The Hubble Tension as a Structural Prediction
of the K=12 FCC Vacuum
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
May 2026
Abstract
The locally measured Hubble constant (73
.
50
±
0
.
81 km/s/Mpc, H0DN 2026) exceeds the
CMB-inferred value (67
.
36
±
0
.
54 km/s/Mpc, Planck 2018) by 9.1%. We derive this ratio from
the Selection-Stitch Model (SSM), a
K=
12 face-centered cubic vacuum framework developed
by Kulkarni (arXiv:2603.20294); (Phys. Open 27, 100423). Gravitational stress from clustered
matter stretches bonds in cosmic voids. When a bond exceeds 2
/
√
3
(twice the metric wall
set by the in-plane circumradius of a bounding triangular face of the cuboctahedral cluster),
a new node nucleates at the midpoint. The new node demands
K=
12 coordination at the
equilibrium bond length, pushing surroundings outward. Per cycle the effective lattice extent
grows by 2
1/3
(node count doubles, 3D scaling) while applied cosmic stretching grows by
2/
√
3. The ratio is
H
local
0
H
CMB
0
=
√
3
2
2/3
= 1.09112 . . . ,
equal to the observed 1
.
0912
±
0
.
0149 to better than 0
.
01
σ
. A direct simulation of the
cycle confirms the analytic result (steady-state mean 1
.
077
±
0
.
088 across cycles 2–10 in
a free-boundary
L=
4 lattice; finite-size effects account for the residual). The prediction
contains no free parameters. Every constant comes from the cuboctahedral kissing-number
geometry and the metric wall already established in the framework.
Keywords: Hubble tension; quantum error correction; FCC vacuum; cosmic voids; structural
cosmology.
1 Introduction
The Hubble constant
H
0
is measured today by two independent classes of probe that disagree
at the 9% level. Type Ia supernovae calibrated through Cepheids in the local universe give
73
.
50
±
0
.
81 km/s/Mpc [
2
]. Sound horizon physics applied to the cosmic microwave background,
under ΛCDM, gives 67
.
36
±
0
.
54 km/s/Mpc [
7
]. Time-delay cosmography in strongly lensed
quasars at z ≈ 0.5 sides with the local value [9], which rules out simple systematic error in the
distance ladder as the source.
Proposed resolutions fall into early-universe modifications (early dark energy, modified recombi-
nation) that change the inferred sound horizon, and late-universe modifications (interacting dark
energy, modified gravity, void effects) that change the post-recombination expansion history
[
8
,
3
]. None of these gives a closed-form prediction at the observed magnitude without fitting at
least one parameter.
1
This paper derives the observed ratio
H
local
0
/H
CMB
0
=
√
3/
2
2/3
≈
1
.
0911 from the Selection-Stitch
Model (SSM), a
K=
12 face-centered cubic (FCC) vacuum framework. Two structural facts from
the framework do the work: the kissing-number coordination
K=
12 of the FCC lattice [
4
], and
the geometric metric wall at
L
eq
/
√
3
from the in-plane circumradius of a bounding triangular
face of the cuboctahedral cluster [6].
The mechanism is local. Galaxies in the cosmic web exert gravitational stress on the FCC
vacuum in surrounding voids. When a stretched bond reaches twice the metric wall, the lattice
nucleates a new node at the midpoint, and the new node’s demand for
K=
12 coordination at
the equilibrium bond length pushes neighbors outward. The cycle has a closed-form expansion
ratio.
The paper proceeds as follows. Section 2 reviews the Selection-Stitch Model and the prior
static results that this paper builds on. Section 3 describes the cycle. Section 4 derives the
ratio. Section 5 confirms it by simulation. Section 6 compares to data. Section 7 lists testable
predictions.
2 The Selection-Stitch Model
The Selection-Stitch Model (SSM) describes the physical vacuum as a discrete three-dimensional
crystalline lattice produced by a kinematic phase cascade [
6
]. Starting from a single Bell-pair
entanglement bond at unit distance
L
eq
, the lattice grows under exactly two operators. Stitch
places a new node at the equilateral apex above an existing edge, adding a bond to two existing
nodes and growing a planar triangulated sheet. Lift places a new node at the regular-tetrahedral
apex above an existing triangle, at height
h
=
p
2/3 L
eq
, adding a bond to three existing nodes.
Stitch and lift form a complete kinematic basis for graph growth in three-dimensional Euclidean
space at fixed bond length, since a 4-sphere intersection in 3D is generically empty. Lift is
exponentially suppressed relative to stitch by
P
lift
=
e
−3
≈
5%, the codimension difference
between the two operators’ solution manifolds [6].
The cascade closes at
K=
12. The sequence Bell pair (
K=
1)
→
stitched sheet (
K=
6)
→
tetrahedral foam (
K=
4, geometrically frustrated)
→
FCC ABC-stacked lattice (
K=
12) reaches
its geometric endpoint at
K=
12, where the kissing number saturates the upper bound on rigid
sphere packing in three dimensions [
4
]. The
K=
4 tetrahedral foam is unstable because regular
tetrahedra cannot tile three-dimensional Euclidean space; the system selects the unfrustrated
K=
12 FCC descendant through continued action of stitch and lift. We set
L
eq
= 1 throughout
this paper.
The lattice carries a [[192
,
130
,
3]] Calderbank-Shor-Steane (CSS) quantum-error-correcting code,
with weight-12 stabilizers on the cuboctahedral cluster of nearest neighbors around each node
[
4
]. The macroscopic metric of spacetime emerges as the coarse-grained average of bond lengths
in the equilibrium lattice.
The metric wall. A central input to this paper is the metric wall at
R
min
=
L
eq
√
3
≈ 0.5774,
the in-plane circumradius of any bounding triangular face of the cuboctahedral cluster. Geomet-
rically,
R
min
is the distance from the centroid of an equilateral triangle of edge
L
eq
to each of its
vertices. A node placed closer than
R
min
to such a face would couple to all three vertices and
over-coordinate beyond
K=
12. The matter paper established this wall through a simulation
sweep over exclusion radii, demonstrating a sharp geometric phase transition at
R
ex
=
L/
√
3
[
6
,
§2.4].
Matter as incomplete crystallization. If the
K=
4
→ K=
12 phase transition is incomplete
2
and a single
K=
4 node remains trapped at the centroid of an FCC tetrahedral void, the trapped
node bonds to the four surrounding vertices and recreates a local
K=
4 tetrahedron embedded in
the
K=
12 bulk. The matter paper [
6
] identifies this trapped node as a baryon. From this single
geometric defect, with no adjustable parameters, the matter paper derives fractional electric
charges
−
1
/
3 and +2
/
3 from regular-tetrahedron geometry and integer winding, three color
charges from the three skew-edge pairs of the bounding tetrahedron (
c
skew
= 3, with no fourth
color possible since
C
(4
,
2)
/
2 = 3), linear confinement from the metric wall preventing extraction
of the trapped node, and the proton-to-electron mass ratio
m
p
/m
e
= (
K
+1)
K
2
−c
skew
K
= 1836,
matching observation to 0.008%.
Particle spectrum. A companion paper [
5
] extends the framework systematically by treating
each particle as a topological defect class in the [[192
,
130
,
3]] CSS code and computing its
fault-tolerant verification cost
C
x
(the classical bit overhead of syndrome extraction). Filtering
25 candidate defect geometries through four strict axioms (Minimum Topological Dimension,
Sector Completeness, Boundary Closure, and Kinematic Shedding) selects exactly five stable
states. Their verification costs match the empirical mass ratios m
x
/m
e
to within 0.12%:
Particle C
x
m
x
/m
e
(obs.) Deviation
Electron 1 1.000 0.000%
Muon 207 206.768 0.11%
Pion π
±
273 273.132 0.05%
Proton 1836 1836.153 0.008%
Neutron 1839 1838.683 0.017%
The proton entry agrees with the matter paper’s defect-counting formula (
K
+ 1)
K
2
−c
skew
K
=
1836; the two derivations are arithmetically equivalent because
K
= 4
c
skew
in the cuboctahedral
geometry [
6
]. The remaining four particles come only from the verification-cost framework. No
parameters are fitted in either paper.
Cosmological extension: the Insert operator. All matter-paper and MEI-paper results
follow from the equilibrium lattice and static defects relative to it. This paper extends the
framework dynamically by introducing a third kinematic operator, parallel in status to stitch
and lift but admissible only under tensile stress. We name it Insert:
•
Insert places a new node at the midpoint of an existing edge when that edge has been
stretched to 2
R
min
= 2
/
√
3
, then displaces the two endpoints outward to restore
K=
12
coordination at L
eq
locally.
Insert differs from stitch and lift in one essential respect: it is non-rigid. Stitch and lift place
new nodes without disturbing existing ones; Insert places a midpoint and then displaces the two
original endpoints outward to maintain
K=
12 around the new node. That displacement costs
geometric work, which the tensile stretching of the bond supplies. Without tensile stress (bonds
at
L
eq
), Insert has no energy source and cannot fire. This is why Insert is admissible only when
bonds have been stretched to the metric wall.
The mechanism by which Insert produces the Hubble tension follows. Gravitational tidal forces
from clustered matter pull bonds in cosmic voids past
L
eq
. When a bond reaches 2
R
min
, Insert
fires there, adding one node and locally restoring equilibrium. The resulting expansion rate
enhancement gives the observed local-to-CMB ratio.
3
3 The Gravity-Driven Cycle
Cosmic voids fill about 80% of late-universe volume. The cosmic web (galaxies, clusters, filaments)
fills the rest. Matter in the web exerts gravitational tidal stress on the FCC lattice in surrounding
voids. Bonds in voids stretch beyond L
eq
under this stress.
We model the response of the lattice as a four-step cycle (Figure 1).
L
eq
= 1
A B
(a) Equilibrium
L
crit
=
2
p
3
A B
(b) Stretched to threshold
R
min
R
min
A B
M
(c) Midpoint nucleated
1 1
A B
M
(d) Push-outward to equilibrium
Figure 1: Cycle mechanism. (a) Equilibrium bond at
L
eq
= 1. (b) Cosmic stretching pulls bond to
L
crit
= 2
/
√
3 ≈
1
.
155. (c) Insertion: a new node M nucleates at the midpoint, at distance
R
min
from
each of A and B. (d) M demands
K=
12 coordination at
L
eq
; A and B push outward to distance 1
.
0 from
M. The pair separation grows from 1.155 to 2.0.
The cycle steps in detail:
(a) Equilibrium. A and B are nearest neighbors at distance L
eq
= 1.
(b) Stretching. Cosmic expansion stretches the bond uniformly. When bond length reaches
L
crit
= 2R
min
=
2
√
3
≈ 1.155,
the midpoint sits at exactly the metric wall from each endpoint.
(c) Insertion. The Insert operator fires: a new node M appears at the midpoint, at distance
R
min
from A and B.
(d) Push-outward relaxation. M is a bulk vacuum site and demands K=12 with all twelve
nearest neighbors at
L
eq
. The two endpoints A and B, currently at
R
min
from M, move outward
to distance
L
eq
. After relaxation, A and M are bonded at length 1, M and B are bonded at
length 1, and A and B sit at distance 2 along the original bond direction.
The cycle nets one new node and replaces one stretched bond with two equilibrium-length bonds.
4 Analytic Derivation
The per-cycle ratio of effective expansion to applied cosmic stretching follows from elementary
counting. The result is the dimensionless gain of the cycle viewed as a steady-state amplifier:
cosmic expansion enters as the input rate, and the lattice extent emerges as the output rate,
with the gain set by FCC geometry alone. Because both input and output carry units of inverse
time, their ratio is dimensionless and equals the structural factor with no normalization step.
Applied stretching. Bonds run from
L
eq
to
L
crit
. The applied linear scale factor per cycle is
a
app
=
L
crit
L
eq
=
2
√
3
.
Node count. Each cycle adds one midpoint per existing node on average:
N →
2
N
. The
justification is geometric, enforced by the metric wall. Consider two bonds at node A, namely
A-B and A-C, where B and C are nearest neighbors of each other (A, B, C form an equilateral
triangle of edge
L
eq
). After cosmic stretching by 2
/
√
3
, the triangle has edge
L
crit
, and the
4
midpoints
M
AB
and
M
AC
sit at distance
L
crit
/
2 =
R
min
from each other (the midsegment length
is half the third side of the stretched triangle, which has length
L
crit
). Both midpoints would lie
exactly on the metric wall. The framework rejects any configuration with node-node separation
at or below
R
min
as topologically inadmissible, so the two midpoints cannot co-nucleate. In
FCC cuboctahedral geometry, every nearest-neighbor pair of A shares exactly 4 common nearest
neighbors with A. Firing one bond A-B therefore forbids simultaneous firing of A-X for any of
the 4 NN that A and B share. Combined with the push-outward relaxation, which drops the
remaining non-fired bonds incident on A below
L
crit
once the first firing completes, the system
settles on exactly one firing per node per cycle. Globally,
N
new midpoints nucleate per cycle
and the node count doubles.
The rule is falsifiable. If two midpoints per node could nucleate per cycle, the node count would
triple (
N →
3
N
), giving a per-cycle ratio 3
1/3
/
(2
/
√
3
)
≈
1
.
249 rather than
√
3/
2
2/3
≈
1
.
091.
The observed ratio 1.0912 ± 0.0149 is consistent only with the single-firing rule.
Effective linear extent. With node count doubled at equilibrium spacing
L
eq
, the 3D linear
extent grows by
a
eff
= 2
1/3
≈ 1.2599.
Ratio. The per-cycle enhancement is
a
eff
a
app
=
2
1/3
2/
√
3
=
√
3
2
2/3
= 1.09112 . . . (1)
The factors have direct geometric origin. The
√
3
is 1
/R
min
, the inverse of the metric wall. The
2
2/3
is the 3D inverse-cube-root mapping from node-count doubling back to linear scale. The
product is the cleanest structural cosmological prediction available from the framework.
From scale ratio to rate ratio. Equation
(1)
is dimensionless. It can equal a Hubble
ratio (also dimensionless) directly, with no conversion or normalization. The cycle acts as a
steady-state amplifier: cosmic expansion enters at rate
H
CMB
0
, the lattice extent grows at rate
H
local
0
, and the gain is set by FCC geometry alone:
H
local
0
=
√
3
2
2/3
H
CMB
0
.
Both quantities carry units of inverse time, the structural factor is a pure number, and the
equation balances on units without any fitted constant. The
√
3
traces to the metric wall
(
R
min
=
L
eq
/
√
3
), and the 2
2/3
to the 3D inverse-cube-root scaling from node-count doubling.
The structural number is the rate ratio.
The identification of the per-cycle linear ratio with the rate ratio follows from the physical state
of cosmic voids, not from any fit to data. A typical void contains on the order of 10
60
bonds.
Each bond reaches
L
crit
at a different moment under cosmic expansion, with no global clock to
synchronize them. Insert fires asynchronously, on a per-bond basis, distributed continuously in
time. The lattice therefore contains bonds at every stretching stage simultaneously, and the cycle
is continuous by construction. The per-cycle linear gain then acts as a multiplicative factor on the
instantaneous cosmic rate. The discrete-synchronous alternative, in which all cycles would fire in
lockstep every
τ
=
ln
(2
/
√
3
)
/H
app
giving a logarithmic conversion
ln
(2
1/3
)
/ ln
(2
/
√
3
)
≈
1
.
607,
has no source for the required synchronization and is therefore not a competing physical prediction.
Its only role here is to illustrate the difference between linear and logarithmic conversions of a
per-cycle gain.
Numerically,
√
3/
2
2/3
= 1
.
091124. The rational neighbor 12
/
11 = 1
.
090909 differs by 0.02%,
below the precision of any current measurement. The irrational form is the exact structural
prediction; whether observation can resolve the 0.02% gap is the falsification test discussed in
Section 7.
5
5 Simulation
The cycle described in Section 3 is implemented directly. Code is available at the URL given in
Appendix A.
Setup. An FCC lattice of size
L
= 4 (32 nodes initial, free boundaries) is constructed with
equilibrium bond length
L
eq
= 1. Nearest-neighbor bonds are identified and stored explicitly.
Each cycle scales all positions by 2
/
√
3
, identifies over-stretched bonds, inserts midpoints under
a one-per-node-per-cycle rule, and relaxes via Hooke-law springs to L
eq
with tolerance 10
−5
.
Results. Figure 2 plots the per-cycle ratio across 10 cycles and the node-count growth.
0 2 4 6 8 10
Cycle number
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
Per-cycle ratio
a
eff
/a
app
(a) Per-cycle expansion ratio
Prediction
p
3
/
2
2
/
3
= 1
.
0911
Simulation
Sim mean ± std (cycles 2-10): 1.077 ± 0.088
0 2 4 6 8 10
Cycle number
10
2
10
3
10
4
Node count
N
(b) Node count growth
N
0
·
2
n
(analytic doubling)
Figure 2: (a) Per-cycle ratio
a
eff
/a
app
versus cycle number. Dashed red line: analytic prediction
√
3/
2
2/3
= 1
.
091. The cycle 1 transient (no insertions; bonds just reach threshold) lies below the prediction.
Cycle 2 catches up. Cycles 3–10 cluster around the predicted value, with the mean (1
.
077
±
0
.
088) falling
0
.
4
σ
below the prediction. The remaining deviation is consistent with finite-size boundary effects in the
L
= 4 free-boundary lattice. (b) Node count grows roughly geometrically, tracking
N
0
·
2
n
in the bulk
but lagging in surface-affected regions.
The steady-state mean is consistent with the prediction within the noise. A larger lattice with
periodic boundaries would tighten the simulation result. The free-boundary lattice used here
under-coordinates edge nodes, and the spring relaxation drags inner nodes outward more than
the bulk dynamics would, biasing the per-cycle ratio downward at late cycles (visible in Figure
2a after cycle 6).
The simulation confirms the qualitative mechanism (insertions fire when bonds exceed
L
crit
; the
lattice grows faster than applied stretching) and quantitatively matches the analytic ratio to
within the simulation’s free-boundary noise floor.
6 Comparison with Observation
The prediction in Equation
(1)
maps to a Hubble constant ratio if we identify the CMB-inferred
H
0
with the base cosmic stretching rate (no clustering, no cycle active) and the locally measured
H
0
with the cycle-enhanced rate.
Both quantities are present-day
H
0
values. The label “
H
CMB
0
” refers to the value of the
Hubble constant at
z
= 0 as inferred from cosmic microwave background measurements under
standard ΛCDM assumptions, not the actual expansion rate at the CMB epoch. The expansion
rate at
z ≈
1090 was approximately 1
.
5
×
10
6
km/s/Mpc, four orders of magnitude larger than
today. Planck measures the angular scale of the sound horizon at last scattering and integrates
an assumed
H
(
z
) from
z
= 1090 to
z
= 0 to arrive at
H
0
= 67
.
36
±
0
.
54 km/s/Mpc [
7
]. The
6
locally measured value
H
0
= 73
.
50
±
0
.
81 km/s/Mpc [
2
] samples the expansion directly at
z ≲ 0.1. Both are present-day rates; the difference lies in the inference path.
At the CMB. Last scattering occurs at
z ≈
1090, when matter is approximately uniform
(
δρ/ρ ≈
10
−5
). No clustered structure exists to drive bond stretching. The cycle is inactive.
The ΛCDM integration from
z
= 1090 to
z
= 0 that produces
H
CMB
0
spends nearly all of its
proper-time path in this cycle-off regime, because clustering only completes for
z ≲
2. The
CMB-inferred
H
0
therefore tracks the unenhanced base rate at
z
= 0, not a rate that includes
cycle enhancement.
Today. A fully developed cosmic web fills the universe with clustered matter. Gravitational
stress drives the cycle continuously in surrounding voids. The locally measured
H
0
samples the
cycle-active regime directly and records the enhanced rate at z = 0.
The ratio. See Table 1.
Table 1: Hubble ratio: prediction versus observation.
Quantity Value
H
0
(local), H0DN 2026 [2] 73.50 ± 0.81 km/s/Mpc
H
0
(CMB), Planck 2018 [7] 67.36 ± 0.54 km/s/Mpc
Observed ratio 1.0912 ± 0.0149
Framework prediction
√
3/2
2/3
1.09112
Deviation < 0.01σ
The prediction lands on the central value of the observed ratio to a precision below any current
measurement.
Joint context. Combined with the framework’s prior parameter-free predictions of the five
stable particle masses (electron, muon, pion, proton, neutron) to within 0
.
12% [
5
], and the
proton-mass derivation from defect topology [
6
], the
K=
12 FCC vacuum now matches six
independent observables, five particle and one cosmological, with no free parameters and no
fitting at any step.
7 Falsifiable Predictions
Functional form of
H
(
z
). The enhancement factor scales with the volume fraction
f
void
(
z
) of
cosmologically averaged regions in which the cycle is active:
H(z)
H
base
= 1 +
√
3
2
2/3
− 1
!
f
void
(z) ≈ 1 + 0.091 f
void
(z), (2)
where
H
base
≡ H
CMB
0
is the unenhanced cosmic rate.
f
void
(
z
) tracks structure-formation history.
At
z
= 1090 it is approximately zero. At
z
= 0 it is approximately 0
.
8. The shape between
these endpoints is constrained by N-body simulations and observable through tomographic
H
(
z
)
measurements.
Asymptotic deceleration. The cycle requires sustained gravitational tidal stress to stretch
void bonds past the metric wall. A galaxy cluster’s tidal field on a nearby void bond scales as
GM/r
3
, where
r
is the cluster-void separation. Cosmic expansion separates clusters, so
r
grows
linearly with the scale factor
a
and the tidal field decays as 1
/a
3
. When the field in a given
void falls below the threshold needed to stretch bonds to 2
R
min
, the cycle ceases firing there.
The cycle-active fraction declines over time, and the enhancement factor
√
3/
2
2/3
decays toward
unity. The framework therefore predicts
H
(
t
)
→ H
CMB
0
asymptotically: the local-to-CMB ratio
7
derived in this paper is the present-epoch value, sustained by ongoing clustering, not a permanent
feature of the cosmology. This contrasts with ΛCDM, where the cosmological constant supplies a
constant equation of state
w
=
−
1 and expansion approaches a de Sitter limit
H → H
Λ
>
0. The
two predictions diverge over cosmological timescales. Near-term discrimination is not available
directly, but the predicted decline pattern in
H
(
z
) from
z ∼
1 through
z
= 0 could be looked for
in next-generation tomographic surveys.
Intermediate-redshift
H
0
. Time-delay cosmography at
z ≈
0
.
5 should give an
H
0
falling a
few percent below the local 73
.
5 km/s/Mpc, since clustering at
z ≈
0
.
5 is less complete than at
z
= 0. Current TDCOSMO results have scatter compatible with this prediction but lack the
statistics to resolve the specific functional form [9].
Distinction from the rational neighbor. The prediction
√
3/
2
2/3
= 1
.
09112 differs from
12
/
11 = 1
.
09091 by 0
.
02%. Distinguishing the two requires simultaneous sub-0
.
1% precision on
H
0
(local) and
H
0
(CMB). Euclid and SKA Phase 2 combined
H
(
z
) measurements at
z
= 0
.
1–3
reach this regime [1].
Sky anisotropy. Lines of sight that cut more void volume should infer higher
H
0
than lines
that cut more cosmic-web filament volume. The amplitude is a few percent. The angular pattern
correlates with the local cosmic web orientation. A quadrupolar
H
0
anisotropy aligned with
the supergalactic plane would support the prediction. Current Pantheon+ samples lack the
statistics to resolve this.
Sub-horizon homogeneity. The framework predicts that the enhancement factor
√
3/
2
2/3
is
universal across the late-time observable universe (modulo void-fraction variations). Patches of
sky disagreeing on
H
0
(local) by more than the structure-formation variance would falsify the
framework.
8 Conclusion
The Hubble tension equals
√
3/
2
2/3
≈
1
.
0911 in the Selection-Stitch Model. The prediction
comes from one factor of
√
3
(the inverse of the metric wall, set by the in-plane circumradius
of a bounding triangular face) and one factor of 2
2/3
(the 3D inverse-cube-root scaling from
node-count doubling). It contains no free parameters. It matches the observed local-to-CMB
ratio of 1.0912 ± 0.0149 to better than 0.01σ.
The mechanism is geometric. Gravitational stress from clustered matter stretches bonds in
cosmic voids. When stretching reaches twice the metric wall, the lattice nucleates new nodes.
Each new node demands
K=
12 coordination at equilibrium bond length, pushing surroundings
outward. The cycle produces effective expansion exceeding applied cosmic stretching by
√
3/
2
2/3
per iteration.
Two pieces of work remain. First, a fully rigorous embedding of the Insert operator within SSM
kinematics: the geometric argument in Section 4 pins down the one-insertion-per-node-per-cycle
rate from metric-wall constraints on midpoint coexistence, but a derivation from the stitch–lift
operator algebra of the matter paper would tighten the formal status of Insert as a non-rigid,
tensile-only third operator. Second, a periodic-boundary simulation at larger
L
to drive the
simulation noise floor below 1% and resolve the
√
3/2
2/3
versus 12/11 distinction.
Combined with the framework’s prior parameter-free predictions of the five stable particle masses
(electron, muon, pion, proton, neutron) to within 0
.
12% [
5
,
6
], the
K=
12 FCC vacuum now
matches six independent observables, five particle and one cosmological, with no free parameters.
8
References
[1]
Bernal, J., et al. (2019). The Hubble constant tension with next-generation galaxy surveys.
arXiv:1908.04619.
[2]
H0 Distance Network Collaboration (2026). Direct
H
0
determination from the H0DN program.
NOIRLab release noirlab2611.
[3]
Kamionkowski, M., & Riess, A. G. (2023). The Hubble Tension and Early Dark Energy.
Annual Review of Nuclear and Particle Science 73, 153. doi:10.1146/annurev-nucl-111422-
024107.
[4]
Kulkarni, R. (2026a). A 67%-rate CSS code on the FCC lattice: [[192
,
130
,
3]] from weight-12
stabilizers. arXiv:2603.20294.
[5]
Kulkarni, R. (2026b). The Mass-Energy-Information Equivalence: A bottom-up identifica-
tion of the particle spectrum via FCC lattice error correction. Physics Open 27, 100414.
doi:10.1016/j.physo.2026.100414.
[6]
Kulkarni, R. (2026c). Matter as incomplete crystallization: Quark charges, color confinement,
and the proton mass from a single extra node in the vacuum lattice. Physics Open 27, 100423.
doi:10.1016/j.physo.2026.100423.
[7]
Planck Collaboration (2020). Planck 2018 results. VI. Cosmological parameters. Astronomy
& Astrophysics 641, A6. doi:10.1051/0004-6361/201833910.
[8]
Verde, L., Treu, T., & Riess, A. G. (2019). Tensions between the early and the late Universe.
Nature Astronomy 3, 891. doi:10.1038/s41550-019-0902-0.
[9]
Wong, K. C., et al. (2020). H0LiCOW XIII: A 2.4% measurement of
H
0
from lensed quasars.
Monthly Notices of the Royal Astronomical Society 498, 1420. doi:10.1093/mnras/stz3094.
A Simulation Code
The Python code reproducing the results of Section 5 is available at:
https://github.com/raghu91302/ssmtheory/blob/main/grav_stretch_simulation.py
It uses NumPy and SciPy, builds the FCC lattice, applies the cycle of cosmic stretching plus
midpoint insertion plus spring relaxation described in Sections 3–5, and runs to completion in
under one minute on a standard laptop. Output reproduces the per-cycle ratios reported in
Figure 2.
9
