Holographic FCC Vacuum: A Universal
Three-Term Formula
for the Fine Structure Constant and the
Higgs Mass
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
April 2026
Abstract
We derive the fine structure constant and obtain a geometric estimate of the
Higgs mass from a single geometric framework: the quantum vacuum modelled as
a K = 12 Face-Centred Cubic (FCC) lattice of Bell-pair bonds at the Planck scale.
The FCC cuboctahedron has two distinct plaquette types (Figure 1): S
TOR
= 8
triangular faces encoding SU(3) and S
TR
= 4 square faces encoding SU(2) × U(1),
giving K = 12 total Standard Model gauge bosons exactly. The FCC can also
be viewed as three hexagonal sheets stacked in the A-B-C sequence (Figure 2):
electromagnetism is absent within each 2D sheet and emerges holographically only
at inter-sheet interfaces. Both results share two dynamical inputs derived once from
FCC geometry—the bare SU(3) coupling α
s
(E
L
) = 1/(16π) from the Wilson lattice
action and the A
4
group tower, and the ABC tunnelling amplitude P
lift
= e
−D
= e
−3
from WKB tunnelling through the 3-layer bulk—and both take the same three-term
form (Figure 3):
1
α
= 16πe + 8e
−3
+
e
−3
8π
= 137.036 025 (1.9 × 10
−7
from CODATA 2018),
m
H
= K
4
m
e
K − N
c
e
−D
−
1
16π
= 125.36 GeV (1.5σ from PDG 2023).
All symbols are integers derived from FCC geometry, or P
lift
, α
s
, and the electron
mass m
e
(which sets the energy scale for m
H
). No tunable free parameters enter.
1 Introduction
The fine structure constant α ≈ 1/137 and the Higgs boson mass m
H
≈ 125.2 GeV are
among the most precisely measured quantities in physics, yet neither has been derived
from a geometric principle. Quantum electrodynamics takes α as an empirical input; the
Higgs mass in the Standard Model (SM) is a free parameter whose radiative instability
constitutes the hierarchy problem.
This paper works within the Selection Stitch Model (SSM), in which the quantum
vacuum is a K = 12 FCC Bravais lattice of Bell-pair bonds at the Planck scale [1]. The
1
FCC lattice is the unique densest sphere-packing in three dimensions and carries the
A
3
∼
=
SU(4) root system, giving rise to an A
4
subgroup tower that yields the SM colour
structure.
The two key geometric observations are:
1. The FCC cuboctahedron has two plaquette types that map exactly onto the SM
gauge group (Section 2; Figure 1).
2. The FCC decomposes into three hexagonal sheets stacked in the A-B-C sequence:
electromagnetism is absent in each 2D sheet and emerges holographically at inter-
sheet interfaces (Section 3; Figure 2).
From these two observations we derive two dynamical inputs (Section 4) and obtain
α (Section 5) and m
H
(Section 6) through the same three-term structure (Section 7;
Figure 3).
2 FCC Geometry: Two Plaquette Types, Two Sec-
tors
The FCC cuboctahedron has K = 12 nearest-neighbour bond vectors ˆn
j
=
1
√
2
(±1, ±1, 0)
and cyclic permutations, decomposing as:
S
TR
= 2
D−1
= 4 (in-plane, square-face bonds), (1)
S
TOR
= 2
D
= 8 (out-of-plane, triangular-face bonds), (2)
with K = S
TR
+ S
TOR
= 12. The rank-2 structure tensor is exactly isotropic, S
µν
= 4δ
µν
,
and the rank-3 tensor vanishes by antipodal symmetry.
The plaquette-to-gauge identification is exact and follows from two facts: (i) S
TOR
= 8
equals the dimension of the SU(3) adjoint representation; (ii) S
TR
= 4 equals the number
of SU(2) × U(1) generators (T
1
, T
2
, T
3
, Y ) (Table 1).
Table 1: FCC plaquette types and Standard Model gauge content.
Plaquette Count CSS sector SM gauge group Gauge bosons
Triangular S
TOR
= 8 Z-type (magnetic) SU(3) 8 gluons
Square S
TR
= 4 X-type (electric) SU(2)×U(1) W
+
, W
−
, W
0
, B
Total K = 12 complete SM gauge sector 12 bosons
3 ABC Stacking and Holographic Electromagnetism
The FCC lattice is the union of three planar hexagonal sheets stacked in the A-B-C-
A-. . . sequence along the [111] direction. Each 2D hexagonal sheet (K
2D
= 6) has only
triangular minimal closed loops; a gauge-invariant U(1) flux requires a 4-bond (square)
loop [3]. Since no square loops exist within a single sheet, electromagnetism is absent in
2D. Square plaquettes arise only where two sheets meet (Figure 2), so:
2
Figure 1: The FCC cuboctahedron. The S
TOR
= 8 triangular faces (orange) host
the SU(3) magnetic stabilizers (8 gluons). The S
TR
= 4 square faces (red) host the
SU(2)×U(1) electric stabilizers (W
+
, W
−
, W
0
, B). The total K = S
TOR
+ S
TR
= 12
equals the number of SM gauge bosons exactly.
Every photon-fermion vertex requires the photon to traverse an inter-sheet square
plaquette, i.e. to penetrate the 3D bulk.
One complete A-B-C period spans exactly D = 3 inter-layer gaps, setting the
fundamental tunnelling scale.
4 Two Dynamical Inputs from FCC Geometry
4.1 Bare SU(3) coupling: α
s
(E
L
) = 1/(16π)
The 12 FCC bond vectors realise the alternating group A
4
(|A
4
| = 12) with normal
subgroup tower {e} ⊂ V
4
⊂ A
4
, where |V
4
| = 4 = S
TR
and |A
4
/V
4
| = 3 = N
c
[4]. This
gives the exact group-theory identity
K = S
TR
× N
c
= 4 × 3 = 12. (3)
The SU(N
c
) Wilson action on the S
TOR
= 8 triangular plaquettes yields bare coupling
g
2
= 1/S
TR
= 1/4 [3], giving
α
s
(E
L
) =
g
2
4π
=
1
16π
≈ 0.01989. (4)
3
Figure 2: ABC stacking and holographic emergence of electromagnetism. Each 2D hexag-
onal sheet contains only triangular plaquettes (orange), supporting SU(3) only—no U(1)
EM exists in 2D (no closed 4-bond loops within a sheet). Square plaquettes (red), which
carry U(1) flux in the Wilson sense [3], appear exclusively at the A-B and B-C inter-sheet
interfaces. Accordingly, every photon-fermion vertex requires traversal of the 3D bulk,
and the ABC period (D = 3 inter-layer gaps) sets the tunnelling amplitude.
This derivation adopts the standard continuum-matching convention in which the Wilson
plaquette action reduces to (1/2g
2
)
P
µ<ν
F
2
µν
with one independent link direction per
spatial axis [4]; the resulting normalisation condition g
2
= 1/S
TR
is a consequence of the
A4 factorisation K = S
TR
× N
c
and is not an additional free choice.
4.2 ABC tunnelling amplitude: P
lift
= e
−3
A fermion localised on sheet A must traverse the D = 3-layer ABC bulk to reach an inter-
sheet square plaquette. The WKB tunnelling amplitude through D identical unit-height
barriers (V − E =
1
2
, natural lattice units, κ = 1 per bond) is [5]
P
lift
= e
−Dκ
= e
−3
. (5)
The FCC lattice also supports a [[192, 130, 3]] CSS quantum error-correcting code [2] with
code distance d = 3 (derived from K: n = K · S
2
TR
= 192, k = (K − 2)(K + 1) = 130,
d = D = 3). In a CSS code, a logical error requires traversing at least d = 3 bonds; the
corresponding logical error amplitude is e
−d
= e
−3
, confirming P
lift
by an independent
route [6].
4
5 Fine Structure Constant
The inverse EM coupling is the partition sum over non-perturbative topological processes
of increasing depth into the ABC bulk, 1/α = W
0
+W
1
+W
2
, where each term W
n
= w
n
Γ
n
is a geometric degeneracy times a topological weight.
Why 1/α, not α, receives additive contributions. In the Wilsonian effective field
theory framework, integrating out high-energy or bulk degrees of freedom generates the
gauge kinetic term of the boundary theory: S
eff
[A] =
1
2e
2
R
F
2
µν
+ . . . [4]. The coeffi-
cient 1/e
2
∝ 1/α of this kinetic term is what receives additive contributions from each
integrated-out process — not the coupling α itself. In holographic theories this is the stan-
dard result: each bulk instanton trajectory contributes to the coefficient of the boundary
U(1) kinetic term, and these contributions add. The coupling α = 1/
P
n
W
n
then follows
from the coefficient as usual. This is entirely analogous to Wilsonian renormalisation of
the gauge kinetic term in lattice QCD, where β ∝ 1/g
2
(not g
2
) receives additive plaquette
contributions [3].
Note on the IR value of α. The formula reproduces 1/α = 137.036 — the Thomson-
limit (infrared) value — rather than the UV value 1/α ≈ 128 at the Z pole. This is
expected: because U(1) electromagnetism emerges holographically at the 2D inter-sheet
boundaries (Section 3), it is a boundary theory, not a 3D bulk theory. In topological field
theories and quantum Hall systems, boundary couplings are protected from conventional
3D perturbative renormalisation-group running by the topological structure of the bulk.
The value 1/α = 137.036 is accordingly the topologically protected infrared fixed point of
the boundary U(1) theory, locked by the geometry rather than running with energy. The
standard QED running from 1/α = 137 to 1/α ≈ 128 at the Z pole arises from 3D bulk
matter loops, which the holographic boundary coupling does not experience.
W
0
= 16πe — leading holographic process
In path-integral language, each tunnelling process carries a topological weight e
−S
where S
is the Euclidean action of the trajectory. The minimal process is a single forward traversal
of one inter-sheet gap. In the Planck-scale FCC lattice, the stabilizer Hamiltonian takes
the form H = −J
P
s
A
s
, where the coupling J defines the bond energy scale. A single
bond traversal (one binary stabilizer operation) lasts for an imaginary time τ = 1/J, so
the Euclidean action is S
0
= J ·τ = J ·(1/J) = 1 exactly — this is the definition of natural
lattice units, in which one bond operation carries one unit of action. The value S
0
= 1 is
therefore not a free normalisation choice but a consequence of choosing the bond energy J
as the unit of energy. The WKB tunnelling amplitude is then P
lift
= e
−DS
0
= e
−3
, which
coincides with the QEC logical error amplitude e
−d
= e
−3
(code distance d = D = 3).
The coincidence confirms that both descriptions share the same natural unit system; it
is not a circular argument. With S
0
= 1, the topological weight for the forward path
is e
−S
0
= e
−1
. The S
TOR
= 8 torsional channels each contribute one full 2π U(1) phase
winding: w
0
= S
TOR
× 2π = 16π. With α
tree
= α
s
(E
L
) · e
−S
0
= α
s
· e
−1
:
W
0
=
1
α
tree
=
1
α
s
· e
−1
= 16πe ≈ 136.636. (6)
The appearance of Euler’s number e is therefore not numerological: it is the non-perturbative
topological weight e
−S
0
= e
−1
of the minimal single-layer instanton trajectory, standard
5
in path-integral treatments of tunnelling [4].
W
1
= 8e
−3
— backward instanton
A backward virtual traversal of the complete D = 3 ABC stack incurs amplitude P
lift
=
e
−3
, summed over S
TOR
= 8 SU(3) channels:
W
1
= S
TOR
· P
lift
= 8e
−3
≈ 0.398. (7)
W
2
= e
−3
/(8π) — second cuboctahedral shell
Each of the N
phys
= S
TR
/2 = 2 transverse photon polarisations couples to the second
cuboctahedral shell via the SU(3)–U(1) mixing amplitude α
s
· P
lift
:
W
2
= N
phys
· α
s
· P
lift
= 2 ·
1
16π
· e
−3
=
e
−3
8π
≈ 0.002. (8)
Convergence
The series terminates at n = 2 by geometric exhaustion: third-shell cuboctahedra are
centred at distance 2a or
√
3 a from the central site, both strictly greater than the face-
diagonal bond length a, so no square plaquette connects them to the central U(1) mode,
forcing w
3
= 0 identically. The amplitude bound gives
P
n≥3
W
n
≲ 10
−6
.
1
α
= 16πe + 8e
−3
+
e
−3
8π
= 136.636 + 0.398 + 0.002 = 137.036 025. (9)
The CODATA 2018 experimental value is 1/α
exp
= 137.035 999; the deviation is 1.9×10
−7
.
6 Higgs Mass
6.1 Code structure and Higgs identification
The [[192, 130, 3]] CSS code carries the full SM content: the k = 130 logical qubits is
consistent with the SM matter sector via k = 5 × 2 × (K + 1) (5 Weyl doublets per gen-
eration, SU(2) dimension 2, (K + 1) = 13 states per bond). The Higgs boson is identified
as the unique scalar mode at the boundary of the X- and Z-type stabilizer sectors, with
minimum stabilizer weight d = D = 3. After electroweak symmetry breaking, three of the
S
TR
= 4 EW generators are eaten by the massive W
±
and Z, one survives as the massless
photon, and one radial mode becomes the physical Higgs: N
phys,H
= S
TR
− N
c
= 1.
The Higgs self-coupling follows from the boundary-to-bulk coordination ratio of the
code:
N
V
= K
3
= 1728, N
S
= K
2
− 3K = 108, λ =
2N
S
N
V
=
1
8
[exact]. (10)
6.2 Three-term formula
T
1
= K
5
m
e
— bare mass from TQFT defect classification
In topological lattice models, the spectrum of physical masses is governed by the di-
mensionality of stabilizer defects [6]. Within the [[192, 130, 3]] CSS code, excitations are
6
classified by the weight of the operator that creates them. We identify the electron as
the minimal weight-1 point defect of the FCC vacuum—the lowest-energy topological
excitation of the stabilizer Hamiltonian. Its mass therefore sets the fundamental energy
gap:
∆
min
= m
e
. (11)
The value of ∆
min
is measured rather than derived from the FCC geometry alone; what
the TQFT classification provides is the scaling relationship between ∆
min
and heavier
excitations.
The Higgs boson, as the global scalar field that couples coherently to every site in the
coordination network, is a boundary-bulk coupled collective excitation. In the K = 12
framework, its bare energy scales as the fundamental gap amplified by the full geometric
coordination volume. The coordination structure has two natural layers: the bulk volume
K
3
= N
V
= 1728 (all coordination states per site) and the bounding surface shell K
2
=
144 (the surrounding cuboctahedral layer). A scalar excitation spanning both layers
therefore carries energy
m
bare
H
= K
3
× K
2
× ∆
min
= K
5
m
e
= 127.153 GeV. (12)
This establishes the structural relationship: the electron does not dynamically generate
the Higgs mass, but both are geometrically locked scale limits of the same FCC vacuum
Hamiltonian. The electron (weight-1 point defect, energy ∆
min
) and the Higgs (boundary-
bulk coupled scalar, energy K
3
×K
2
×∆
min
) occupy opposite ends of the topological exci-
tation hierarchy: the electron couples to a single bond, while the Higgs couples coherently
to both the K
3
-coordination volume and its K
2
bounding surface.
−T
2
— top-quark loop (QEC estimate)
The top quark is the heaviest SM fermion (y
t
= m
t
√
2/v ≈ 0.994 ≈ 1) and, within the
QEC picture, occupies the weight-1 sector of the matter code. Because the code distance
is d = 3, weight-1 operators are unprotected and traverse the D = 3 ABC bulk with
amplitude P
lift
. With N
c
= 3 colour channels:
δ
top
= N
c
K
4
m
e
P
lift
= 3 × 10.596 GeV × e
−3
= 1.583 GeV. (13)
Fermion loops carry a negative sign relative to the scalar self-energy.
Limits of the QEC approximation. The standard one-loop top-quark contribution
in the Wilsonian effective theory involves (3y
2
t
/8π
2
) m
2
t
ln(Λ/m
t
) and differs in functional
form from the formula above (no π
2
denominator, no logarithm of the cutoff). The QEC
formula provides a geometric estimate derived from the topological structure of the code;
establishing a formal connection to the Wilsonian loop integral is a natural direction for
future theoretical development.
−T
3
— SU(3) gluon loop (QEC estimate)
The SU(3) gluon loop contributes one unit of the bare coupling α
s
= 1/(16π) per bond
energy. This is the leading-order coupling strength at the Planck scale times the MEI
bond energy:
δ
SU(3)
= K
4
m
e
α
s
= 10.596 GeV ×
1
16π
= 0.211 GeV. (14)
7
The derivation of this term from first principles within the Wilsonian effective field theory
framework is a direction for future work.
m
H
= K
4
m
e
K − N
c
e
−D
−
1
16π
= 125.36 GeV. (15)
Table 2: Three-term Higgs mass formula. T
1
follows from TQFT defect classification; T
2
,
T
3
are QEC-motivated loop estimates.
Term Physical origin Contribution Cumulative
K
5
m
e
TQFT gap scaling: K
5
∆
min
+127.153 GeV 127.153 GeV
−N
c
K
4
m
e
e
−D
Top quark (3 colours, ABC traversal) −1.583 GeV 125.570 GeV
−K
4
m
e
/(16π) SU(3) gluon loop −0.211 GeV 125.360 GeV
PDG 2023 experiment 125.200 ± 0.110 GeV
Deviation +0.160 GeV (1.5σ)
The remaining 1.5σ residual is consistent with the electroweak gauge loop δ
EW
=
N
W
K
4
m
e
α
W
P
lift
/N
c
≈ 0.017 GeV, where α
W
= 13α/3 and sin
2
θ
W
= N
c
/(K + 1) = 3/13
from the A
4
tower. This term is left to future work.
7 Universal Three-Term QEC Structure
Figure 3: Universal three-term QEC structure. Left: the 1/α partition sum; three positive
contributions build to 137.036. Right: the Higgs mass; the bare value 127.15 GeV is
reduced by two loop corrections. In both panels Term 2 uses P
lift
= e
−3
and Term 3 uses
α
s
= 1/(16π). Red dashed lines mark the experimental values.
Table 3 shows the exact correspondence. The universal pattern is:
Observable = (bare/tree) ± (fermion loop via P
lift
) ± (gauge loop via α
s
).
Both formulas use the same two Planck-scale inputs derived once from FCC geometry.
8
Table 3: Universal three-term QEC structure for 1/α and m
H
.
Role Input In 1/α In m
H
Term 1 (bare/tree) geometry S
TOR
· 2πe = 16πe K
5
m
e
Term 2 (fermion, P
lift
) e
−D
+S
TOR
· e
−3
= 8e
−3
−N
c
K
4
m
e
e
−3
Term 3 (gauge, α
s
) 1/(16π) +N
phys
α
s
e
−3
−K
4
m
e
/(16π)
Result 137.036 025 125.36 GeV
Experiment 137.035 999 125.20 ± 0.11 GeV
Precision 1.9 × 10
−7
1.5σ
8 All Inputs from Five FCC Integers
Every symbol in both formulas is an exact function of five integers from FCC geometry
with D = 3 (Table 4). The electron mass m
e
sets the energy scale for m
H
; it does not
appear in 1/α.
Table 4: Five integers determine all inputs to both formulas.
Symbol Value Origin In 1/α In m
H
K 12 D(D + 1); 3D kissing number S
TOR
= K − 4 K
5
, K
4
D 3 spatial dimensions P
lift
= e
−D
P
lift
= e
−D
N
c
3 |A
4
/V
4
|; colour charges N
phys
= 2 N
c
P
lift
α
s
1
16π
Wilson+A
4
Term 3 Term 3
P
lift
e
−3
WKB; d = D Terms 2,3 Term 2
9 Conclusion
We have derived the fine structure constant and obtained a geometric estimate of the
Higgs mass from a single framework—the K = 12 FCC vacuum lattice—without tunable
free parameters beyond the electron mass m
e
as energy unit. Both results share the same
two dynamical inputs P
lift
= e
−3
and α
s
= 1/(16π), and a parallel three-term structure:
1
α
= 16πe + 8e
−3
+
e
−3
8π
= 137.036 025 (1.9 × 10
−7
from CODATA 2018), (16)
m
H
= K
4
m
e
K − N
c
e
−3
−
1
16π
= 125.36 GeV (1.5σ from PDG 2023). (17)
The two formulas reflect different levels of theoretical completion: the α formula is a fully
convergent partition sum with a proven termination condition, while the Higgs formula
applies the TQFT defect-classification framework for the bare term and QEC-motivated
geometric estimates for the loop corrections. Establishing the formal bridge between the
QEC estimates and the Wilsonian effective field theory is a well-defined open problem
that the present framework motivates and partially constrains.
9
Falsifiable prediction. The A
4
group has |A
4
| = K = 12 non-trivial symmetry opera-
tions acting on the 4-element square-bond set. After electroweak symmetry breaking, one
U(1) mode (the photon) survives as a neutral singlet, adding one state to the count; the
electroweak tower therefore contains K + 1 = 13 total states. Of these, N
c
= |A
4
/V
4
| = 3
carry the A
4
/V
4
hypercharge. The hypercharge fraction directly gives the electroweak
mixing angle:
sin
2
θ
W
=
N
c
K + 1
=
3
13
= 0.2308. (18)
The PDG 2023 value in the MS scheme at the Z pole is sin
2
θ
MS
W
= 0.23122 ± 0.00003, a
deviation of 0.18% (pointing to a higher-order EW correction of order α
W
∼ 0.03 from the
A
4
tower). In the effective leptonic scheme the PDG value is sin
2
θ
eff
W
= 0.23153 ±0.00016
(2.8σ). Equation (18) constitutes a parameter-free geometric prediction; its sub-percent
deviation defines the precision target for the next correction within the FCC framework.
The overarching pattern—(bare/tree) ± (fermion channel via P
lift
) ± (gauge coupling
via α
s
)—determined entirely by the FCC ABC topology and the Wilson lattice action, is
proposed as an organising principle of the FCC vacuum code that should apply to further
SM observables.
References
[1] T. C. Hales, “A proof of the Kepler conjecture,” Ann. of Math. 162, 1065–1185 (2005).
[2] R. Kulkarni, “A 67%-Rate CSS Code on the FCC Lattice: [[192,130,3]] from Weight-12
Stabilizers,” arXiv:2603.20294 (2026).
[3] K. G. Wilson, “Confinement of quarks,” Phys. Rev. D 10, 2445 (1974).
[4] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory,
Addison-Wesley (1995), Ch. 15.
[5] L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd
ed., Pergamon Press, Oxford (1977),
§
50.
[6] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information,
Cambridge University Press (2000), Ch. 10.
[7] Particle Data Group, “Review of Particle Physics,” Prog. Theor. Exp. Phys. 2022,
083C01 (2022).
[8] E. Tiesinga, P. J. Mohr, D. B. Newell, and B. N. Taylor, “CODATA recommended
values of the fundamental physical constants: 2018,” Rev. Mod. Phys. 93, 025010
(2021).
A Numerical Verification
The following Python script reproduces both results from the five geometric integers.
import numpy as np
10
K, D, N_c, S_TR, S_TOR = 12, 3, 3, 4, 8
m_e = 0.511e-3 # GeV
P_lift = np.exp(-D) # = e^{-3} = 0.049787
alpha_s = 1/(16*np.pi) # = 0.019894 [Wilson + A4]
# -- Fine structure constant ------------------------------
W0 = S_TOR * 2*np.pi * np.e # 136.6357 tree-level
W1 = S_TOR * P_lift # 0.3983 backward instanton
W2 = (S_TR/2) * alpha_s * P_lift # 0.0020 2nd shell
alpha_inv = W0 + W1 + W2
print(f"1/alpha = {alpha_inv:.6f}") # 137.036025
print(f"CODATA = 137.035999 dev = {abs(alpha_inv-137.035999)/137.035999:.1e}")
# -- Higgs mass -------------------------------------------
E1 = K**4 * m_e # 10.5961 GeV bond energy unit
T1 = K**5 * m_e # 127.1532 GeV bare
T2 = N_c * E1 * P_lift # 1.5826 GeV top loop
T3 = E1 * alpha_s # 0.2108 GeV SU(3) loop
m_H = T1 - T2 - T3
print(f"m_H = {m_H:.4f} GeV") # 125.3597 GeV
print(f"PDG 2023 = 125.200 +/- 0.110 GeV ({(m_H-125.20)/0.11:.1f} sigma)")
11
