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
March 27, 2026
Abstract
Does information possess physical mass? Modeling the physical vacuum as a
substrate-free quantum error-correcting code suggests that an elementary parti-
cle’s mass is simply its fault-tolerant verification cost. We test this Mass-Energy-
Information (M/E/I) equivalence on the Face-Centered Cubic (FCC) lattice, track-
ing defects within a [[192, 130, 3]] CSS code. Through a bottom-up classification of all
possible defect geometries, we filter 15 mathematical candidate states through a set
of strict thermodynamic and topological axioms: Minimum Topological Dimension,
Sector Completeness, Boundary Closure, and Kinematic Shedding. Exactly 5 phys-
ically stable states survive this sieve. Their verification costs—1, 207, 273, 1836, and
1839—match the empirical mass ratios of the electron, muon, pion, proton, and neu-
tron to within 0.12%. No parameters are fitted. The rejected configurations violate
specific physical constraints and match no known particles. This offers highly con-
strained macroscopic evidence that inertial rest mass is the thermodynamic shadow
of quantum error correction overhead.
Keywords: quantum error correction; FCC lattice; mass-energy-information equivalence;
standard model; lattice QCD
1 Introduction
Landauer’s 1961 theorem [1] cemented the idea that information has a physical footprint.
Erasing or manipulating a bit dissipates at least E ≥ kT ln 2 of energy. Coupling this with
Einstein’s mass-energy equivalence [2] implies a literal Mass-Energy-Information (M/E/I)
equivalence, where a bit of data carries a fundamental mass of m = kT ln 2/c
2
.
Measuring this mass in a laboratory remains practically impossible. The Landauer mass
of a standard bit sits near 10
−35
kg at room temperature—a value completely drowned
out by the heavy atomic lattices of modern storage drives [3].
1
We can investigate the M/E/I equivalence purely by analyzing a system lacking an inde-
pendent material substrate: the physical vacuum itself [5]. If we model spacetime not as
a continuous manifold but as a discrete quantum error-correcting network—specifically
a CSS stabilizer code on the Face-Centered Cubic (FCC) lattice [4]—particles emerge
simply as localized syndrome patterns. Without a physical hardware substrate beneath
them, their informational verification cost is their absolute source of mass [6,7].
This paper presents a bottom-up enumeration of all topological defects on the FCC lattice.
We map established string-junction geometries from lattice QCD [8–10] to discrete error-
correction costs. By passing these candidates through strict topological axioms, we find
exactly five stable states are permitted to exist. Their extraction costs seamlessly identify
the Standard Model mass spectrum [11, 12].
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.
2 Thermodynamic Costs in a Substrate-Free Vacuum
Any localized excitation x in a topological vacuum requires continuous verification to
prevent code collapse. We define the fault-tolerant verification cost C
x
as the total number
of classical bits generated by checking the quantum constraints associated with the defect.
By Landauer’s principle, the resulting mass is:
m
x
= C
x
×
kT ln 2
c
2
(1)
When calculating the mass ratio between two different particle states, the environmental
variables (local temperature T , Boltzmann’s constant k, and the speed of light c) factor
out entirely:
m
x
m
y
=
C
x
C
y
(2)
If the M/E/I equivalence holds, fundamental mass ratios must act as pure topological
invariants dictated by the vacuum network’s geometry.
3 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
(the origin plus 12 nearest neighbors). This K = 12 coordina-
tion decomposes uniquely into three orthogonal 2D sheets containing 4 neighbors each.
Topological properties are defined by their f-vectors (f
0
, f
1
, f
2
):
The syndrome extraction circuit relies on the coupling sub-matrix B
sector
∈ {0, 1}
E
s
×C
s
.
The constraint set C
s
maps edges to specific stabilizer types: full (V + F ), confined
(V + F
△
), electromagnetic (F
□
), or trivial.
2
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: 1-sheet (1D
lines), 2-sheet (2D planes), and 3-sheet (3D volume). Each higher dimension introduces
emergent constraints.
Table 1: The f-vectors of the FCC coordination sub-structures.
Sub-structure Vertices (V ) Edges (E) Triangles (F
△
) Squares (F
□
) Total Faces (F )
1-sheet 5 4 0 0 0
2-sheet 9 16 8 2 10
3-sheet (full) 13 36 32 6 38
4 The Axioms of Topological Mass
To determine which mathematical sub-matrices constitute physically viable particles, we
apply a strict topological sieve. Violating any of these axioms allows quantum flux to
leak, immediately deleting the defect from the code space.
Axiom 1 (Minimum Topological Dimension). The extraction circuit must obey the princi-
ple of least thermodynamic action. It scales from 1D (edge) to 2D (plane) to 3D (volume)
strictly based on the defect’s structural footprint. A 1D lepton cannot arbitrarily trigger
3D cross-sheet stabilizers, and a 3D baryon cannot be verified by a 2D sub-matrix.
Axiom 2 (Sector Completeness). The active sub-matrix must fully cover the stabilizers
capable of detecting the defect. The circuit cannot selectively ignore constraints that in-
tersect the defect’s spatial footprint. A baryon spanning the full 3D volume must trigger
all V + F constraints; it cannot ignore the square faces to save energy.
Axiom 3 (Boundary Closure). The sub-matrix must form a closed gauge boundary.
Checking vertices but ignoring faces within an active sector allows topological flux to leak.
Furthermore, for a confined color pair on a 2D sheet, the boundary remains open until
closed by a 1D string to prevent monopole leakage.
Axiom 4 (Kinematic Shedding for Rest Mass). Mass represents verification cost at rest.
If a closed defect is deconfined (spatially extended but not pinned by color), the lattice’s
d
2
translational checks measure kinematic momentum across the d = 3 extraction rounds.
These d
2
= 9 dynamic trajectory checks must be subtracted to isolate rest mass. Kine-
matic shedding applies only when dim(B
sector
) > d
2
; a minimal (weight-1) defect has no
3
extended footprint to generate inter-round redundancy. Conversely, if a defect is perfectly
neutral and topologically closed, its boundary mimics the vacuum; the circuit must probe
its internal d = 3 geometry.
5 The 15-Candidate Sieve
Table 2 details the complete enumeration of the 15 candidate states generated by per-
muting the possible structural footprints, force sectors, and dynamic states on the FCC
lattice. The dim(B) column represents the baseline matrix dimension prior to any dynam-
ical boundary corrections. We pass these mathematical candidates through the thermo-
dynamic axioms. A candidate must satisfy all axioms to manifest as a physical mass. The
Verdict column provides the final extraction cost for the survivors, or flags the specific
axiom that destroys the rejected configurations.
Table 2: The 15 candidate combinations filtered by the thermodynamic axioms. Only 5
physically closed, minimal states survive the sieve.
Valid? Footprint Sector Dynamics dim(B) Cost (C) Verdict / Axiom Violated
✓ 1-sheet trivial minimal 1 1 Electron (1D, minimal ground
state)
X 1-sheet full minimal 20 20 Ax1: 1D object cannot bound
faces (F = 0)
X 1-sheet full moving 20 11 Ax1/4: 1D object lacks extended
footprint
✓ 2-sheet confined string 272 273 Pion (π
±
) (+1 closes gauge flux)
X 2-sheet full static 304 304 Ax1: Baryon requires 3 sheets,
not 2
X 2-sheet EM moving 32 23 Ax1: Deconfined leptons spread
to all sheets
X 2-sheet confined static 272 272 Ax3: Requires string to close
boundary
X 2-sheet confined moving 272 263 Ax4: Color-confined objects are
pinned
✓ 3-sheet full static (charged) 1836 1836 Proton (Closed 3D cage)
✓ 3-sheet full static (neutral) 1836 1839 Neutron (Neutral requires +d
probe)
✓ 3-sheet EM (F
□
) moving 216 207 Muon (Deconfined rest mass
sheds −d
2
)
X 3-sheet confined static 1620 1620 Ax2: 3D bounding incomplete
without squares
X 3-sheet vertex static 468 468 Ax3: Unclosed boundary (faces
ignored)
X 3-sheet full moving 1836 1827 Ax4: Confined 3D states cannot
move freely
X 3-sheet EM (F
□
) static 216 216 Ax4: Colorless objects must shed
kinematics
6 The Physical Particle Spectrum
The 5 surviving states map naturally to the fundamental Standard Model particles.
4
6.1 Electron (C
e
= 1)
A single edge error is a 1D topological object. A 1D network mathematically cannot
bound a face (Axiom 1). Without faces, there is no cross-network hook error propagation
and no extended kinematic footprint. The verification circuit only requires the syndrome
change of the single edge itself: C
e
= 1.
6.2 Pion / Meson (C
π
= 273)
A meson is a quark-antiquark pair [8, 9], geometrically requiring exactly two coupled
sheets. The 2-sheet sub-network yields V = 9, E = 16. In a confined state, the electro-
magnetic syndrome is identically zero, decoupling the square faces. The active confined
constraints are V (9) + F
△
(8) = 17. The base cage is 16 × 17 = 272. To satisfy Boundary
Closure (Axiom 3), a 1D internal string must connect the pair to prevent gauge leakage,
adding +1. C
π
= 272 + 1 = 273.
6.3 Muon (C
µ
= 207)
The muon is a colorless, deconfined particle spanning the 3D lattice, triggering the EM
sector (the F
□
= 6 bipartite square faces). The sub-matrix is E(36) × F
□
(6) = 216.
Because the state is spatially extended but unpinned by color confinement, the lattice
registers its kinematic momentum. By Kinematic Shedding (Axiom 4), the d
2
= 3
2
= 9
trajectory redundancies are subtracted to find the rest mass. C
µ
= 216 − 9 = 207.
6.4 Proton (C
p
= 1836)
The QCD baryon maps to a Y-junction defect [10] coupling all three orthogonal sheets.
The vacuum deploys the full 3D verification matrix (E = 36, V + F = 51). Because the
proton is charged, it emits a global topological flux, satisfying Boundary Closure (Axiom
3) via its bounding cage alone. C
p
= 36 × 51 = 1836.
6.5 Neutron (C
n
= 1839)
Structurally identical to the proton, the neutron utilizes the same 3D matrix (1836).
Because it is neutral, however, its closed boundary perfectly mimics the empty vacuum.
To prevent the extraction circuit from deleting the state, it must locally probe the d = 3
internal color flux arms constituting the junction (Axiom 4). C
n
= 1836 + 3 = 1839.
7 Comparison with Experiment and Limitations
The five derived integer costs match the experimental CODATA [11] values for the lightest
non-strange particles with high precision.
5
Table 3: Comparison of derived topological costs against empirical mass ratios.
Particle Predicted Cost (C
x
) Experimental Ratio (m
x
/m
e
) Deviation
Electron 1 1.000 0%
Muon 207 206.768 0.11%
Pion (π
±
) 273 273.132 0.05%
Proton 1836 1836.153 0.008%
Neutron 1839 1838.684 0.017%
The Proton-Neutron Splitting: While absolute mass ratios are accurate to within
0.02%, the integer model predicts a mass splitting of m
n
− m
p
= 3m
e
. The experimental
splitting is 2.531m
e
, an 18.5% error on the splitting scale. This highlights a fundamental
limitation: discrete topological verification costs govern the integer baseline, but they
cannot capture sub-integer QFT corrections generated by electromagnetic self-energy and
bare quark mass differences.
Falsifiability and Strange Hadrons: No strange hadron (Kaon, Λ, Σ, Ξ, Ω) is matched
by any combination in the first coordination shell. This framework predicts that heavier
strange quarks fundamentally probe into the second coordination shell of the FCC lattice.
Finding a sixth stable, non-strange particle matching an integer in Table 2 would extend
the theory. Discovering a stable non-strange particle with no topological match would
cleanly falsify it.
8 Conclusion
Discarding the continuous geometric vacuum in favor of a discrete topological CSS code
allows us to identify five consistent defect states whose verification costs exactly match
fundamental particle masses. Applying strict thermodynamic axioms filters 15 mathe-
matical candidates down to exactly 5 physically viable states. The precise alignment of
these survivors with the empirical Standard Model mass spectrum provides robust, falsifi-
able evidence that inertial mass is simply the thermodynamic footprint of quantum error
correction.
Declarations
Conflict of Interest: The author declares no conflicts of interest. No external funding
was received for this research.
Data Availability: Mathematical parameters and derivations are contained within the
manuscript. Interactive visualization source code is available at https://github.com/
raghu91302/ssmtheory.
6
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