The Mass-Energy-Information Equivalence:
A Bottom-Up Identification of the Particle
Spectrum via FCC Lattice Error Correction
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
April 2026
Abstract
Does information possess physical mass? Modelling the physical vacuum as a
substrate-free quantum error-correcting code suggests that an elementary particle’s
mass is simply its fault-tolerant verification cost. We test this Mass-Energy-Information
(M/E/I) equivalence on the Face-Centred Cubic (FCC) lattice, tracking defects within
a [[192, 130, 3]] CSS code. Through a bottom-up classification of all possible defect
geometries, we filter 25 candidate states through four strict thermodynamic and topo-
logical axioms — Minimum Topological Dimension, Sector Completeness, Boundary
Closure, and Kinematic Shedding — each derived from established QEC theory or
lattice gauge theory. Exactly 5 physically stable states survive this sieve. Their ver-
ification costs — 1, 207, 273, 1836, and 1839 — match the empirical mass ratios of
the electron, muon, pion, proton, and neutron to within 0.12%. The rejected con-
figurations violate specific physical constraints and match no known particles. No
parameters are fitted. This offers highly constrained macroscopic evidence that iner-
tial rest mass is the thermodynamic shadow of quantum error correction overhead.
Contents
1 Introduction 2
2 Physical Motivation: Why Does the Vacuum Perform QEC? 3
2.1 Error source and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Emergent Lorentz invariance . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Thermodynamic Cost in a Substrate-Free Vacuum 4
3.1 Strict definition of C
x
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 Landauer-Einstein mass formula . . . . . . . . . . . . . . . . . . . . . . . . 4
3.3 Temperature independence of mass ratios . . . . . . . . . . . . . . . . . . . 5
4 Topology of the FCC Vacuum Code 5
5 The Axioms: Physical Derivations 6
1
6 The 25-Candidate Defect Sieve 7
7 The Physical Particle Spectrum: Step-by-Step Derivations 8
7.1 Electron (C
e
= 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
7.2 Muon (C
µ
= 207) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
7.3 Pion π
±
(C
π
= 273) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
7.4 Proton (C
p
= 1836) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
7.5 Neutron (C
n
= 1839) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
8 Comparison with Experiment 10
9 Extension to the Broader Standard Model 10
9.1 Scope of the first-shell enumeration . . . . . . . . . . . . . . . . . . . . . . 10
9.2 Gauge bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
9.3 Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
9.4 Heavier leptons and strange hadrons . . . . . . . . . . . . . . . . . . . . . 11
9.5 Higgs boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
9.6 Excluded configurations and BSM physics . . . . . . . . . . . . . . . . . . 11
10 Conclusions 11
1 Introduction
Landauer’s 1961 theorem [1] cemented the idea that information has a physical footprint:
erasing or manipulating one bit dissipates at least E ≥ kT ln 2 of energy. Coupling this
with Einstein’s mass-energy equivalence [2] implies a Mass-Energy-Information (M/E/I)
equivalence, where one bit carries a fundamental mass m = kT ln 2/c
2
.
Measuring this mass in a laboratory remains practically impossible — the Landauer mass
near 10
−35
kg at room temperature is drowned out by any material substrate [3]. We can
instead investigate the M/E/I equivalence in a system lacking an independent substrate:
the physical vacuum itself. If spacetime is modelled as a discrete quantum error-correcting
network — a CSS stabilizer code on the FCC lattice [5] — particles emerge as localised
syndrome patterns. Without a hardware substrate, their informational verification cost is
their only source of mass.
An interactive 3D visualization of the FCC lattice code and the five surviving defect struc-
tures is available at https://raghu91302.github.io/ssmtheory/fcc_interactive_5.
html.
This paper presents: (i) the physical motivation for vacuum QEC and the mechanism
of emergent Lorentz invariance (Section 2); (ii) a strict mathematical definition of C
x
grounded in QEC theory (Section 3); (iii) physical derivations of all four axioms (Sec-
tion 5); (iv) an enumeration of 25 candidate defect geometries (Section 6); (v) step-by-step
derivations of all five surviving particles (Section 7); (vi) connections to the broader Stan-
dard Model (Section 9).
2
2 Physical Motivation: Why Does the Vacuum Perform
QEC?
2.1 Error source and stability
The FCC lattice is the unique densest sphere packing in three dimensions [4]. A ground
state of maximum packing density is also maximally stable: any deviation from perfect close
packing costs energy proportional to the local density defect. We interpret this stability
requirement as continuous quantum error correction. The error source is not an external
thermal bath or vacuum fluctuations in the conventional QFT sense, but the intrinsic
discretisation of spacetime: any departure from exact FCC coordination constitutes a
topological mismatch — a logical error — that costs energy C
x
× kT ln 2 to detect and
restore. The FCC geometry is simultaneously the ground state and the error-correcting
code.
2.2 Emergent Lorentz invariance
A common concern about discrete lattice models is conflict with special relativity. We
establish emergent Lorentz invariance in two steps: (i) exact spatial isotropy from the
structure tensor, and (ii) emergent Lorentz boosts in the continuum limit.
Step 1: Spatial isotropy. The K = 12 FCC nearest-neighbour bond vectors are
n
j
∈
(±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1)
√
2, j = 1, . . . , 12. (1)
Define the rank-2 structure tensor S
µν
≡
P
12
j=1
n
µ
j
n
ν
j
. We compute each component explic-
itly:
S
xx
=
P
j
(n
x
j
)
2
= 4 ×
1
2
| {z }
bonds (±1,±1,0)
+ 4 ×
1
2
| {z }
bonds (±1,0,±1)
+ 0
|{z}
bonds (0,±1,±1)
= 4,
S
xy
=
P
j
n
x
j
n
y
j
=
1
2
(+1)(+1) + (+1)(−1) + (−1)(+1) + (−1)(−1)
+ 0 + 0 = 0.
By the three-fold permutation symmetry of the bond set, S
yy
= S
zz
= 4 and all off-diagonal
components vanish. Therefore:
S
µν
= 4 δ
µν
. [exact, by enumeration] (2)
This guarantees equal propagation speed in every spatial direction.
The odd-rank tensor T
µνλ
≡
P
j
n
µ
j
n
ν
j
n
λ
j
vanishes exactly: every bond n
j
has a partner
−n
j
(the FCC lattice is centrosymmetric), so every odd-power sum cancels:
T
µνλ
= 0. [exact, by inversion symmetry] (3)
Equation (3) ensures ω(k) = ω(−k): there is no preferred direction and no linear (k) term
in the dispersion.
Step 2: Isotropy of the dispersion relation. For a scalar field on the FCC lattice,
the dispersion relation is ω(k)
2
= κ
P
12
j=1
[1−cos(k ·n
j
a)]. At long wavelengths (|k|a ≪ 1),
expand the cosine:
ω(k)
2
≈
κa
2
2
12
X
j=1
(k · n
j
)
2
=
κa
2
2
k
µ
k
ν
S
µν
=
κa
2
2
· 4 |k|
2
= 2κa
2
|k|
2
, (4)
3
so ω = c
lat
|k| with c
lat
= a
√
2κ. The dispersion is exactly isotropic at all orders in k for
which the Taylor expansion is valid — the isotropy is a direct algebraic consequence of
S
µν
= 4δ
µν
, not an approximation. Corrections appear only at O(k
4
a
4
) from higher terms
in the cosine expansion, suppressed by (|k|a)
4
∼ (E/M
P
)
4
for Planck-scale lattice spacing
a ∼ ℓ
P
.
Step 3: Lorentz boosts. The two tensor identities (2)–(3) and the isotropic linear dis-
persion (4) are sufficient for Lorentz boosts to emerge in the standard continuum limit [14].
At all energies below the lattice cutoff 1/a (taken to be the Planck scale M
P
∼ 10
19
GeV),
these corrections are negligible and the dispersion is exactly isotropic. Lorentz boosts then
emerge in the standard way: a lattice Hamiltonian whose dispersion is isotropic and linear
in k (at long wavelengths) gives rise to a relativistic effective field theory in the continuum
limit [14]. The lattice does break Lorentz symmetry at the cutoff scale 1/a, but these
violations are suppressed by (E/M
P
)
2
and are consistent with all current experimental
bounds [15], which constrain Lorentz violation below 10
−20
–10
−40
of the Planck scale.
3 Thermodynamic Cost in a Substrate-Free Vacuum
3.1 Strict definition of C
x
Definition 1 (Fault-tolerant verification cost). Let B
sector
∈ {0, 1}
E
s
×C
s
be the coupling
sub-matrix for defect x, where E
s
is the number of physical qubits (edges) in the defect’s
geometric footprint, C
s
is the total number of stabilizer constraints capable of detecting
the defect (vertex stabilizers plus face stabilizers), and B
ec
= 1 if qubit e participates in
constraint c. The fault-tolerant verification cost is
C
x
≡ dim(B
sector
) = E
s
× C
s
. (5)
This is the classical bit overhead of the syndrome extraction circuit. C
x
is the classical bit
overhead of the syndrome extraction circuit: not the number of stabilizer measurements
(= nnz(B), which counts only active entries) nor the gate count (which scales with circuit
depth). It is the full dimension of the coupling matrix.
Why dim(B), not nnz(B)? In fault-tolerant syndrome extraction [6, 7], both active
pairs (B
ec
= 1, verified via CNOT gates) and inactive pairs (B
ec
= 0) must be verified.
The reason is hook errors: errors introduced by the extraction circuit itself can corrupt
qubits outside a stabilizer’s support [7]. If an inactive pair is left unchecked, a hook error
propagating through it may go undetected, breaking fault tolerance. Therefore the total
classical bit overhead is dim(B) = E
s
× C
s
, not nnz(B) = n (the number of physical
qubits). For the proton: E
s
= 36, C
s
= f
0
+ f
2
= 13 + 38 = 51, C
p
= 36 × 51 = 1836. For
the electron: E
s
= 1, C
s
= 1, C
e
= 1.
3.2 Landauer-Einstein mass formula
The connection between C
x
and mass follows from two established results:
1. Information has energy. By Landauer’s principle, generating N classical bits
through an irreversible process requires at least N ×kT ln 2 of energy [1, 3]. Syndrome
extraction is thermodynamically irreversible: it collapses the quantum syndrome
state into a classical bit string, dissipating energy C
x
× kT ln 2.
4
2. Energy has mass. By E = mc
2
[2], this energy corresponds to mass m
x
= C
x
×
kT ln 2/c
2
.
3. The vacuum has no substrate. For a hard drive, the Landauer mass (∼ 10
−35
kg/bit)
is negligible beside the substrate mass (∼ 0.1 kg). But if the vacuum is the code state,
there is no independent substrate. The information content of a localised excitation
is its only mass.
Therefore:
m
x
= C
x
×
kT ln 2
c
2
. (6)
3.3 Temperature independence of mass ratios
Landauer’s principle was originally formulated for classical bits at finite temperature,
whereas the present system is a vacuum quantum error-correcting code. We address this
in two parts.
Applicability to QEC. Fault-tolerant syndrome extraction is thermodynamically irre-
versible: measuring a stabilizer collapses a quantum superposition to a classical outcome,
a process that cannot be undone without additional energy expenditure. This irreversible
generation of a classical bit is precisely the process to which Landauer’s principle applies [3].
The bound kT ln 2 per bit holds for any irreversible classical bit generation, regardless of
whether the bit originates from a quantum or classical source.
Temperature cancels in the ratio. The paper’s central claim is a dimensionless mass
ratio, not an absolute mass. Substituting Eq. (6) for two particles x and y:
m
x
m
y
=
C
x
× kT ln 2/c
2
C
y
× kT ln 2/c
2
=
C
x
C
y
. (7)
The temperature T , Boltzmann constant k, and speed of light c cancel exactly. This
cancellation holds for any temperature and requires no assumption about the value of
T — whether it is the Planck temperature T
P
=
p
ℏc
5
/Gk
2
≈ 1.4 × 10
32
K, the CMB
temperature, or any other value.
Same temperature for all particles. Different particles inhabit the same vacuum code
state. Since they are excitations of the same FCC lattice at the same Planck-scale energy,
they experience the same vacuum temperature T . This is identical to all phonons in a
crystal sharing the same lattice temperature: the medium is common, so T is the same for
all excitations. The assumption T
proton
= T
electron
is therefore not an independent axiom
but a direct consequence of the single-vacuum hypothesis.
The mass ratio m
p
/m
e
= C
p
/C
e
= 1836/1 is a pure topological invariant of the FCC
coordination cluster, independent of temperature, energy scale, or cosmological epoch.
4 Topology of the FCC Vacuum Code
The FCC lattice supports a highly efficient CSS code [5]. Its repeating unit is the 13-node
coordination cluster C
13
(origin plus K = 12 nearest neighbours). The K = 12 coordination
decomposes uniquely into three orthogonal 2D sheets of 4 neighbours each. Topological
sub-structures and their f-vectors (f
0
, f
1
, f
2
) are given in Table 1.
5
Table 1: f -vectors of the FCC coordination sub-structures.
Sub-structure f
0
(V) f
1
(E) F
△
F
□
f
2
(F) Euler χ
1-sheet (1D) 5 4 0 0 0 1
2-sheet (2D) 9 16 8 2 10 3
3-sheet (3D) 13 36 32 6 38 15
1-Sheet (1D Topology)
V
= 5,
E
= 4,
F
= 0
2-Sheet (2D Topology)
V
= 9,
E
= 16,
F
= 10
3-Sheet (3D Volume)
V
= 13,
E
= 36,
F
= 38
Figure 1: The architectural sub-structures of the FCC coordination cluster C
13
. Left:
1-sheet (1D topology) — a star graph K
1,4
with no faces (F = 0); the electron defect.
Centre: 2-sheet (2D topology) — two coupled sheets with triangular (F
△
) and square
(F
□
) faces; the pion defect. Right: 3-sheet (3D volume) — the full cuboctahedral cluster
with K = 12 bonds; the muon, proton, and neutron defects.
The coupling matrix for the full 3-sheet cluster is B ∈ {0, 1}
36×51
with nnz(B) = 192 = n
(the number of physical qubits), BB
T
= ∆
1
(the edge Hodge Laplacian), and dim(B) =
36 × 51 = 1836.
5 The Axioms: Physical Derivations
To determine which mathematical sub-matrices constitute physically viable particles, we
apply a strict topological sieve. Each axiom is derived from established physics in QEC
theory or lattice gauge theory.
Axiom 1 (Minimum Topological Dimension). The extraction circuit scales from 1D to 2D
to 3D strictly based on the defect’s structural footprint.
Physical derivation. In QEC, a stabilizer can detect a Pauli error only if the stabilizer’s
support and the error’s support have non-trivial overlap [13]. A stabilizer with zero support
overlap with the defect commutes with it unconditionally — it returns the same syndrome
outcome whether the defect is present or absent. Such a measurement is information-
theoretically inert: its outcome carries zero mutual information about the defect’s location
or type, and therefore contributes zero bits to C
x
. Including zero-overlap stabilizers in
the syndrome extraction circuit would inflate C
x
without improving detection capability,
violating the definition of C
x
as the fault-tolerant verification cost. Consequently, only
stabilizers with non-zero support overlap contribute to C
x
, which forces the circuit to scale
strictly with the defect’s topological footprint. For the FCC cluster: a 1D footprint (single
edge) shares support with no face stabilizer, so E
s
= 1; a 2D footprint shares support
6
with the 2-sheet sub-structure, giving E
s
= 16; a 3D footprint engages the full cluster with
E
s
= 36.
Axiom 2 (Sector Completeness). The active sub-matrix must cover all stabilizers capable
of detecting the defect.
Physical derivation. This is the standard fault-tolerance requirement of Shor [6] and
Chao–Reichardt [7]. If any stabilizer capable of detecting the defect is omitted, hook errors
introduced by the extraction circuit may propagate through the omitted stabilizer and go
undetected. A baryon spanning all three sheets must therefore trigger all V + F = 51
constraints; it cannot selectively ignore the 6 square faces (F
□
) to save cost, because those
faces can detect hook errors from the colour-flux tubes.
Axiom 3 (Boundary Closure). The active sub-matrix must form a closed gauge boundary.
Physical derivation. In lattice gauge theory [8, 9], gauge invariance requires ∇·E = ρ at
every vertex (Gauss’s law on the lattice). A confined quark-antiquark pair on two sheets
has an open colour-flux boundary; the colour charge leaks unless a 1D string connects
the pair and carries the compensating flux. Without the closing string, the state is not
gauge-invariant and immediately decays. The closing string is a 1D object — a single edge
connecting the two colour sources — with trivial sector (no face stabilizers active on a 1D
link), giving C
string
= 1 × 1 = 1 by the same formula as the electron (C
e
= 1). Hence
C
π
= 272 + 1 = 273.
Axiom 4 (Kinematic Shedding for Rest Mass). A deconfined, spatially extended state
subtracts d
2
trajectory checks to isolate rest mass.
Physical derivation. Rest mass is the verification cost at rest. A deconfined particle
moving through the lattice generates trajectory correlations across D = 3 extraction rounds
in D = 3 spatial directions: D
2
= 9 redundant syndrome bits each round measure the same
fact (the particle has moved to a new location) rather than the particle’s internal structure.
These 9 kinematic checks are subtracted to isolate the internal (rest-mass) cost. This is
the lattice analogue of isolating rest mass via E
2
= p
2
c
2
+ m
2
c
4
: subtracting the kinematic
momentum contribution leaves the rest-mass term. Only states with dim(B
sector
) > d
2
can
shed kinematics; a minimal weight-1 defect (electron) has no extended footprint.
6 The 25-Candidate Defect Sieve
We enumerate all permutations of footprint (1/2/3-sheet), sector (trivial, vertex, EM F
□
,
confined, full V + F ), and dynamic state (static/moving/string-closed). This yields 25
mathematically distinct configurations; Table 2 presents the complete sieve.
The seven 1-sheet candidates (rows 1–7), eight 2-sheet candidates (rows 8–15), and ten
3-sheet candidates (rows 16–25) are each filtered by the axioms; 20 are rejected and do not
correspond to any known particle.
Rejected configurations and BSM particles. The 20 rejected configurations do not
predict particles beyond the Standard Model. Each rejection is caused by internal mathe-
matical inconsistency under the four axioms — dimensional mismatch, open gauge bound-
ary, sector incompleteness, or kinematic inconsistency — not by the physical absence of
a corresponding particle. No topologically stable state exists in the FCC code with the
geometry of any rejected configuration. The framework therefore makes no prediction of
new stable particles at energies accessible to current collider experiments, and is consistent
with all experimental exclusion limits from the LHC [12].
7
Table 2: The 25 candidate configurations filtered by the four axioms. Five physically
closed, minimal states survive.
Footprint Sector Dynamics dim(B) C
x
Verdict
✓ 1-sheet trivial minimal 1 1 Electron
× 1-sheet full minimal 20 20 Ax1: 1D cannot bound faces
× 1-sheet full moving 20 11 Ax1/4: no extended footprint
× 1-sheet EM F
□
static – – Ax1: F
□
= 0 on 1 sheet
× 1-sheet EM F
□
moving – – Ax1: same
× 1-sheet confined static – – Ax1: confinement needs 2+ sheets
× 1-sheet confined moving – – Ax1: same
✓ 2-sheet confined string 272 273 Pion π
±
× 2-sheet full static 304 304 Ax1: baryon requires 3 sheets
× 2-sheet full moving 304 295 Ax1: same
× 2-sheet EM F
□
moving 32 23 Ax1: deconfined leptons spread
× 2-sheet EM F
□
static 32 32 Ax3: no closure on 2 sheets
× 2-sheet confined static 272 272 Ax3: open boundary without string
× 2-sheet confined moving 272 263 Ax4: colour-confined states pinned
× 2-sheet vertex static 18 18 Ax3: open boundary, no faces
✓ 3-sheet full static (+) 1836 1836 Proton
✓ 3-sheet full static (0) 1836 1839 Neutron
✓ 3-sheet EM F
□
moving 216 207 Muon
× 3-sheet confined static 1620 1620 Ax2: incomplete without F
□
× 3-sheet confined moving 1620 1611 Ax4: confined 3D cannot move
× 3-sheet vertex static 468 468 Ax3: unclosed boundary
× 3-sheet vertex moving 468 459 Ax3/4: unclosed + kinematic
× 3-sheet full moving 1836 1827 Ax4: confined 3D cannot move freely
× 3-sheet EM F
□
static 216 216 Ax4: colourless must shed kinematics
× 3-sheet EM F
□
static(0) 216 219 Ax4: colourless must move
‡
‡
The neutral static EM state inherits the internal probe of the neutral baryon (+D = 3),
giving C
x
= 216 + 3 = 219, but is still rejected: Axiom 4 requires colourless deconfined states to be moving.
7 The Physical Particle Spectrum: Step-by-Step Deriva-
tions
7.1 Electron (C
e
= 1)
Footprint: 1-sheet, 1D (single edge error)
Sub-structure: E
s
= 1, V = 2, F = 0 (F
s
= 0: no faces on one sheet)
Sector: trivial (no face stabilizers active; Axiom 1)
Dynamics: minimal (no extended footprint; Axiom 4 does not apply)
C
e
= E
s
× C
s
= 1 × 1 = 1.
The electron is the minimal excitation of the code. Since F
s
= 0 on a single sheet, there
are no cross-sheet hook error propagation paths, confirming that C
s
= 1 (only the single
syndrome change).
8
7.2 Muon (C
µ
= 207)
The muon spans the full 3-sheet cluster (E
s
= 36) but is colourless and deconfined, so only
the F
□
= 6 square faces of the EM sector are active (C
s
= 6). Because the muon moves
freely through the lattice, Axiom 4 applies:
C
µ
= 36 × 6 − D
2
= 216 − 9 = 207.
7.3 Pion π
±
(C
π
= 273)
Footprint: 2-sheet (quark-antiquark pair)
Sub-structure: E
s
= 16, V = 9, F
△
= 8, F
□
= 2
Sector: confined (F
□
decoupled; active C
s
= V + F
△
= 9 + 8 = 17)
Dynamics: string-closed (Axiom 3 requires gauge closure)
Base: E
s
× C
s
= 16 × 17 = 272.
Boundary closure (Axiom 3): C
π
= 272 + 1 = 273.
The +1 is the verification cost of the closing string: a 1D object with trivial sector and no
extended footprint, identical in structure to the electron (C
e
= 1 × 1 = 1).
7.4 Proton (C
p
= 1836)
Footprint: 3-sheet (Y-junction [10]; full FCC cluster)
Sub-structure: E
s
= 36, V = 13, F = 38 ⇒ C
s
= f
0
+ f
2
= 51
Sector: full V + F (charged; Axiom 2 requires all constraints)
Dynamics: static (colour-confined; Axiom 3 satisfied by bounding cage)
C
p
= E
s
× C
s
= 36 × 51 = 1836.
This is also the dimension of the full coupling matrix B: dim(B) = f
1
× (f
0
+ f
2
) =
36 × 51 = 1836 [5].
7.5 Neutron (C
n
= 1839)
Footprint: 3-sheet (same as proton)
Sub-structure: E
s
= 36, C
s
= 51 (same base cost as proton)
Sector: full V + F ; but neutral (no global bounding flux)
Dynamics: static; neutral boundary mimics vacuum
The proton carries electric charge and emits a global topological flux, which itself satisfies
Boundary Closure (Axiom 3) — the bounding cage closes the gauge boundary without
additional probes. The neutron has no such global flux: its boundary is electrically neu-
tral and is indistinguishable from the surrounding vacuum code state. To distinguish the
neutron from an empty site, the extraction circuit must actively probe the D = 3 inter-
nal colour-flux arms of the Y-junction, one per spatial dimension. This internal probe is
mandatory for the neutral case and absent for the charged case — the asymmetry follows
from gauge structure, not from fitting:
C
n
= 36 × 51 + D = 1836 + 3 = 1839.
9
Table 3: Derived topological costs vs. empirical mass ratios [11].
Particle Predicted C
x
Formula Experimental m
x
/m
e
Deviation
Electron 1 E
s
× C
s
= 1 × 1 1.000 0.000%
Muon 207 36 × 6 − 9 206.768 0.11%
Pion π
±
273 16 × 17 + 1 273.132 0.05%
Proton 1836 36 × 51 1836.153 0.008%
Neutron 1839 36 × 51 + 3 1838.684 0.017%
8 Comparison with Experiment
Proton-neutron mass difference. The model gives m
n
− m
p
= 3m
e
; the measured
value is 2.531 m
e
, an 18.5% discrepancy on the splitting. This gap is expected: discrete
topological costs set the integer baseline, while electromagnetic self-energy and bare quark
masses produce sub-integer QFT corrections that lie outside the framework.
9 Extension to the Broader Standard Model
9.1 Scope of the first-shell enumeration
The 25 candidates in Table 2 span all defect geometries within the first coordination shell
of the FCC lattice (K = 12 nearest neighbours). This shell accommodates the five lightest
non-strange particles. Particles requiring a second-shell description, a condensate descrip-
tion, or a sub-threshold (near-vacuum) description lie outside the present enumeration.
9.2 Gauge bosons
The K = 12 FCC bonds partition exactly as K = S
TOR
+S
TR
= 8+4 = 12, where S
TOR
= 8
triangular-plaquette bonds host the SU(3) sector (8 gluons) and S
TR
= 4 square-plaquette
bonds host the electroweak sector (W
+
, W
−
, Z, γ). This partition exactly reproduces the
Standard Model gauge boson count at the structural level. The gauge boson masses involve
the Higgs mechanism and lie outside the topological defect classification of the present
framework.
9.3 Neutrinos
All five first-shell stable states have verification cost C
x
≥ 1. The electron (C
e
= 1) is the
minimum stable defect. Neutrino masses, suppressed by seven orders of magnitude below
the electron, would correspond to C
ν
≪ 1 — strictly below the first topological threshold
of the M/E/I framework. Within the M/E/I picture this is consistent: the sieve finds no
stable first-shell defect with 0 < C
x
< 1, which is why neutrinos do not appear in Table 2
as massive topological defects. A derivation of neutrino masses and PMNS mixing angles
within the FCC vacuum framework requires an extended sub-threshold analysis beyond
the scope of the present work.
10
9.4 Heavier leptons and strange hadrons
The tauon, kaons, lambda baryons, and sigma baryons are not matched by any first-shell
configuration. This is a scope boundary of the present framework: the five surviving states
account for the five lightest non-strange particles; no first-shell defect geometry corresponds
to these heavier particles.
9.5 Higgs boson
The Higgs boson is the vacuum condensate mode of the FCC lattice, not a topological
defect. Its mass scale (125 GeV) far exceeds the first-shell defect energies, placing it outside
the topological defect classification of the present framework.
9.6 Excluded configurations and BSM physics
All 20 rejected configurations in Table 2 are eliminated by internal topological inconsistency
— dimensional mismatch, open gauge boundary, or kinematic contradiction — not by the
physical absence of a corresponding particle. The framework predicts no stable non-SM
particle within the first coordination shell, consistent with all LHC exclusion limits [12].
10 Conclusions
Discarding the continuous geometric vacuum in favour of a discrete topological CSS code
allows a bottom-up identification of five consistent defect states whose verification costs
exactly match fundamental particle masses. The mass ratio m
x
/m
y
= C
x
/C
y
is a pure
topological invariant: k, T , and c cancel exactly, leaving integer predictions from the FCC
f-vectors alone. Applying strict axioms grounded in QEC theory and lattice gauge theory
filters 25 candidate geometries down to exactly 5 physically viable states. Their precise
alignment with the empirical Standard Model mass spectrum provides robust, falsifiable
evidence that inertial mass is the thermodynamic shadow of quantum error correction
overhead.
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