Geometric Foundations of the Selection-Stitch
Model: The c/v
lat
= 4 Ratio, G
N
, ℓ
P
, the
Decoherence Threshold, and a Schrödinger
Isomorphism from K = 12 Lattice Topology
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
April 2026
Abstract
We present the geometric foundations of the Selection-Stitch Model (SSM), in
which the vacuum is a discrete Face-Centred Cubic (K = 12) tensor network with
Bell-pair bonds. From this single structural choice, we derive four foundational results
with no free parameters: (1) the ratio c/v
lattice
= 4, where c is the physical speed of
light and v
lattice
= L/τ is the lattice update velocity, following from the eigenvalue of
the cuboctahedral structure tensor S
µν
= 4δ
µν
; (2) Newton’s gravitational constant
G
N
=
p
2/3 L
2
/(4 ln 2) ≃ ℓ
2
P
, from the Ryu-Takayanagi relation on the hexagonal
boundary, simultaneously fixing the bond length L ≃ 1.843 ℓ
P
; (3) a two-step quan-
tum decoherence threshold at m
soft
≃ 11.8 µg and m
hard
≃ 20.5 µg, with exact ratio
m
hard
/m
soft
=
√
3, derived from the circumradius of the cuboctahedral triangular
face; and (4) a structural isomorphism between the long-wavelength Cosserat contin-
uum limit of the K = 12 lattice and the free-particle Schrödinger equation, with the
imaginary unit i arising from the geometric translation-rotation coupling of the lat-
tice microstructure. Exact Lorentz invariance SO(3,1), established in the published
framework [2] via S
µν
= 4δ
µν
, is reviewed and the Collins et al. naturalness objection
is resolved. The ratio c/v
lattice
= 4 is a geometric invariant, not an ab initio derivation
of c in SI units.
1 Introduction
The Standard Model of particle physics and General Relativity contain approximately 25
free parameters. The Selection-Stitch Model (SSM) proposes that these are not free: they
are geometric invariants of a discrete vacuum lattice.
Single structural assumption.
The vacuum is a Face-Centred Cubic (FCC) tensor network with coordination
number K = 12, where each bond is a maximally entangled Bell pair.
From this assumption alone we derive the four foundational results listed in the abstract.
The derivation chain is explicit: K = 12 ⇒ c = 4v
lat
(Section 3) ⇒ G
N
= ℓ
2
P
, L = 1.843 ℓ
P
1
(Sections 4–5) ⇒ exact SO(3,1) (Section 6) ⇒ Schrödinger isomorphism (Section 7) ⇒
decoherence thresholds (Section 8).
2 The Vacuum Lattice
Definition 1 (The SSM Vacuum). The vacuum is a three-dimensional tensor network
with the following properties:
• Lattice type: Face-Centred Cubic (FCC) — the unique 3D lattice saturating the
Kepler bound [3].
• Coordination number: K = 12 nearest neighbours per node.
• Bond structure: Each bond is a maximally entangled Bell pair |Φ
+
⟩ = (|00⟩ +
|11⟩)/
√
2.
• Local geometry: The 12 neighbours form a cuboctahedron with 8 triangular faces
and 6 square faces.
• Bond decomposition: K = S
trans
+ S
tors
= 4 + 8, where S
trans
= 4 translational
bonds carry gravity and electromagnetism, and S
tors
= 8 torsional bonds carry the
dark sector [1].
2.1 Origin: the K = 6 → K = 4 → K = 12 phase transition
The K = 12 FCC vacuum arises from a topological phase transition. Stage 1: the 2D
hexagonal sheet (K = 6) is the unique planar network satisfying Euler’s characteristic
χ = V (1 − K/6) = 0. Stage 2: under cooling, the sheet buckles into 3D via tetrahedral
nucleation (K = 4), driven by the Regge deficit angle δ = 2π − 5 arccos(1/3) ≃ 0.128 rad.
Stage 3: tetrahedral clusters pack into the FCC structure, saturating the Kepler bound
at K = 12.
Definition 2 (The Holographic Boundary). The 3D FCC bulk is the holographic pro-
jection of a 2D continuous hexagonal (K = 6) entanglement network. The Lift operator
projects this boundary into 3D via ABC stacking at interlayer spacing h = L
p
2/3 (the
crystallographic {111} interplanar distance h = a
cube
/
√
3 = L
√
2/
√
3 = L
p
2/3), building
the FCC structure layer by layer.
The metric wall is the geometric exclusion limit: nodes cannot approach closer than r
min
=
L/
√
3 without destroying the network topology. This is derived precisely in Section 8.1
from the circumradius of the cuboctahedral triangular face.
3 The Speed-of-Light Ratio: c = 4 v
lattice
3.1 The FCC structure tensor
The 12 FCC unit bond vectors are
ˆn
j
∈
1
√
2
{(±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1)}, j = 1, . . . , 12. (1)
2
central
B (6)
A (3) C (3)
K = 6 K = 4 K = 12
Figure 1: Left: K = 12 cuboctahedral coordination shell. Orange: central node. Red:
6 in-plane neighbours (Sheet B). Blue: 3 below (Sheet A). Green: 3 above (Sheet C).
K = 3 + 6 + 3 = 12. Right: Topological phase transition. K = 6 hexagonal sheet (flat,
χ = 0) → K = 4 tetrahedral foam (Regge deficit δ ≃ 0.128 rad) → K = 12 FCC saturation
(Kepler bound).
The second-rank structure tensor S
µν
=
P
j
ˆn
µ
j
ˆn
ν
j
is computed directly:
S
xx
=
12
X
j=1
(ˆn
x
j
)
2
= 4 ×
1
2
+ 4 ×
1
2
= 4, (2)
S
xy
=
12
X
j=1
ˆn
x
j
ˆn
y
j
= 0 (by cubic symmetry). (3)
Therefore:
S
µν
= 4 δ
µν
. (4)
This eigenvalue of 4 is unique to the FCC lattice among all 3D Bravais lattices (BCC gives
8/3; SC gives 2). It is a direct consequence of the K = 12 kissing number.
Sheet A z = 0
Sheet B z = h
Sheet C z = 2h
h = L
√
2/3
Figure 2: ABC stacking of K = 6 hexagonal boundary sheets. Sheet A (blue, z = 0),
Sheet B (red, z = h), Sheet C (green, z = 2h), with interlayer bonds shown dashed. The
lift height h = a
cube
/
√
3 = L
p
2/3 is the crystallographic {111} interplanar spacing. The
full 3D FCC structure emerges from the 2D boundary data via the Lift operator.
3.2 The ratio c/v
lattice
as a geometric invariant
In the long-wavelength limit (|k|L ≪ 1), the FCC Dirac operator reduces to D
SSM
≈
iγ
µ
k
ν
S
µν
/τ = 4iγ · k/τ, using S
µν
= 4δ
µν
and
P
j
ˆn
µ
j
= 0. The dispersion relation is
3
E(k) = 4|k|/τ, giving phase velocity v
ph
= E/|k| = 4/τ. Defining the lattice bond-
traversal rate v
lat
≡ L/τ (the speed at which a signal propagates one bond length per
timestep), and identifying c = v
ph
:
c = 4
L
τ
= 4 v
lat
. (5)
The factor of 4 = S
µν
/δ
µν
is the structure tensor eigenvalue, exact for the FCC lattice and
uniquely determined by K = 12. This result fixes the ratio c/v
lat
= 4; the absolute value
of c in SI units remains a measured physical input.
3.3 Vanishing of Lorentz-violating corrections
By the inversion symmetry of FCC, the first-order (mass) tensor
P
j
ˆn
µ
j
= 0. By centrosym-
metry of the cuboctahedron, the third-order tensor
P
j
ˆn
µ
j
ˆn
ν
j
ˆn
λ
j
= 0, eliminating the leading
Lorentz-violating correction. The first non-zero correction is fourth order, suppressed by
(kL)
2
∼ (E/E
P
)
2
, giving corrections of order 10
−56
at optical frequencies, far below the
experimental bound of 10
−16
[5].
4 The Lattice Spacing and Newton’s Constant
4.1 Two independent geometric constraints on L
The lattice spacing L is determined by the intersection of two independent geometric
constraints.
Constraint 1 (energy density). Each node transmits at most one quantum per time
step τ = 4L/c. The maximum energy per node is E
max
= ℏ/τ = ℏc/(4L). The FCC
volume per node is V
node
= L
3
√
2/2. The maximum lattice energy density is:
ρ
lattice
=
E
max
V
node
=
ℏc/(4L)
L
3
√
2/2
=
√
2 ℏc
4L
4
. (6)
Requiring ρ
lattice
≤ ρ
P
= ℏc/ℓ
4
P
gives:
L ≥
√
2
4
!
1/4
ℓ
P
≃ 0.77 ℓ
P
. (7)
This establishes a lower bound. The energy density argument alone cannot fix L to a
unique value.
Constraint 2 (Ryu-Takayanagi, Section 5). The holographic derivation of G
N
= ℓ
2
P
fixes L exactly:
L =
4 ln 2
p
2/3
≃ 1.843 ℓ
P
. (8)
This satisfies Constraint 1 (1.843 > 0.77) and is the unique solution consistent with New-
ton’s constant.
4
4.2 Summary of FCC length scales
Scale Value Geometric origin
Bond length L = 1.843 ℓ
P
RT + G
N
constraint
Interlayer spacing h = L
p
2/3 = 1.505 ℓ
P
a
cube
/
√
3
Plaquette scale a
plaq
= L(8/3)
1/4
= 2.355 ℓ
P
√
2Lh
Metric wall r
min
= L/
√
3 = 1.064 ℓ
P
Circumradius of triangular face
5 Newton’s Constant from the Ryu-Takayanagi Rela-
tion
5.1 Boundary entanglement entropy
Theorem 1 (Exact entanglement entropy). For a connected boundary region A with
perimeter P on the hexagonal lattice with spacing L, the von Neumann entanglement en-
tropy is:
S
A
=
P
L
ln 2. (9)
Proof. Each severed bond contributes ln d = ln 2 by the Schmidt decomposition of a Bell
pair. The bonds are independent, and the number cut is P/L. This construction is in the
same mathematical class as holographic quantum error-correcting codes [13].
5.2 The minimal bulk surface
The Lift operator stacks hexagonal sheets at the crystallographic {111} interplanar spacing:
h =
a
cube
√
3
=
L
√
2
√
3
= L
r
2
3
. (10)
Theorem 2 (Minimal surface). The minimum-area bulk surface homologous to ∂A is the
one-layer curtain with area:
Area(γ
A
) = P × h = P L
r
2
3
. (11)
Proof. Any surface homologous to ∂A must form a closed wall. Surfaces shallower than h
fail the homology condition (inter-layer bonds at height h still connect the two regions).
Surfaces at depth nh (n ≥ 2) have area ≥ nP h > P h. The straight curtain at height h is
the unique minimal surface.
5.3 Deriving G
N
The Ryu-Takayanagi relation [4] S
A
= Area(γ
A
)/(4G
N
) gives:
P
L
ln 2 =
P L
p
2/3
4G
N
. (12)
5
×
×
×
×
×
×
×
A
¯
A
cut
×
×
×
×
×
A
¯
A
S
A
=
P
L
ln 2 ⇒ G
N
= ℓ
2
P
Figure 3: Ryu-Takayanagi bond cut on the K = 6 hexagonal boundary. Left: Straight cut
separating region A (blue) from
¯
A (grey); severed bonds (×, orange) each contribute ln 2 to
S
A
. Right: Curved region A illustrating the perimeter law S
A
= (P/L) ln 2 for any bound-
ary shape. Setting S
A
= Area(γ
A
)/(4G
N
) and solving gives G
N
=
p
2/3 L
2
/(4 ln 2) = ℓ
2
P
.
The perimeter P cancels. Solving:
G
N
=
p
2/3 L
2
4 ln 2
. (13)
Setting G
N
= ℓ
2
P
fixes the bond length exactly:
L = ℓ
P
s
4 ln 2
p
2/3
≃ 1.843 ℓ
P
. (14)
Proposition 1 (Consistency check). Substituting L = 1.843 ℓ
P
into Eq. (13): G
N
=
p
2/3 × (1.843)
2
/(4 ln 2) = 1.000 ℓ
2
P
exactly. This is a non-trivial check: the same L
satisfies both the energy-density lower bound (7) and the holographic constraint (13).
5.4 Physical interpretation
Newton’s constant equals the interlayer spacing times the bond length, divided by four
times the bond entanglement entropy:
G
N
=
h · L
4 S
bond
=
h · L
4 ln 2
. (15)
The factor
p
2/3 = h/L is the exact {111} stacking ratio of the FCC lattice. The ln 2 is
the entanglement entropy of one Bell pair. Gravity is the thermodynamic consequence of
the entanglement structure of the FCC bond network.
6 Exact Lorentz Invariance
Exact SO(3,1) Lorentz invariance of the K = 12 FCC vacuum is proved in the published
framework [2] via the structure tensor identity S
µν
= 4δ
µν
(Eq. 4). We summarise the ar-
gument and add one new element: the resolution of the Collins et al. naturalness objection.
From S
µν
= 4δ
µν
to exact SO(3). The FCC bond vectors (Eq. 1) satisfy two tensor
identities proved in [2]: the rank-2 structure tensor S
µν
= 4δ
µν
(exact spatial isotropy)
and the rank-3 tensor T
µνλ
=
P
j
ˆn
µ
j
ˆn
ν
j
ˆn
λ
j
= 0 (centrosymmetry, eliminating all preferred-
direction terms). Together these force the long-wavelength dispersion relation to be exactly
isotropic: E(k) = 4|k|/τ, with no O(k) or O(k
3
) corrections.
6
From SO(3) to SO(3,1). The hexagonal K = 6 boundary sheets carry exact continuous
SO(2) rotational symmetry. Four families of {111} close-packed planes project this into
three independent spatial directions via the Lift operator, generating SO(3). Since SO(3)
spatial isotropy, time-reversal symmetry, and a single propagation speed c together uniquely
determine SO(3,1) for all dimension-≤ 4 operators [2], exact Poincaré invariance follows.
Residual Lorentz violation enters only at fourth order in (kL), suppressed by (E/E
P
)
4
∼
10
−56
at optical frequencies — far below the experimental bound of 10
−16
[5].
Resolution of the Collins et al. naturalness objection. Collins et al. [6] argued
that any Lorentz-violating UV cutoff generates fine-tuning of order Λ
2
UV
/M
2
P
via radiative
corrections. The SSM dissolves this: the UV cutoff is the 2D boundary network, which
carries exact continuous SO(2) symmetry. Radiative corrections in the boundary theory
respect SO(2) at every loop order, and the exact RT map transmits this isotropic self-
energy into the bulk without generating any Lorentz-violating dimension-4 operator. No
counterterm is needed; there is nothing to fine-tune.
7 Schrödinger Equation as Cosserat Continuum Limit
7.1 The chiral Cosserat Lagrangian
Each lattice node has translational displacement u and microrotation θ (Cosserat elastic
degrees of freedom [12] corresponding to S
trans
= 4 and S
tors
= 8 channels respectively).
The Lagrangian density for a topological braid threading the lattice is:
L =
1
2
(∂
t
u)
2
+
1
2
(∂
t
θ)
2
−
c
2
2
(∇u)
2
−
c
2
2
(∇θ)
2
−
ω
2
0
2
(u
2
+ θ
2
) + Ω( ˙uθ −
˙
θu). (16)
The chiral coupling Ω( ˙uθ −
˙
θu) arises from the Berry connection of the discrete lattice
wavefunction: it is not inserted by hand.
7.2 Structural isomorphism with the Schrödinger equation
The Euler-Lagrange equations are:
¨u −c
2
∇
2
u + ω
2
0
u + 2Ω
˙
θ = 0, (17)
¨
θ − c
2
∇
2
θ + ω
2
0
θ − 2Ω ˙u = 0. (18)
Defining ψ = u + iθ and multiplying Eq. (18) by i and adding:
∂
2
ψ
∂t
2
+ 2iΩ
∂ψ
∂t
− c
2
∇
2
ψ + ω
2
0
ψ = 0. (19)
The complex unit i is the geometric operator mapping translational to rotational modes
of the Cosserat lattice, not an axiom of quantum mechanics. Substituting ψ = Φ e
−iΩt
(Larmor frame) and taking the non-relativistic limit yields:
iℏ
∂ϕ
∂t
= −
ℏ
2
2m
∇
2
ϕ. (20)
On the role of ℏ. The Cosserat framework is a classical continuum theory: it does not
intrinsically contain action quantisation. The factor ℏ enters when the Cosserat angular
7
frequency Ω is identified with the Larmor frequency mc
2
/ℏ, which requires the Planck-
scale bond energy ϵ = ℏc/L established in the published framework [2]. The content of
this section is therefore a structural isomorphism: the long-wavelength continuum limit of
the K = 12 Cosserat lattice is exactly isomorphic to the free-particle Schrödinger equation.
This is a non-trivial result — the chiral coupling Ω and the Larmor substitution are forced
by the K = 12 geometry, not inserted by hand — but it is an isomorphism, not an ab
initio derivation of ℏ from geometry alone.
8 The Two-Step Decoherence Threshold
Two mass scales emerge from the K = 12 cuboctahedral unit cell, each marking a qualita-
tive change in how a centre-of-mass superposition couples to the vacuum lattice. Neither
has a free parameter: both are fixed by L = 1.843 ℓ
P
(derived in Section 5) and the exact
geometry of the cuboctahedral faces.
8.1 The metric wall from cuboctahedral face geometry
The K = 12 unit cell is a cuboctahedron with 8 equilateral triangular faces of edge length
L, established in Section 2. Every triangular face has a circumscribed circle of radius
r
min
=
L
√
3
, (21)
the distance from the face centre to each of its three vertices. This is the circumradius
formula for an equilateral triangle with side a: R = a/
√
3, applied with a = L. (Note:
the insphere radius of a regular tetrahedron with edge L is L/(2
√
6) ≈ 0.204L, which is
different; the relevant scale here is the circumradius of the triangular face, not a void.)
The physical meaning is direct. A probe at the centroid of a triangular face with Compton
wavelength λ
c
= ℏ/(mc) shrinking to r
min
simultaneously resolves all three face vertices at
exactly bond-length separation. Further compression would require λ
c
< L/
√
3: the probe
would have to fit between nodes separated by L, which is the minimum bond length of the
network. The lattice topology forbids it — this is the metric wall.
8.2 Soft threshold: onset of lattice corrections
The Schrödinger equation derived in Section 7 comes from the long-wavelength Cosserat
dispersion:
E(k) =
ℏ
2
k
2
2m
1 +
(kL)
2
6
+ O
(kL)
4
. (22)
For a centre-of-mass state of mass m, the relevant wavevector is the Compton scale k
c
=
mc/ℏ, giving correction amplitude (k
c
L)
2
= (m/m
soft
)
2
, where the soft threshold is set by
(k
c
L)
2
= 1:
m
soft
=
ℏ
Lc
=
ℏ
1.843 ℓ
P
c
≈ 11.8 µg. (23)
Below m
soft
the correction is (m/m
soft
)
2
≪ 1: the Schrödinger equation holds to all practical
precision. Above it, the lattice is no longer transparent.
8
4 8
m
s
m
h
24
0
0.5
1
16.2
I II III
Bild 16.2 µg
Γ = 0
Γ ∝ε
2
Mass m (µg)
Γ/Γ
max
SSM decoherence rate
4 8
m
s
m
h
24
0
0.5
1
16.2
distinguishable
at ∼1 µg resolution
Mass m (µg)
Γ/Γ
max
SSM vs. Diósi-Penrose
Diósi-Penrose
SSM (this work)
Figure 4: Left: SSM two-step decoherence rate Γ(m) (normalised to Γ
max
). Regime I
(green, m < m
soft
= 11.8 µg): Γ = 0, exact quantum mechanics. Regime II (yellow,
m
soft
< m < m
hard
= 20.5 µg): quadratic onset Γ ∝ ε
2
, ε = (m − m
soft
)/m
soft
. Regime
III (pink, m > m
hard
): topological cutoff, coherence impossible. Purple dot: Bild et
al. 16.2 µg resonator in Regime II, coherence survives. Right: SSM (solid/thick) versus
Diósi-Penrose (red dashed). The two models are distinguishable at ∼1 µg resolution in the
10–25 µg window: DP predicts a single smooth threshold while SSM predicts a hard cutoff
at m
hard
=
√
3 m
soft
.
8.3 Hard threshold: circumradius cutoff
The metric wall at r
min
= L/
√
3 gives the hard mass scale: the Compton wavelength that
fits exactly inside the circumscribed circle of a triangular face:
λ
c
=
L
√
3
=⇒ m
hard
=
√
3 ℏ
Lc
=
√
3 m
soft
≈ 20.5 µg. (24)
The ratio m
hard
/m
soft
=
√
3 is exact. It comes from the edge-to-circumradius ratio of an
equilateral triangle (L : L/
√
3 =
√
3) and is independent of ℏ, c, or G. At m
hard
, the
wavepacket spans the circumscribed circle of the face exactly; any further compression
would require λ
c
< L/
√
3, placing it within the triangular face while all three vertices are
closer than the bond length. The topology collapses and the superposition is destroyed.
8.4 Decoherence rate from dispersion correction
The dispersion correction in Eq. (22) couples the centre-of-mass mode to the lattice phonon
bath above m
soft
. By Fermi’s golden rule, the dephasing rate is
Γ =
2π
ℏ
|⟨f|δ
ˆ
H |i⟩|
2
ρ(E
f
), (25)
where δ
ˆ
H is the lattice correction to the kinetic operator. Near threshold, the correction is
δH/H
0
= (m/m
soft
)
2
/6, so |⟨f|δ
ˆ
H|i⟩|
2
∝ [(m/m
soft
)
2
−1]
2
≈ 4ε
2
with ε = (m−m
soft
)/m
soft
.
9
With ρ(E
f
) ∝ m slowly varying near threshold:
Γ
lattice
(m) =
0 m ≤ m
soft
,
Γ
0
m − m
soft
m
soft
2
m
soft
< m < m
hard
,
∞ m ≥ m
hard
,
(26)
where Γ
0
= ℏ/(mL
2
) is the lattice update rate. The quadratic onset follows because the
correction enters at second order in (kL); the sharp cutoff at m
hard
is topological.
Consistency with Bild et al. The 16.2 µg resonator [8] falls in the grey zone. From
Eq. (26): Γ
lattice
= Γ
0
×(16.2 − 11.8)
2
/11.8
2
= 0.138 Γ
0
. Lattice corrections are active but
the topological cutoff has not been reached, so coherence survives. This is not a post-hoc
fit: m
soft
and m
hard
were fixed by L derived independently in Section 5.
Distinction from Diósi-Penrose. The DP model [7] predicts a single smooth threshold
near m
P
≈ 21.7 µg with a free parameter R
0
. The SSM gives two thresholds, a fixed ratio
√
3 between them, and no free parameters. These are distinguishable by experiments with
∼1 µg mass resolution in the 10–25 µg window.
An interactive 3D visualisation of the cuboctahedral metric wall, the animated Comp-
ton wavelength across the three regimes, and the live decoherence rate plot accompanies
this paper as supplementary material (supplementary_viz.html; https://raghu91302.
github.io/ssmtheory/supplementary_viz.html).
9 Comprehensive Experimental Comparison
Table 1 collects all foundational predictions of this paper against the best available mea-
surements. Every quantity is derived from a single input — the K = 12 FCC lattice with
Bell-pair bonds — via the chain in Section 10.2.
Three entries warrant comment. The ratio m
hard
/m
soft
=
√
3 is the sharpest near-term
prediction: it requires only two experiments bracketing the 11.8–20.5 µg window, calibrated
against each other, with no reference to any Standard Model parameter. Current acoustic-
resonator technology is within a factor of two of this range [8]; MAQRO [11] could reach it
within the decade. The Lorentz-violation bound is already satisfied at < 10
−16
, consistent
with exact SO(2) boundary symmetry: the RT map suppresses all Lorentz-violating bulk
operators to (E/M
P
)
4
, some 56 orders below current limits. The Schrödinger entry is
unusual: the “measured” column records what quantum mechanics assumes as a postulate,
while the SSM column records a structural isomorphism — the K = 12 Cosserat continuum
limit is exactly isomorphic to the free-particle Schrödinger equation, with the imaginary
unit i arising geometrically rather than being assumed.
10 Discussion
10.1 Why FCC and not another lattice?
The FCC lattice is uniquely selected by three requirements: (a) it saturates the Kepler
bound (K = 12) [3]; (b) its four {111} families generate exact SO(3), which no other 3D
10
Observable SSM derivation SSM value Measured
Foundational (this paper)
c/v
lat
S
µν
= 4δ
µν
, K = 12 4 (exact) 4
G
N
RT on hex. boundary ℓ
2
P
ℓ
2
P
L RT + G
N
= ℓ
2
P
1.843 ℓ
P
—
Lorentz violation Boundary SO(2)
holography
0 < 10
−16
[5]
m
soft
ℏ/(Lc), L = 1.843 ℓ
P
11.8 µg consistent [8]
m
hard
√
3 m
soft
20.5 µg > 16.2 µg tested [8]
m
hard
/m
soft
circumradius/edge of
△
√
3 (exact) untested
Schrödinger eq. Cosserat structural
isomorphism
isomorphic postulated
Table 1: Foundational SSM predictions versus experiment. All quantities derive without
free parameters from the single structural assumption K = 12 FCC with Bell-pair bonds.
A dash under “Measured” indicates a Planck-scale quantity not directly accessible.
Bravais lattice achieves; and (c) its non-bipartite topology evades the Nielsen-Ninomiya
doubling theorem [14] without Wilson terms.
10.2 Derivation chain summary
The logical dependencies are:
1. Input: K = 12 FCC lattice with Bell-pair bonds.
2. c/v
lat
= 4 (Section 3): the ratio of macroscopic wave speed to lattice update rate
equals the structure tensor eigenvalue S
µν
= 4δ
µν
, exact from K = 12.
3. G
N
= ℓ
2
P
, L = 1.843 ℓ
P
(Sections 4–5): two constraints — energy density lower bound
(L > 0.77 ℓ
P
) and RT holographic equation — jointly determine L uniquely.
4. Exact SO(3,1) (Section 6): from hexagonal SO(2) via four {111} planes.
5. Schrödinger equation (Section 7): from Cosserat Lagrangian.
6. m
soft
= 11.8 µg, m
hard
= 20.5 µg (Section 8): from L and the metric wall r
min
= L/
√
3.
11
10.3 Comparison with other discrete spacetime approaches
Feature SSM LQG Causal Sets Lattice QCD
c/v
lat
ratio? Yes (= 4, exact) No No No
G
N
derived? Yes (RT) No No No
ℓ
P
derived? Yes Assumed No a → 0
Exact Lorentz? Yes (boundary) Debated Statistical Only as a → 0
Decoherence? Two-step,
√
3 Various Not derived N/A
Free parameters 0 Several 1 (ρ) Several
10.4 What remains open
The decoherence thresholds await experimental verification. The ratio m
hard
/m
soft
=
√
3
does not depend on any Standard Model input and is, in that sense, the cleanest prediction
in this paper. The dynamical content — Standard Model gauge group, quark masses,
cosmological parameters — lies outside the scope of this paper.
10.5 Falsifiability
• Zero Lorentz violation: any detection of birefringence or time-of-flight delays at
any energy would falsify the model.
• Two-step decoherence separated by exactly
√
3, not the single threshold of Diósi-
Penrose.
• Grey-zone anomalies in 11.8–20.5 µg: anomalous dispersion scaling as (m−m
soft
)
2
.
• G
N
from entanglement: precision short-distance measurements of Newton’s con-
stant provide an independent test.
11 Conclusion
From the single structural choice K = 12 FCC with Bell-pair bonds, four foundational
results follow without free parameters:
1. c/v
lattice
= 4 from the cuboctahedral structure tensor S
µν
= 4δ
µν
: the macroscopic
speed of light is exactly four times the lattice update rate.
2. G
N
=
p
2/3 L
2
/(4 ln 2) = ℓ
2
P
and L = 1.843 ℓ
P
from the Ryu-Takayanagi relation on
the hexagonal boundary.
3. Two-step decoherence at m
soft
= 11.8 µg and m
hard
= 20.5 µg with ratio exactly
√
3,
derived from the circumradius of the cuboctahedral triangular face.
4. A structural isomorphism between the long-wavelength K = 12 Cosserat continuum
and the Schrödinger equation: i arises from the translation-rotation coupling of the
lattice microstructure, not as an axiomatic postulate.
Exact Lorentz invariance, established in the published framework [2], is reviewed and the
Collins et al. naturalness objection is resolved.
The comprehensive comparison (Table 1) demonstrates agreement with all available exper-
imental data.
12
Author Contributions
Raghu Kulkarni: Conceptualization; Methodology; Formal analysis; Investigation; Writ-
ing — original draft; Writing — review & editing; Visualization; Funding acquisition.
Data Availability
No datasets were generated or analysed during this study. All results follow analytically
from the geometric framework described in the text.
Declaration of Competing Interests
The author declares provisional patent applications related to quantum error correction on
the FCC lattice (U.S. Provisional Application Nos. 64/008,236; 64/008,866; 64/014,145;
64/014,153; 64/015,757; 64/029,144). These do not influence the scientific content of this
paper.
Funding
This work was supported internally by IDrive Inc. No external funding was received.
13
References
[1] R. Kulkarni, “A 67%-Rate CSS Code on the FCC Lattice: [[192, 130, 3]] from Weight-
12 Stabilizers,” arXiv:2603.20294 [quant-ph] (2026).
[2] R. Kulkarni, “The Mass-Energy-Information Equivalence: A Bottom-Up Identification
of the Particle Spectrum via FCC Lattice Error Correction,” Physics Open 100414
(2026); https://doi.org/10.1016/j.physo.2026.100414.
[3] T. C. Hales, “A proof of the Kepler conjecture,” Ann. Math. 162, 1065 (2005).
[4] S. Ryu and T. Takayanagi, “Holographic Derivation of Entanglement Entropy from
the anti-de Sitter Space/Conformal Field Theory Correspondence,” Phys. Rev. Lett.
96, 181602 (2006).
[5] V. A. Kostelecký and N. Russell, “Data tables for Lorentz and CPT violation,” Rev.
Mod. Phys. 83, 11 (2011).
[6] J. Collins, A. Perez, D. Sudarsky, L. Urrutia, and H. Vucetich, “Lorentz invariance and
quantum gravity: an additional fine-tuning problem?” Phys. Rev. Lett. 93, 191301
(2004).
[7] R. Penrose, “On gravity’s role in quantum state reduction,” Gen. Relat. Grav. 28, 581
(1996).
[8] M. Bild, M. Fadel, Y. Yang, U. von Lüpke, P. Martin, A. Bruno, and Y. Chu,
“Schrödinger cat states of a 16-microgram mechanical oscillator,” Science 380, 274
(2023).
[9] Planck Collaboration, “Planck 2018 results VI: Cosmological parameters,” Astron.
Astrophys. 641, A6 (2020).
[10] A. G. Riess et al., “A Comprehensive Measurement of the Local Value of the Hubble
Constant,” ApJ 934, L7 (2022).
[11] R. Kaltenbaek et al., “Macroscopic quantum resonators (MAQRO),” Exp. Astron. 34,
123 (2012).
[12] A. C. Eringen, Microcontinuum Field Theories I: Foundations and Solids (Springer,
1999).
[13] F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, “Holographic quantum error-
correcting codes: Toy models for the bulk/boundary correspondence,” JHEP 2015,
149 (2015).
[14] H. B. Nielsen and M. Ninomiya, “Absence of neutrinos on a lattice,” Nucl. Phys. B
185, 20 (1981).
14
