Geometric Emergence of Spacetime Scales:
Speed of Light Renormalization, the Planck
Scale, and the Two-Step Mass Limit of
Quantum Decoherence
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
March 8, 2026
Abstract
Discretizing spacetime without breaking Lorentz invariance remains an open
problem [1, 2]. Building on the Selection-Stitch Model (SSM) [3], we model the
vacuum as an emergent Face-Centered Cubic (FCC, K = 12) tensor network. In
this framework, the foundational scales of relativity and quantum mechanics emerge
from the geometric limits of a saturated lattice. First, the macroscopic speed of
light (c) acts as a geometric renormalization of the underlying lattice hopping speed
(c = 4v
lattice
), where the factor 4 is the translational eigenvalue of the FCC structure
tensor. Second, we derive the fundamental lattice spacing L = 1.84l
P
via the Ryu-
Takayanagi holographic entanglement map [6,7]. Third, by modeling the vacuum as
a Chiral Cosserat continuum with translational and rotational degrees of freedom,
we show that the angular momentum constraint of a topological defect threading
the lattice generates a chiral coupling whose Euler-Lagrange equations yield the
complex Schr¨odinger equation. Fourth, by intersecting this wave mechanics with
the absolute L/
√
3 kinematic exclusion limit (the metric wall) [4], we establish a two-
step decoherence limit: dimension-8 structural corrections onset at m
soft
≈ 11.8µg
and an absolute structural hard cutoff occurs at m
hard
=
√
3m
soft
≈ 20.5µg. We
compute the dimension-8 phase shift accumulated by a superposed mass in the
transition window and compare it to existing optomechanical data [8].
1 Introduction
The continuous spacetime assumed by Quantum Field Theory leads to ultraviolet diver-
gences when applied to gravity. Discrete spacetime models provide natural regulariza-
tion [9], but rigid grids typically destroy Lorentz invariance at microscopic scales [1]. The
Selection-Stitch Model (SSM) proposes a geometric resolution. Unlike hypercubic models,
the SSM uses the Face-Centered Cubic (FCC) lattice (K = 12), representing the densest
possible sphere packing [10].
1
Computational verification of this network’s kinematics [4] establishes a geometric
exclusion limit—a metric wall at 1/
√
3 of the lattice spacing, beyond which vacuum
nodes cannot compress. This single framework simultaneously generates four results: (i)
the invariant speed of light as a geometric renormalization, (ii) the absolute Planck scale
from holographic entanglement, (iii) the complex Schr¨odinger equation from Cosserat
lattice mechanics, and (iv) a two-step mass threshold for objective quantum decoherence
with quantitative predictions for optomechanical experiments.
2 Geometric Renormalization of Relativity
2.1 The Geometric Boost of Light Speed (c = 4v
lattice
)
Field propagation through the FCC vacuum is governed by a discrete hopping opera-
tor summed over the 12 nearest-neighbor vectors of the cuboctahedral unit cell. The
normalized lattice vectors are:
ˆn
j
∈
1
√
2
(±1, ±1, 0),
1
√
2
(±1, 0, ±1),
1
√
2
(0, ±1, ±1)
(1)
The discrete Dirac operator D
SSM
in momentum space is:
D
SSM
(k) =
1
τ
12
X
j=1
(γ
µ
ˆn
j,µ
)e
ik·(Lˆn
j
)
(2)
In the long-wavelength continuum limit, we Taylor-expand the phase factor. The
zeroth-order term vanishes by the inversion symmetry of the FCC lattice: for every vector
ˆn
j
, there exists −ˆn
j
, so
P
ˆn
j
= 0. The first-order kinetic term depends on the rank-2
structure tensor:
12
X
j=1
ˆn
j,µ
ˆn
ν
j
= 4δ
ν
µ
(3)
This is an exact geometric identity of the FCC bond directions, verified by direct com-
putation. Substituting back into the expanded operator yields the effective continuum
Dirac momentum:
D
SSM
(k) ≈ i
4
L
τ
γ
µ
k
µ
(4)
This identifies the macroscopic speed of light:
c = 4
L
τ
= 4v
lattice
(5)
Light propagates as a collective excitation moving four times faster than the underlying
node-to-node lattice update rate. The factor 4 is the translational eigenvalue of the
FCC structure tensor, not an adjustable parameter. The third-order Lorentz-violating
tensor
P
ˆn
µ
j
ˆn
ν
j
ˆn
λ
j
also vanishes by centrosymmetry. The first nonzero Lorentz-violating
contributions appear at O(L
2
), consistent with current experimental bounds on Lorentz
violation [2].
2
2.2 The Holographic Origin of the Planck Scale (L ≈ 1.84l
P
)
The FCC bulk emerges from a 2D hexagonal (K = 6) boundary tensor network. Each
boundary bond acts as a maximally entangled Bell pair contributing ln 2 to the von
Neumann entropy. For a boundary region with perimeter P , the entanglement entropy
is S
A
= (P/L) ln 2. The FCC bulk is constructed by ABC stacking of hexagonal layers
with interlayer height h =
p
2/3L. The minimal bulk surface (a curtain one layer deep)
has area P ×
p
2/3L. The Ryu-Takayanagi (RT) relation [6], validated for flat isometric
tensor networks by the HaPPY framework [7], equates these:
P
L
ln 2 =
P
p
2/3L
4G
N
(6)
The macroscopic perimeter P cancels. Solving for Newton’s constant:
G
N
=
p
2/3L
2
4 ln 2
(7)
Equating G
N
= l
2
P
(in natural units c = ℏ = 1) and solving for L:
L =
q
2
√
6 ln 2 l
P
≈ 1.8427l
P
(8)
The Planck length is not an arbitrary regulator. It is the macroscopic shadow of the
fundamental 1.84l
P
entanglement spacing of the discrete tensor network.
3 The Geometric Derivation of Quantum Mechanics
3.1 The Chiral Cosserat Action
Standard elasticity models lattice nodes as point masses, yielding real-valued phonons
[11,12]. The SSM extends this by treating the vacuum as a Chiral Micropolar (Cosserat)
continuum, where each node possesses both translational (u) and rotational (θ) degrees
of freedom. The lattice Lagrangian density 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(u
˙
θ − θ ˙u) (9)
The first five terms represent standard massive Cosserat elasticity with equal translational
and rotational stiffness. The final term, Ω(u
˙
θ − θ ˙u), is the chiral coupling. Its origin is a
topological angular momentum constraint, derived as follows.
3.2 Origin of the Chiral Coupling from Lattice Angular Mo-
mentum
A topological defect (knot or braid) threading the lattice introduces a nonzero winding
number. The angular momentum of this defect is conserved and locked to the lattice
microrotation field θ. For a node at position r carrying angular momentum J ∝ (u × ˙u),
conservation of J under parallel transport around the defect generates a gauge connection:
A
0
= ⟨ψ|∂
t
|ψ⟩ = Ω(u
˙
θ − θ ˙u) (10)
3
where Ω is the angular velocity of the defect’s frame dragging. This is the standard
minimal coupling of a Berry connection to a rotating coordinate system [13].
In the absence of topological defects (Ω = 0), the u and θ fields decouple and propagate
as independent real phonon modes—no quantum mechanics. The chiral coupling switches
on only when a topological defect with nonzero winding number is present, linking the
translational and rotational modes into complex wave mechanics.
3.3 Complexification and the Schr¨odinger Limit
The Euler-Lagrange equations for u and θ from Eq. 9 are two coupled real differential
equations. Defining the complex field ψ = u + iθ merges them into a single equation:
∂
2
ψ
∂t
2
+ 2iΩ
∂ψ
∂t
− c
2
∇
2
ψ + ω
2
0
ψ = 0 (11)
The complex unit i is not an abstract axiom. It acts as the geometric operator rotating
between the radial (u) and torsional (θ) lattice modes.
Transforming to the Larmor frame via ψ = Φe
−iΩt
and expanding the time derivatives,
the first-derivative cross-terms cancel exactly, yielding:
∂
2
Φ
∂t
2
− c
2
∇
2
Φ + (ω
2
0
+ Ω
2
)Φ = 0 (12)
This is the Klein-Gordon equation with effective rest mass m
2
c
4
/ℏ
2
= ω
2
0
+ Ω
2
. Factoring
out the rest-mass oscillation via Φ = ϕe
−imc
2
t/ℏ
and applying the non-relativistic envelope
approximation (|∂
2
t
ϕ| ≪ |mc
2
∂
t
ϕ/ℏ|) recovers the free Schr¨odinger equation:
iℏ
∂ϕ
∂t
= −
ℏ
2
2m
∇
2
ϕ (13)
4 The Objective Reduction of Superposition
With continuous wave mechanics derived from the discrete lattice, we can map its breaking
points. The reduced Compton wavelength λ
c
= ℏ/(mc) defines the quantum resolution
scale. Measuring this against the geometric limits (L and L/
√
3) reveals a two-step mass
limit for objective wave collapse.
4.1 The Soft Limit: Dimension-8 Corrections at m
soft
≈ 11.8µg
For a wavefunction to remain consistently represented as a linear oscillation, its wavelength
must exceed the macroscopic lattice spacing L:
λ
c
≥ L ⇒ m
soft
=
ℏ
Lc
=
m
P
p
2
√
6 ln 2
≈ 11.8µg (14)
The holographic map guarantees exact Lorentz invariance for operators of dimension
≤ 4 [5]. The FCC structure tensor is exactly isotropic up to rank 4; discrete artifacts first
appear at rank 6, corresponding to dimension-8 operators in the effective Lagrangian.
At m
soft
, these dimension-8 corrections become O(1). The leading correction modifies
the free-particle dispersion relation:
E
2
= p
2
c
2
+ m
2
c
4
+ α
8
pL
ℏ
4
c
2
p
2
(15)
4
where α
8
is an O(1) coefficient set by the rank-6 structure tensor of the FCC lattice.
For a superposed mass m in a spatial superposition of width ∆x, the accumulated phase
difference between the two branches over time T is:
∆ϕ
8
≈ α
8
m
m
soft
4
∆x
L
4
× (ω
c
T ) (16)
where ω
c
= mc
2
/ℏ is the Compton frequency. For masses below m
soft
, this phase shift
is negligible. For masses approaching m
soft
and ∆x ∼ L, the phase shift reaches O(1)
per Compton period, causing progressive decoherence of the superposition. This is not
an instantaneous collapse but a gradual loss of coherence.
4.2 The Hard Limit: Metric Wall Cutoff at m
hard
≈ 20.5µg
Absolute decoherence occurs when the particle’s Compton wavelength compresses to the
kinematic exclusion limit of the lattice. When λ
c
reaches L/
√
3, the vacuum nodes are
forced against the metric wall. The lattice cannot stretch further to support linear wave
oscillation:
λ
c
≥
L
√
3
⇒ m
hard
=
√
3m
soft
≈ 20.5µg (17)
At this mass, the lattice reaches its kinematic limit. Superposition terminates and the
system reduces to a classical configuration.
4.3 Comparison to the Di´osi-Penrose Model and Current Data
The Di´osi-Penrose (DP) model [14] posits a gravitational instability threshold near the
Planck mass (∼ 21.8µg). The SSM hard limit (20.5µg) agrees to within 6%. However,
the mechanisms differ. The DP model derives its threshold from gravitational self-energy
(∆E ∼ Gm
2
/R) and requires an auxiliary spatial cutoff parameter R
0
to prevent diver-
gence. Recent analysis of a 16.2µg acoustic resonator constrains R
0
≥ 6.2 × 10
−17
m [8].
The SSM hard cutoff is set by L/
√
3 ≈ 1.06l
P
≈ 1.7 ×10
−35
m. This scale differs from
R
0
by 18 orders of magnitude. The discrepancy is not a conflict: R
0
in the DP model
parametrizes the spatial extent of the mass distribution, while L/
√
3 in the SSM sets the
lattice node spacing. The SSM cutoff is a property of the vacuum, not of the superposed
object. The decoherence condition activates when the object’s Compton wavelength—a
quantum property inversely proportional to mass—crosses the lattice scale, regardless of
the object’s physical size. This eliminates the R
0
ambiguity.
The SSM is consistent with the 16.2µg resonator data [8]: the resonator crosses the
11.8µg soft limit, so dimension-8 corrections modify the dispersion relation at O(1). How-
ever, the acoustic mode’s spatial delocalization is ∼ 1.9 × 10
−18
m, and the total mass re-
mains below the 20.5µg hard cutoff. The lattice is not forced against the metric wall, so the
state survives without objective collapse. To trigger geometric collapse, next-generation
levitated optomechanics experiments must achieve a wide spatial superposition (compa-
rable to the object’s diameter) at the 20.5µg threshold.
5 Conclusion
We have shown that the foundational scales of physics emerge from the geometric limits
of a K = 12 discrete lattice. The invariant speed of light (c = 4v
lattice
) is the translational
5
Model Threshold Free Parameters Mechanism
Di´osi-Penrose ∼ 21.8µg R
0
(spatial cutoff) Gravitational self-energy
SSM soft limit 11.8µg None Dimension-8 lattice corrections
SSM hard limit 20.5µg None Metric wall (λ
c
= L/
√
3)
Table 1: Comparison of decoherence thresholds. The SSM produces two parameter-free
thresholds that bracket the single DP threshold.
eigenvalue of the FCC structure tensor. The lattice spacing (L ≈ 1.84l
P
) is fixed by
the RT holographic entanglement map. The complex Schr¨odinger equation arises from
the chiral coupling of translational and rotational lattice modes, activated by topological
defects. Intersecting this wave mechanics with the L/
√
3 metric wall yields a two-step de-
coherence mechanism: dimension-8 corrections onset at 11.8µg (Eq. 14) with quantifiable
phase shifts (Eq. 16), and absolute collapse occurs at 20.5µg (Eq. 17). Both thresholds
are parameter-free, and the hard limit coincides with the Di´osi-Penrose gravitational de-
coherence scale to within 6%. Experimental tests require wide spatial superpositions of
masses in the 11.8 − 20.5µg window.
A Self-Contained SSM Summary
A.1. K = 12 Lattice Saturation. The FCC lattice is the unique solution to the Kepler
conjecture [10]. The densest packing of identical spheres in 3D has coordination K = 12.
Each node has 12 nearest-neighbor bonds of length L/
√
2.
A.2. The Metric Wall at 1/
√
3L. The FCC unit cell’s deepest void lies along
the (111) body diagonal. For hard spheres of diameter L, the minimum center-to-center
distance along this diagonal is L/
√
3. No physical process can compress adjacent nodes
below this separation [4].
A.3. Isometric Tensor Network and Lorentz Invariance. The 3D bulk lattice
is a quasilocal isometric projection of a 2D continuous boundary, following the RT pre-
scription. Because the 2D boundary has exact continuous symmetry, the bulk inherits
macroscopic Lorentz invariance for operators of dimension ≤ 4 [5].
A.4. Holographic Origin of G
N
. The RT formula maps boundary entangle-
ment entropy (S
A
= (P/L) ln 2) to the minimal bulk surface area. This derives G
N
=
p
2/3L
2
/(4 ln 2). Equating this to the Planck area fixes L = 1.84l
P
[6, 7].
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