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 error-correcting code, and the
mass of a particle is the cost of verifying it. This paper asks how such a cost becomes an energy.
We assume that a field excitation that fails the code’s checks gains a local mass equal to the
cost of those checks. The quantum Zeno effect then links verification to confinement. If strong
checks fail everywhere outside a region, they hold the excitation inside it, and its cost equals
the region’s lowest mode energy,
xℏc/L
. We test this link in three ways. First, for a massless
quark the penalty must be a mass. Only a mass gives the confining wall of the bag model, while
a potential or a projector penalty does not. Second, on the code’s face-centered-cubic lattice, a
lone color fails two checks, while a full set of three colors fails none. Adding a cost
h
for flipping
bonds to the check cost
λ
maps the code onto a transverse-field Ising model. Quantum Monte
Carlo then finds free colors below
h/λ ≈
0
.
09 and confined colors above 0
.
11. Third, letting
failed checks move turns the code into a field. In one dimension this field is exactly a Majorana
fermion, whose mass is the check cost measured from a critical value. In three dimensions it is a
scalar field that becomes massless at the confinement transition. A coupling to the quark field
is not derived. Finally, we apply the link to the proton. Its charge radius fixes the size of its
13-site verification region. Three quarks in their lowest state there have a mass of 1059 MeV,
2.4% below the mean of the nucleon and ∆ masses.
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
]. In Ref. [
2
] this cost is 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.
The link rests on one assumption. A field excitation that fails the code’s checks gains a local
mass equal to the cost of those checks. This applies the SSM’s rule that mass is verification cost,
point by point. We then define the verification cost of a state as an energy. It is the rise in the lowest
energy of the excitation 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. The two
sides relate as a record and a wave do in particle–wave duality (Section 3).
Six results follow.
1.
Which penalty holds a quark. Checks of position do not hold a massless quark, and neither
does an electric potential. A projector penalty holds the quark but not the antiquark, with a
different wall. A verification mass outside the region holds both and 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. Above 0
.
11 the potential grows linearly, and the string tension holds
as the box grows. Under the working rule of Section 7, fractions are confined in this phase
and complete patterns, such as baryons, are not. A dark-matter defect in an octahedral void
fails twelve checks around it and shows no string cost. The calculation gives it no baryon-like
cavity energy (Section 7).
5.
From the code to the field. A failed check is its own antiparticle, so its cost cannot enter a
field as a potential, and a mass treats both alike. With a term that moves failed checks, the
one-dimensional code is exactly a Majorana field. Its mass is
λ − λ
c
, the check cost measured
from a critical value, and its cavity levels follow the bag and the Zeno law. In three dimensions,
quantum Monte Carlo of the FCC code gives a scalar field with
m
2
∝ λ − λ
c
. It becomes
massless at the same critical ratio, 0
.
090, at which color confinement sets in. The fermion
coupling in three dimensions is not derived (Section 8).
6.
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 1059 MeV for three quarks in their lowest
state, with the wall fixed by the lattice. This is 2
.
4% below the mean of the nucleon and ∆
masses, and its shape factor,
xρ ≈
1
.
50, barely depends on the cavity. Motion of the cluster
can only lower it. The cavity has no spin interaction, which is why the comparison is with
that mean. The SSM’s count of the same region’s checks gives 938 MeV for the nucleon, a
second estimate of the same verification cost (Section 9).
2
(a)
Z
-check on a node (b)
X
-check on an octahedral void
(c) the 13-site cluster
Figure 1: The SSM code [
1
,
2
]. (a) A
Z
-check on the twelve bonds of one node. (b) An
X
-check on
the twelve edges of one octahedral void. (c) The 13-site cluster, a node and its twelve neighbors. In
the confinement code this cluster is the proton’s verification region.
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
(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. In the SSM 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
of order 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,
3
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)
For a Dirac field the full ground-state energy includes the filled negative-energy states, so we work
in the one-particle sector. There
E
0
(
H
) is the lowest positive level of the one-particle Hamiltonian,
and
E
0
(
H
field
) is the same without the verification mass, zero for a massless field in open space.
A many-body version would need the vacuum energy subtracted with and without the term. 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.
Why a mass. A failed check is a
Z
2
charge. It is its own antiparticle, so a particle and its
antiparticle at the same point fail the same checks and pay the same cost. A potential couples to a
charge density, which changes sign between a particle and its antiparticle. A mass does not. A cost
that is the same for both therefore cannot enter as a potential. This does not single out the scalar
mass. A pseudoscalar mass also treats both alike and gives the same wall (Table 4). Equation
(1)
takes the scalar mass. Section 8 derives the mass form from the code exactly in one dimension, for
a Majorana field. In three dimensions it gives a scalar analogue, not the fermion coupling of Eq.
(1)
.
It also shows that the mass vanishes at a balance point of the check cost.
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 term separates the inside of the region
from 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. This gives the duality:
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.
It is the statement of Section 1, and it follows from Eqs.
(1)
and
(2)
. Section 5 shows that it holds
for quarks, and that the term must be a mass to give the bag wall.
Records and waves. The two sides of the duality are a record and a wave. The code keeps a
discrete record of where an excitation is. Between checks, the field spreads as a wave. The Zeno
effect is the contest between the two. Just after a check, the chance that the state has changed
4
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
, in units of
2
mL
2
/ℏ
, 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).
grows as the square of the elapsed time. Frequent checks therefore reset the state before it can
spread. This is the complementarity behind particle–wave duality. In a two-slit experiment, a record
of the path destroys the interference. Here, a record of the region keeps the wave from leaving it.
The uncertainty principle sets the price. A wave held in a region of size
L
carries momentum of
order
ℏ/L
, so a massless field pays an energy of order
ℏc/L
. This is the form
x ℏc/L
of the duality.
In this reading, error correction plays the part of the particle, and the field plays the part of the
wave. Two differences matter. In particle–wave duality one object shows both behaviors. Here the
record and the field are different objects, and the duality says that they carry one energy. The kind
of record also matters. Section 5 shows that a record of position does not hold a massless quark,
while a cost of mass type does.
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
opposite case. It persists, it exists at zero temperature, and it does not depend on
T
. Landauer’s
principle therefore cannot turn a verification count into a mass.
5
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
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. Ref. [
5
] gives two reasons why Landauer’s bound does not apply here. Nothing is erased,
because the defect is never repaired and its syndrome is constant. And there is no environment to
take up heat. Landauer’s principle supplies only the precedent that information and energy convert.
Ref. [
5
] states the rule as a postulate, not as a theorem. The rule does not depend on temperature,
and the energy stays as rest energy. Ref. [
6
] adds the dynamics. Suppose 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
, so 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 9.
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 Which penalty 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
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
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. In one dimension the argument is exact. The
quark’s two chiral parts move at
±c
, and a check of position does not mix them. After a time
L/c
the quark has left, however often it is checked.
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. Of the three, 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
]). A penalty on a quark’s presence is not
enough. To give the bag wall, 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
. The
wall is not a check on the quark inside. It is the verified vacuum outside, which does not carry a
lone color. Section 8 shows that the mass of the field vanishes at a balance point of the check cost,
not at zero cost, and the open questions of Section 10 return to this.
6 The strength of the wall and a Zeno law
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.
We now estimate
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
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
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. The wall is not hard. It lies near the softest case tested below.
In three dimensions we repeated the cavity calculation of Section 9 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 9 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. 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 they do not, as Section 5 shows exactly in one dimension. 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). The
second is half the reduced Compton wavelength,
ℏ/M c
, of a particle whose rest energy
Mc
2
equals the wall’s strength
K
. Both are of 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
. The wall for a massless
particle thus 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).
8
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. For
a
= 1
the penalty is 2λ times the projector of Table 2.
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
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
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
.
09 (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. It uses 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.
7 Which species are confined
Checks hold only what they 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. 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
]
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.
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. The two descriptions agree.
A color singlet, one site of each color, therefore 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 baryon. 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). 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
. The two terms compete. Systems of this kind have two
phases [
17
,
18
]. When the checks dominate, a lone color moves at the fixed cost 2
λ
and is free.
10
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)
. The failed octahedral checks are therefore 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
, and
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
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
.
09. The mass gap of the same Ising model places it at 0
.
090
±
0
.
002
(Section 8). 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
.
09, and a lone color’s cost grows with distance. The fine code is in the free phase,
h/λ ≲
0
.
09. 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.
Confinement by fraction. In the confined phase we use a working rule. A species is confined
by the cluster if it is a fraction of a minimal verified pattern, and free if it is a complete one. The
11
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.
sectors computed here support the rule for a lone color and a singlet. Its use for the other species
below extends it and does not classify them. A lone color is a third of a tetrahedral cell, so quarks
are confined. Gluons carry color without being singlets, so under the rule they are confined too.
Baryons are complete tetrahedral patterns and pay no string cost. 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.
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.
Dark matter. In the SSM, dark matter is a node trapped in an octahedral void [
5
]. It bonds to
the void’s six corners. The corners are joined by the twelve edges of the octahedral cell, the graph
K
2,2,2
. It has no color, because the mechanism that gives a baryon its colors has no counterpart in
this cell [
5
]. We applied
Z
to these twelve edges in the periodic FCC code. Each edge also belongs
to one neighboring octahedral cell. The pattern therefore fails exactly twelve checks, those of the
twelve octahedral cells that share an edge with it. Their centers lie at distance
L
c
and form a
cuboctahedron. No part of the pattern is free of failed checks, so it contains no baryon-like piece.
The same pattern placed elsewhere costs the same twelve failed checks, with no string between the
two positions. This shows only that the pattern itself carries no linear cost. It does not show that
the defect moves. In Eq.
(6)
the failed octahedral checks are conserved, so the twelve failures stay
where they are. A mobile defect would need an added term that translates the whole syndrome at
bounded cost, and we have not studied one. We checked separation with explicit operators. For a
given set of failed checks, the fewest flipped bonds that produce it follow from a minimum matching
of the failures by distance. The twelve failures of the intact pattern need six flipped bonds. Moving
12
one of them a distance
r
along a cube axis needs 5 +
√
2 r/L
c
bonds, out to
r
= 7
.
1
L
c
in a box of
6912 nodes. Moving two neighboring failures together needs six bonds at every distance. In the
confined phase each flipped bond costs about
h
. The first cost therefore grows linearly, and the
second stays bounded. This is the same bond-term estimate used for a lone color. It is not a Monte
Carlo result. Applied as
X
instead, the same twelve edges form an octahedral check, which costs
nothing. Only the
Z
pattern carries a cost, and that cost is the standing syndrome of its twelve
checks. The checks hold this syndrome fixed, as the Zeno effect freezes a measured state. Here the
Zeno effect holds a pattern in place instead of holding a field in a region. We find no fraction of the
pattern held in a region, so the present calculation gives no baryon-like cavity contribution to the
mass. We have not computed the spectrum of fluctuations around the defect. It may have localized
or internal modes of other kinds. As a control, the lowest bag mode of an octahedral cell of edge
L
c
costs 606 MeV. Three such modes, as for a baryon, would give 1
.
48 GeV, 14% below the counted
1
.
719 GeV [
5
]. We find no confined species in the cell, so we do not use this energy for the mass.
The counted value rests on the standing-information postulate of Ref. [
5
] (Section 4). The twelve
failed checks found here do not test that value. The same split appears in standard physics. Lattice
QCD computes a hadron’s mass from its confined fields. In most models the mass of a colorless
dark-matter candidate is a free parameter. In the SSM the count still fixes it.
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 From the code to the field
Equation
(1)
is this paper’s one assumption. This section asks what the code itself gives. In one
dimension it gives the mass form exactly, for a Majorana field. In three dimensions it gives a scalar
analogue of the mechanism, not the fermion coupling. The code of Eq.
(6)
keeps failed octahedral
checks in place. A field needs them to move. We add the simplest term that moves them,
−t
P
b
Z
b
.
Each
Z
b
flips the two octahedral checks that contain bond
b
. It moves a failed check to a neighboring
cell, or it creates or removes two of them. We set h = 0 in this section.
One dimension, exactly. The repetition code is the one-dimensional case. Its checks are
Z
i
Z
i+1
,
each costing
λ
i
to fail, and the moving term is
−t
P
i
X
i
. The Jordan–Wigner transformation turns
this chain into a free Majorana field [
20
]. Its mass is
m
(
x
) =
λ
(
x
)
− λ
c
with
λ
c
= 2
t
, and its speed
is
v
= 2
t
in lattice units. We computed the levels of a region at the balance point,
λ
=
λ
c
, inside
walls with
λ
=
λ
c
+
K
. The levels stand in the ratio 1 : 3 : 5. The lowest tends to
πℏv/
2
L
, the
bag value of Table 1. For finite
K
it follows the Zeno law, Eq.
(5)
, to within 1% once the region
has 100 sites (Fig. 5a). Walls of the other sign, with
λ < λ
c
outside, confine in the same way. The
mapping is exact, so in one dimension Eq.
(1)
is a result, with one correction. The mass is zero at
the balance point, not at zero cost. A region whose checks are switched off has mass
−λ
c
, and a
Majorana zero mode then sits at each of its walls [
21
]. Wilson fermions in lattice QCD show the
same shift of the zero of mass [22].
The FCC code. Each bond lies in exactly two octahedral cells, and the octahedral cells form
an FCC lattice with twelve neighbors each. The product of
Z
b
around any closed path of bonds
through these cells commutes with the Hamiltonian. In the sector where every such product is +1,
13
25 50 100 200 400
region size
L
(sites)
1.44
1.46
1.48
1.50
1.52
1.54
1.56
E
1
L
/
ħ
v
π
/2
(a) one dimension, exact
K
= 1
t
K
= 4
t
K
= 16
t
0.00 0.01 0.02 0.03 0.04 0.05
t
/
λ
0.0
0.2
0.4
0.6
0.8
1.0
mass of a failed check (
λ
)
confinement
transition
(
Fig. 4
)
,
h
/2
λ
(b) FCC code, quantum Monte Carlo
linear theory
m
2
∝
t
c
−
t
(2048 cells)
256 cells
864 cells
2048 cells
Figure 5: From the code to the field. (a) One dimension, exact. The lowest level of a region at the
balance point inside walls of strength
K
, against the size of the region, with the Zeno law of Eq.
(5)
(lines) and the bag value
π/
2. (b) The FCC code, Eq.
(8)
, by quantum Monte Carlo. The mass of a
failed check against
t/λ
on three box sizes, with the linear theory (dashed) and a fit with
m
2
linear
in t (dotted). The shaded band is the confinement transition of Fig. 4, mapped to this model.
we write Z
b
= s
o
s
o
′
with Ising spins on the cells. Each octahedral check then acts as σ
x
o
, and
H = −t
X
⟨oo
′
⟩
s
o
s
o
′
+
λ
2
X
o
1 − σ
x
o
. (8)
A failed check is a flipped
σ
x
o
. At
t
= 0 it costs
λ
. The sector contains a code state, and by the
argument of Appendix A it is the lowest block. Equation
(8)
has no sign problem. We sampled it
by path-integral quantum Monte Carlo with a cluster algorithm on 256, 864, and 2048 cells. The
mass of a failed check follows from the decay of its correlation in imaginary time.
The mass in three dimensions. Figure 5b shows the result. The mass falls from
λ
at
t
= 0 and
vanishes near
t/λ
= 0
.
045. On 2048 cells it is 0
.
566, 0
.
458, 0
.
315, 0
.
240, and 0
.
152
λ
at
t/λ
= 0
.
030,
0
.
035, 0
.
040, 0
.
042, and 0
.
044. Two runs at each point agree to between 0
.
001 and 0
.
006
λ
. The
square of the mass is linear in
t
over this range and vanishes at
t
c
/λ
= 0
.
0448. A straight line in
the mass itself fits five times worse. In three dimensions the failed checks are bosons, so the field
is a scalar. Its transition lies in four Euclidean dimensions, the upper critical dimension of the
Ising model. There
m
2
is linear in the distance from the transition, up to logarithms [
23
]. The
verification cost therefore enters this scalar field as a mass, with
m
2
∝ λ − λ
c
and
λ
c
≈
22
t
. This
is a microscopic scalar analogue of the verification mass. It is not the fermion bilinear of Eq.
(1)
.
The linear theory of the same model gives
m
2
=
λ
(
λ −
24
t
). It agrees with the Monte Carlo far
from the transition and places
λ
c
at 24
t
. In the linear theory we also computed the lowest mode of
a massless region inside strong walls. It extrapolates to 3
.
16 for a sphere and 3
.
95 for the shape
of the 13-site cluster. The hard-wall values of a scalar field are
π
and 3
.
926. The ratio of the two
computed values, 1.251, matches the hard-wall ratio, 1.250.
The massless point is the confinement transition. Equation
(8)
has the form of the vacuum
block of Eq.
(7)
, with
h/
2 replaced by
t
and the nodes replaced by the octahedral cells. Both kinds
14
of check in the FCC code sit on FCC lattices, so the two problems are the same Ising model. The
field becomes massless at 2
t
c
/λ
= 0
.
090
±
0
.
002. With bond verification in place of the moving term,
color confinement therefore sets in at
h
c
/λ
= 0
.
090
±
0
.
002. This sharpens the estimate of Section 7.
One reading follows. The interior of the cluster sits at the transition, where the field is massless.
The vacuum outside sits in the phase where checks dominate. We have not studied the code with
both terms at once.
What the code does not give. The field derived here is a Majorana field in one dimension and
a scalar in three. The one-dimensional result is a fermion mass term. The three-dimensional result
is a gap of a bosonic field, which does not by itself give a mass to a fermion. One bridge would be a
Yukawa coupling
gϕ
¯
ψψ
between the scalar and a quark field, which would give a mass
g⟨ϕ⟩
where
the scalar has an expectation value. Neither
g
nor the quark field follows from the code. The quark
is a Dirac field. Failed checks in three dimensions are bosons, so the spin of the quark does not
follow from this code. Lattice fermions can be written exactly as a
Z
2
gauge theory with a modified
Gauss law [24]. Whether the checks of the SSM code can take that form is open.
9 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 [
25
], 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 higher. The quark mode is 27% higher, and the gluon mode 16 to 24% higher.
The size of the cluster. The measured proton radius,
r
p
= 0
.
841 fm [
26
], 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
. The trapped node sits at the center of its void, held
there by equal bonds to the corners. If the lattice pins the cluster’s center, the radius about that
center is the measured one. Setting ρL
c
= r
p
then gives
L
c
=
r
p
ρ
= 1.45 fm. (9)
No mass enters this step. Table 6 gives the mode energies at this size.
Three quarks in their lowest state. The proton holds three quarks in their lowest mode. Their
energies add to 3
E
. The proton’s mass is the energy of the lowest state of the whole system, and
that includes the motion of the cluster’s center. The trapped node is held at the center of its void
by equal bonds, so the lattice fixes the wall. For three quarks in a fixed wall, the lowest state is
the product of their lowest modes. Its center is not frozen. It moves within the wall, and that
15
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, about the
cavity’s center, 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
zero-point motion is part of 3
E
. The quark density in this state spreads about the void’s center, so
the measured radius is the radius about that center, as in Eq.
(9)
. The energy and the radius then
refer to the same center, and
M = 3xρ
ℏc
r
p
= 4.51
ℏc
r
p
= 1059 MeV. (10)
The cavity has no spin interaction, so it does not separate the nucleon from the ∆. It should be
compared with their mean, 1085 MeV. Equation
(10)
is 2
.
4% below it. It uses the measured radius
and no fitted length.
Motion of the cluster. The proton moves through the lattice, so the fixed wall is a limit. If the
cluster can pass from one void to another, its lowest state spreads over the voids, and its energy
falls below the energy in one void. Equation
(10)
is therefore an upper bound within this model.
The opposite limit is a wall that moves freely with the quarks. There the lowest state has zero total
momentum, and the motion of the center must be removed from both the energy and the radius.
Appendix B does this by projecting the three-quark state onto zero total momentum, as Peierls and
Yoccoz did for nuclei [
27
]. Our wall is a fixed field, so the projected energy depends on how the
wall’s energy is assigned. The two rules of Appendix B give 765 and 569 MeV for the cluster. In
both, the spread about the quarks’ own center is 0
.
72 of the spread about the cavity’s center. The
simpler corrections,
√
6 E
for the energy [
28
] and
p
2/3
for the radius, give 706 MeV. The real case
lies between the fixed and the free wall. Where it lies depends on how the cluster moves. In the
SSM the node cannot leave its void through a face [
7
], so that motion may require the lattice to
restitch around the defect.
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, xρ = 1.49. For the cuboctahedron,
xρ
= 1
.
50. The same product enters every limit above. It is the result that does not depend on
how the motion of the center is treated. If the cavity were the tetrahedral void itself, the same
matching would give
xρ ≈
1
.
57, 4% higher. But it would need a bond length of 4
.
1 fm, far from the
confinement scale. The 13-site region needs 1
.
45 fm. 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. 6). 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 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 for the nucleon. This paper
computes the energy shift of Eq.
(2)
for the cluster and obtains 1059 MeV for the spin-averaged
mass in a fixed wall. 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
16
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
1600
1800
three-quark mass
(
MeV
)
proton
(N+
Δ
)/2
r
p
cuboctahedron, fixed wall
cuboctahedron, free wall
sphere (dashed)
Figure 6: Mass of three confined quarks against the rms radius of their density. Upper curves: the
lowest state in a wall fixed by the lattice, Eq.
(10)
. Lower curves: a wall that moves freely with
the quarks, with the simple corrections of Section 9. Solid: the 13-site cluster (cuboctahedron).
Dashed: the sphere, which nearly coincides with the cluster. The dotted line is the proton’s mass,
the dash-dotted line the mean of the nucleon and ∆ masses, and the vertical line the measured
charge radius.
shift. Relating the two also needs the spin interaction, which the code does not give. Read the other
way, Eq.
(10)
with the spin-averaged mass gives a charge radius of 0
.
820 fm, against the measured
0.841 fm.
What the estimate leaves out. Several terms move the estimate. We compare with the usual
corrections of the bag model, which this paper does not derive from the code. The color-magnetic
part of the shared color field splits the nucleon from the ∆. With the measured splitting, Eq. (10)
would place the nucleon near 912 MeV and the ∆ near 1206 MeV, each within 3%. This uses
the measured splitting, so it is a consistency check, not a prediction. Motion of the cluster lowers
the mass. A volume energy would raise it. In the bag model that term carries a quarter of the
mass [
13
,
25
], and the SSM has no counterpart yet. Each of these terms can reach 10% or more, and
they partly cancel, so the 2
.
4% agreement may be partly fortunate. 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 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 [29].
Two further checks. The same mode gives two more properties of the proton, with no new
parameter. We computed them as we computed
x
, from the exact sphere values of the bag model [
25
]
times the ratio of cluster to sphere on one grid. The axial charge depends only on the shape of the
mode. For three quarks in one mode it is
g
A
=
5
3
⟨
Σ
z
⟩
. This gives 1
.
11 for the cluster and 1
.
09 for
the sphere, against the measured 1
.
275 [
30
]. The magnetic moment also depends on the size. With
the size fixed by the charge radius and the center fixed by the lattice, it is 2
.
17
µ
N
for the cluster,
against the measured 2
.
793
µ
N
[
26
]. The two are 13% and 22% low, as in the bag model. In bag
models a pion cloud raises both, and the code does not give one. These checks show a field that
behaves like a bag-model proton in a region of the right size. They do not test the region more
17
sharply than the mass does.
10 What remains open
The size of the cluster. Equation
(9)
gives
L
c
= 1
.
45 fm. With a freely moving wall and the
simple corrections of Section 9, the cluster would be larger,
L
c
= 1
.
78 fm. Both are larger than
the charge radius, because the charge sits well inside the wall. Both are 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
.
09.
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 rule of Section 7 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 an energy shift. Equa-
tion
(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 agreement of Section 9 into a test.
The form of the wall. Section 7 gives a cost that grows with distance, a linear wall. Sections 6
and 9 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 motion of the cluster. Equation
(10)
holds for a wall fixed by the lattice. The energy
the cluster gains by moving between voids, or by the lattice restitching around it, lowers the mass.
Computing it needs the dynamics of the verified cluster, which this paper does not treat.
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.
The zero of the verification mass. Section 8 puts zero mass at a balance point
λ
c
of the
check cost, and in three dimensions finds
m
2
∝ λ − λ
c
for a scalar analogue. The coupling of the
code to the quark field in three dimensions, for example through a Yukawa term, is not derived.
Equation
(1)
takes the interior of the cluster to be massless, with
m
v
∝ λ
. Whether the interior sits
at the balance point is open. The code with both the moving term and bond verification is also
open. The spin of the quark is not derived from the code.
18
The dark-matter defect. Two questions about the octahedral defect of Section 7 are open.
Whether it moves needs a term that translates its whole syndrome at bounded cost. Whether it
has excited states needs the spectrum of fluctuations around it. The present calculation addresses
neither.
11 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 a
pure mass gives the bag condition, and any penalty must be mostly mass to hold both quarks and
antiquarks. These results combine into a Zeno law for walls. It covers 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
.
09. A dark-matter defect in an octahedral void carries twelve failed
checks and shows no string cost. The present calculation gives it no baryon-like cavity energy, so its
mass in the SSM rests on the count and on the standing-information postulate. A failed check is its
own antiparticle, so its cost cannot enter a field as a potential, and a mass treats both alike. In
one dimension the code gives this mass term exactly, for a Majorana field. In three dimensions it
gives a scalar analogue, whose mass vanishes at a balance point that coincides with the confinement
transition. The fermion coupling of Eq.
(1)
in three dimensions remains an assumption. Applied to
the SSM’s proton, with its size fixed by the charge radius, the duality gives 1059 MeV for the lowest
state with the wall fixed by the lattice, 2
.
4% below the mean of the nucleon and ∆ masses. Motion
of the cluster can only lower it. Counting the same cluster’s checks gives 938 MeV for the nucleon.
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. 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
. 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.
19
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. A sector therefore 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. That block is the
shortest chain.
B The projection onto zero total momentum
The state. Let
u
s
(x),
s
= 1
,
2, be the two degenerate modes of lowest positive energy
e
of the
one-quark Hamiltonian
h = α · p + β
h
M(x) +
r
2
(−∇
2
)
i
, (11)
on the periodic grid of Section 9, with
M
= 0 inside the cavity,
M
=
K
outside, and a Wilson
term with
r
= 1 that removes the lattice doublers. Write
h
=
T
+
βM
, where
T
commutes with
translations. Each quark occupies one of the two modes, and the three-quark state is the product
Φ(x
1
,
x
2
,
x
3
) =
Q
i
u
s
i
(x
i
). Its translate by a is Φ
a
(x
1
,
x
2
,
x
3
) = Φ(x
1
−
a
,
x
2
−
a
,
x
3
−
a). The state
with zero total momentum is [27]
Ψ =
Z
d
3
a Φ
a
. (12)
Overlaps and normalization. Every matrix element between Φ and Φ
a
factorizes into one-quark
overlaps. We average over the two modes,
n(a) =
1
2
X
s
⟨u
s
|u
s,a
⟩, (13)
and the same for the other one-quark matrix elements below. Then
⟨
Φ
|
Φ
a
⟩
=
n
(a)
3
, and
⟨
Ψ
|
Ψ
⟩
=
V
R
d
3
a n
(a)
3
. The volume factor
V
comes from the second integral in Eq.
(12)
and cancels in every
ratio below. The average drops the cross-overlaps
⟨u
1
|u
2,a
⟩
, which are below 10
−7
of the diagonal
ones. On the grid the integrals are sums over the periodic shifts, which we compute by fast Fourier
transforms. At the largest shift the box allows,
|n| ≤
0
.
014, so
n
3
≤
3
×
10
−6
, and the periodic
images do not affect the sums.
20
Energy and the two rules for the wall. For a sum of one-quark terms,
⟨
Φ
|H|
Φ
a
⟩
= 3
h
(a)
n
(a)
2
,
and
E
PY
=
3
R
d
3
a h(a) n(a)
2
R
d
3
a n(a)
3
. (14)
The wall
M
(x) is a fixed field. The Hamiltonian therefore does not commute with translations, and
h(a) needs a rule. We use two.
1.
The wall moves with the quarks. The translated state sits in the translated wall
M
(x
−
a).
The one-quark term between Φ and Φ
a
uses the mean of the two walls,
h(a) = ⟨u|T |u
a
⟩ +
1
2
⟨u|β
M(x) + M(x − a)
|u
a
⟩. (15)
Since (
T
+
βM
)
u
=
e u
and the translated mode is an eigenstate of the translated wall with
the same
e
, this equals
e n
(a) exactly. Equation
(14)
then gives
E
PY
= 3
e
, the unprojected
energy.
2.
Only the kinetic energy is projected. We set
h
(a) =
⟨u|T |u
a
⟩
in Eq.
(14)
and add the wall
energy 3⟨u|βM|u⟩ unprojected. For the walls used here this wall energy is 0.3 to 2% of 3e.
Neither rule comes from a Hamiltonian that commutes with translations. Their difference, about
26% in the energy, measures how much the answer depends on the dynamics of the wall.
Radius. The charge radius is measured from the proton’s own center, R = (x
1
+ x
2
+ x
3
)
/
3. The
quark charges sum to one and the three quarks share one mode, so
r
2
int
=
⟨
Ψ
|
(x
1
−
R)
2
|
Ψ
⟩/⟨
Ψ
|
Ψ
⟩
.
With x
1
−
R = (2x
1
−
x
2
−
x
3
)
/
3, and with
q
(a) =
⟨u|
x
2
|u
a
⟩
and d(a) =
⟨u|
x
|u
a
⟩
averaged as in
Eq. (13),
r
2
int
=
R
d
3
a
2
3
q(a) n(a)
2
− d(a) · d(a) n(a)
R
d
3
a n(a)
3
. (16)
Without projection only a = 0 contributes, d(0) = 0, and
r
2
int
=
2
3
⟨
x
2
⟩
, the factor
p
2/3
of Section 9.
For a Gaussian mode, Eq.
(16)
returns this factor exactly, because the product state then separates
into the motion of the center and the internal motion. The cavity modes have a sharp edge, and the
projection gives r
int
= 0.72 of the radius about the cavity’s center.
Mass at the measured radius.
E
PY
scales as 1
/L
and
r
int
as
L
, so the product
E
PY
r
int
is
independent of the cavity’s size. Setting
r
int
=
r
p
gives
M
= (
E
PY
r
int
)
ℏc/r
p
. As for
xρ
in Section 9,
we correct the grid values by the sphere’s grid error of about 1%.
Checks. The overlap satisfies
n
(0) = 1 to 10
−13
. Rule 1 gives
h
(0) =
e
to 10
−6
. The Fourier-
transform overlaps match directly shifted sums to 10
−15
. The imaginary parts of
n
and
h
are at
most 10
−8
. The Gaussian test above checks Eq. (16).
Convergence. Table 7 gives the results on three grids and three wall strengths. At fixed wall
strength, KL/ℏc = 48, the two masses change by less than 0.1% from a sphere of radius 9 sites to
one of 15 sites. Doubling the wall strength from 48 to 96 lowers them by 0
.
9 and 1
.
4%, and halving
it to 24 raises them by 1
.
4 and 2
.
4%. The cluster agrees between its two grids to 0
.
2%. The values
of Section 9 use the cluster at KL/ℏc = 48 on the finer grid.
21
Table 7: Convergence of the projection. Grid of
n
3
sites, cavity circumradius
L
in sites, wall strength
KL/ℏc
. The kinetic ratio is the projected over the unprojected kinetic energy.
Mr
p
is in units of
ℏc
, before the 1% grid correction, for rule 1 (wall moves with the quarks) and rule 2 (kinetic energy
projected).
cavity n, L KL/ℏc kinetic ratio r
int
/r
cav
Mr
p
, rule 1 Mr
p
, rule 2
sphere 32, 9 24 0.759 0.730 3.231 2.466
sphere 32, 9 48 0.754 0.721 3.185 2.408
sphere 32, 9 96 0.751 0.716 3.157 2.374
sphere 40, 12 48 0.754 0.720 3.186 2.410
sphere 48, 15 48 0.754 0.718 3.186 2.409
cluster 32, 9 48 0.740 0.723 3.229 2.398
cluster 40, 12 48 0.741 0.723 3.230 2.402
Data availability
The scripts that produce every number and figure in this paper are archived as ssmthe-
ory zeno scripts v3.zip in the repository github.com/raghu91302/ssmtheory.
Declarations
Funding. This work received no external funding.
Competing interests. The author declares no competing interests.
Use of artificial intelligence. The author used Claude (Anthropic) to help write and run the
numerical scripts and to draft and edit the manuscript. The author reviewed all results and text
and takes full responsibility for the content.
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