
v
, and the octahedral checks are constants. This is Eq.
(7)
. Each bond lies in
the stars of both its ends, so flipping every star changes nothing:
Q
v
Z
v
= 1. Correspondingly,
s
and
−s
give the same
x
. So the orbit is the even sector,
Q
v
σ
x
v
= +1, of the spin model, with 2
N−1
states for N nodes.
The octahedral constraints. Each corner of an octahedral cell lies on four of its twelve edges,
so
Q
e∈o
s
u
s
w
= 1. The value of check
o
is then
Q
e∈o
J
e
, fixed by
J
alone. For the pair,
J
e
=
−
1 on
a chain of bonds that links the two failed cells through shared bonds. Each cell inside the chain
contains two of its bonds and passes. Each end cell contains one and fails. The program checks this
for every chain it uses.
Other sectors. A sector of the octahedral checks has 2
B−r
X
configurations, where
B
is the
number of bonds and
r
X
the rank of the octahedral checks. Each orbit has 2
r
Z
configurations, with
r
Z
=
N −
1 the rank of the node checks. So a sector splits into 2
B−r
X
−r
Z
= 2
k
orbits, where
k
is the number of logical qubits. For the periodic FCC code we computed
r
Z
=
r
X
=
N −
1 and
B = 6N, so k = 4N + 2. For N = 32 this gives the [[192, 130, 3]] code of Ref. [1]. The orbits differ
in the product of
J
around cycles that are not products of octahedral cycles, such as triangles. The
Hamiltonian never mixes them. For the vacuum, the lowest orbit is
J
e
= +1. In the eigenbasis of
the
σ
x
v
, the couplings act off the diagonal with magnitude
h/
2, and the Perron–Frobenius theorem
then gives
E
0
(
J
)
≥ E
0
(+1) for every
J
. For a pair, no such theorem fixes the lowest orbit. The
calculation uses the orbit of the shortest chain, so its V (r) is an upper bound.
A spectral check. We compared the two descriptions on a fragment of the code. It has the ten
nodes of two octahedral cells that share an edge, with their 23 edges as qubits. It has ten node
checks and two octahedral checks, and each sector has 2
21
states. At
h/λ
= 0
.
15 we found the
following. First, each sector splits into 4096 orbits of 512 states, as counted above. Second, for 30
random orbits, the full spectrum of the original Hamiltonian, built from the 23-bit configurations
and diagonalized completely, equals that of Eq.
(7)
in the even sector to 6
×
10
−14
, multiplicities
included. Third,
Tr H
and
Tr H
2
over each whole sector agree with the sums over all 4096 blocks
to 10
−12
relative. Fourth, the lowest ten levels of each sector, from a sparse eigensolver on the
original Hamiltonian, appear in the pooled spectra of the blocks to within 10
−6
. In the sector with
two failed checks, the lowest block has a single reversed coupling on the shared bond. This is the
shortest chain.
15
Data availability
The scripts that produce every number and figure in this paper are archived as ssmthe-
ory zeno scripts.zip in the repository github.com/raghu91302/ssmtheory.
Declarations
Funding. This work received no external funding.
Competing interests. The author declares no competing interests.
References
[1]
R. Kulkarni, A 67%-rate CSS code on the FCC lattice: [[192
,
130
,
3]] from weight-12 stabilizers
(2026). arXiv:2603.20294
[2]
R. Kulkarni, The mass-energy-information equivalence: a bottom-up identification of
the particle spectrum via FCC lattice error correction, Phys. Open 27, 100414 (2026).
doi:10.1016/j.physo.2026.100414
[3]
B. Misra and E. C. G. Sudarshan, The Zeno’s paradox in quantum theory, J. Math. Phys. 18,
756 (1977). doi:10.1063/1.523304
[4]
P. Facchi and S. Pascazio, Quantum Zeno subspaces, Phys. Rev. Lett. 89, 080401 (2002).
arXiv:quant-ph/0201115
[5]
R. Kulkarni, Dark matter as incomplete crystallization: a geometric construction on the octahe-
dral void of the FCC vacuum lattice, Quantum Rep. 8, 89 (2026). doi:10.3390/quantum8030089
[6]
R. Kulkarni, Vacuum fluctuations as the stiffness of spacetime: induced Regge gravity on the
Selection–Stitch lattice, Quantum Rep. (2026), accepted.
[7]
R. Kulkarni, Matter as incomplete crystallization: quark charges, color confinement, and the
proton mass from a single extra node in the vacuum lattice, Phys. Open 27, 100423 (2026).
doi:10.1016/j.physo.2026.100423
[8]
P. Facchi, S. Pascazio, A. Scardicchio, and L. S. Schulman, Zeno dynamics yields ordinary
constraints, Phys. Rev. A 65, 012108 (2001). arXiv:quant-ph/0101037
[9] P. Exner and T. Ichinose, A product formula related to quantum Zeno dynamics, Ann. Henri
Poincar´e 6, 195 (2005). arXiv:math-ph/0302060
[10]
R. Landauer, Irreversibility and heat generation in the computing process, IBM J. Res. Dev. 5,
183 (1961). doi:10.1147/rd.53.0183
[11]
S.-Y. Lan, P.-C. Kuan, B. Estey, D. English, J. M. Brown, M. A. Hohensee, and H.
M¨uller, A clock directly linking time to a particle’s mass, Science 339, 554 (2013).
doi:10.1126/science.1230767
[12]
N. Margolus and L. B. Levitin, The maximum speed of dynamical evolution, Physica D 120,
188 (1998). doi:10.1016/S0167-2789(98)00054-2
16
[13]
A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F. Weisskopf, New extended model of
hadrons, Phys. Rev. D 9, 3471 (1974). doi:10.1103/PhysRevD.9.3471
[14]
J. Duran and A. Mas, Convergence of generalized MIT bag models to Dirac operators with
zigzag boundary conditions, Anal. Math. Phys. (2024). doi:10.1007/s13324-024-00946-7
[15]
O. Klein, Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen
Dynamik von Dirac, Z. Phys. 53, 157 (1929). doi:10.1007/BF01339716
[16]
N. Arrizabalaga, L. Le Treust, and N. Raymond, On the MIT bag model in the non-relativistic
limit, Commun. Math. Phys. 354, 641 (2017). arXiv:1607.02603
[17]
F. J. Wegner, Duality in generalized Ising models and phase transitions without local order
parameters, J. Math. Phys. 12, 2259 (1971). doi:10.1063/1.1665530
[18]
S. Trebst, P. Werner, M. Troyer, K. Shtengel, and C. Nayak, Breakdown of a topological phase:
quantum phase transition in a loop gas model with tension, Phys. Rev. Lett. 98, 070602 (2007).
doi:10.1103/PhysRevLett.98.070602
[19]
R. Kulkarni, The mass–energy–information equivalence extended:
D
4
lattice, tri-
ality, and the three-generation structure of matter, Phys. Open (2026) 100499.
doi:10.1016/j.physo.2026.100499
[20]
P. J. Mohr, E. Tiesinga, D. B. Newell, and B. N. Taylor, CODATA recommended val-
ues of the fundamental physical constants: 2022, Rev. Mod. Phys. 97, 025002 (2025).
doi:10.1103/RevModPhys.97.025002
[21]
J. F. Donoghue and K. Johnson, Pion and an improved static bag model, Phys. Rev. D 21,
1975 (1980). doi:10.1103/PhysRevD.21.1975
[22]
T. H. Boyer, Quantum electromagnetic zero-point energy of a conducting spherical
shell and the Casimir model for a charged particle, Phys. Rev. 174, 1764 (1968).
doi:10.1103/PhysRev.174.1764
17
