Discrete Wave Mechanics: Deriving the Schr¨odinger Equation and the
Mass Limit of Quantum Superposition from Vacuum Lattice Sintering
Raghu Kulkarni
1, ∗
1
Independent Researcher, Calabasas, CA
(Dated: February 8, 2026)
Standard Quantum Mechanics treats the complex wavefunction and its first-order time evolution
as fundamental postulates. Expanding on the Selection-Stitch Model (SSM), we propose that these
properties are emergent consequences of a discrete, crystallized vacuum. We define a ”particle”
as a stable K = 13 topological defect within a K = 12 Face-Centered Cubic (FCC) lattice. We
model this vacuum as a Chiral Micropolar Continuum, where nodes possess both translational and
rotational degrees of freedom. We explicitly derive the isotropic Laplacian from the 12 nearest-
neighbor forces and introduce a novel Chiral Velocity Coupling arising from the Berry connection
of the defect’s topology. We demonstrate that this coupling naturally generates the complex unit i,
the global U (1) symmetry, and the exact Schr¨odinger equation in the non-relativistic limit. Finally,
using the geometrically renormalized lattice spacing (a ≈ 0.77l
P
) derived in previous work, we
predict a specific mass limit for quantum coherence at m
max
≈ 28µg. This distinguishing prediction
aligns remarkably with the ”Planck Mass” collapse criteria of the Di´osi-Penrose model, identifying
the physical mechanism for the transition from quantum superposition to classical gravity as a
geometric resolution limit of the vacuum.
I. INTRODUCTION
The duality of matter remains the central mystery of quantum foundations. While the Schr¨odinger
equation predicts experimental outcomes with precision, it lacks a mechanical substrate. Ap-
proaches by ’t Hooft [1] and Volovik [2] suggest that quantum behavior emerges from deterministic
discrete dynamics.
The Selection-Stitch Model (SSM) [4] operationalizes this by modeling the vacuum as an elastic
lattice capable of ”sintering” phase transitions (K = 12 → 13). We show that when the vacuum
is treated as a Chiral Cosserat Continuum, the classical lattice dynamics naturally generate the
complex wave equation.
II. GEOMETRIC FRAMEWORK
We adopt the structural foundation of the SSM, which identifies the Face-Centered Cubic (FCC)
lattice (Cuboctahedron coordination) as the mechanically stable ground state of the vacuum [5].
A. Cosserat Mechanics and Chirality
While the lattice geometry is fixed, the mechanics of the nodes determine the emergent physics.
Standard elasticity treats nodes as point masses, yielding real-valued phonons. To recover Quantum
∗
raghu@idrive.com
2
Mechanics, we model the vacuum as a Micropolar (Cosserat) Continuum, where nodes possess both
translational (u) and rotational (θ) degrees of freedom.
• Spin: The rotational mode θ provides the physical basis for particle spin and rest mass.
• Chirality: The K = 13 defect is a topological braid (Trefoil knot). This introduces Chirality,
creating the cross-coupling that generates the complex unit i.
III. DERIVATION FROM THE LATTICE LAGRANGIAN
A. Discrete-to-Continuum Limit (The Isotropic Laplacian)
The Lagrangian formulation requires a continuous spatial derivative operator. We explicitly
derive this from the discrete FCC lattice to justify the use of the isotropic Laplacian ∇
2
.
The potential energy of a node at x is determined by the tension differences with its 12 nearest
neighbors at vectors {e
i
}. The FCC basis vectors are permutations of
a
√
2
(±1, ±1, 0).
The lattice Laplacian is defined as the sum of first differences:
∆
F CC
ψ(x) =
12
X
i=1
[ψ(x + e
i
) − ψ(x)] (1)
Performing a Taylor expansion:
ψ(x + e
i
) ≈ ψ(x) + (e
i
· ∇)ψ +
1
2
(e
i
· ∇)
2
ψ (2)
The linear terms vanish due to the inversion symmetry of the FCC lattice. The quadratic terms
involve the sum
P
(e
i
· ∇)
2
. Explicit calculation shows that for the 12 FCC vectors, the diagonal
terms sum to 8(∂
2
x
+ ∂
2
y
+ ∂
2
z
) while cross-terms cancel. Including the Taylor factor (1/2) and the
lattice constant basis factor (a
2
/2), we obtain:
∆
F CC
ψ ≈
1
2
·
a
2
2
· 8∇
2
ψ = 2a
2
∇
2
ψ (3)
This rigorous convergence justifies the use of ∇
2
in the continuum Lagrangian below.
B. The Chiral Cosserat Action
We define the physics of the defect via a Lattice Lagrangian Density L. The first terms represent
standard Micropolar Elasticity (kinetic energy and elastic potential ω
0
). The final term is the novel
topological contribution: a time-asymmetric coupling representing the defect’s chirality (spin).
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)
| {z }
Chiral Coupling
(4)
Here, Ω is the gyroscopic coupling strength determined by the braid winding number.
3
C. Microscopic Origin of the Chiral Coupling
The coupling term Ω(u
˙
θ − θ ˙u) is not an ad hoc addition but arises fundamentally from the Berry
connection of the discrete lattice wavefunction. As a defect (K = 13) moves through the lattice, the
local basis vectors undergo a rotation R(t). The discrete hopping matrices Γ
j
acquire a geometric
phase factor e
iγ
. In the continuum limit, this phase manifests as a gauge field A
µ
coupled to the
velocity:
A
0
∝ ⟨ψ|∂
t
|ψ⟩ ≈ u
˙
θ − θ ˙u (5)
Thus, the term representing ”Chirality” in the macroscopic Lagrangian is the direct imprint of the
Berry curvature generated by the non-trivial topology of the Trefoil knot defect.
D. Topological Origin of Parameters
The Lagrangian introduces two frequency scales: the elastic stiffness ω
0
and the chiral frequency
Ω. In the SSM, these correspond to the two distinct topological features of the defect:
• ω
0
(Lattice Stiffness): Corresponds to the tension of the K = 12 bulk lattice. This repre-
sents the high-energy hadronic scale (Proton mass equivalent).
• Ω (Gyroscopic Frequency): Corresponds to the winding frequency of the K = 13 braid.
This represents the low-energy leptonic scale (Electron mass equivalent).
The mass of the particle is derived as m ∝
p
ω
2
0
+ Ω
2
. If we identify the bulk stiffness with
the proton mass scale and the braid frequency with the electron mass scale, the model naturally
accommodates the hierarchy of particle masses m
p
/m
e
≈ 1836 as a ratio of bulk-to-defect coupling
strengths [6].
E. Explicit Euler-Lagrange Derivation
The Euler-Lagrange equation is given by:
∂
t
∂L
∂
˙
ϕ
+ ∇ ·
∂L
∂(∇ϕ)
−
∂L
∂ϕ
= 0 (6)
We apply this independently for the fields u and θ.
For the radial field u:
∂L
∂ ˙u
= ˙u − Ωθ,
∂
∂t
∂L
∂ ˙u
= ¨u − Ω
˙
θ (7)
∂L
∂(∇u)
= −c
2
∇u, ∇ · · · · = −c
2
∇
2
u (8)
∂L
∂u
= −ω
2
0
u + Ω
˙
θ (9)
Combining terms:
(¨u − Ω
˙
θ) + (−c
2
∇
2
u) − (−ω
2
0
u + Ω
˙
θ) = 0 (10)
4
⇒ ¨u − 2Ω
˙
θ − c
2
∇
2
u + ω
2
0
u = 0 (11)
For the torsional field θ:
∂L
∂
˙
θ
=
˙
θ + Ωu,
∂
∂t
(. . . ) =
¨
θ + Ω ˙u (12)
∂L
∂θ
= −ω
2
0
θ − Ω ˙u (13)
Combining terms:
(
¨
θ + Ω ˙u) + (−c
2
∇
2
θ) − (−ω
2
0
θ − Ω ˙u) = 0 (14)
⇒
¨
θ + 2Ω ˙u − c
2
∇
2
θ + ω
2
0
θ = 0 (15)
F. Complexification and Geometry
We define the complex field ψ = u + iθ. Multiplying Eq. (15) by i and adding it to Eq. (11):
(¨u + i
¨
θ) + 2Ω(−
˙
θ + i ˙u) − c
2
∇
2
(u + iθ) + ω
2
0
(u + iθ) = 0 (16)
We rewrite the coupling term using the identity −
˙
θ + i ˙u = i( ˙u + i
˙
θ) = i
˙
ψ. Substituting this back,
we obtain the complex wave equation:
∂
2
ψ
∂t
2
+ 2iΩ
∂ψ
∂t
− c
2
∇
2
ψ + ω
2
0
ψ = 0 (17)
This derivation proves that the complex unit i is the geometric operator mapping the chiral rotation
between radial and torsional modes.
G. Unitarity and Probability Conservation
The Lagrangian possesses a global U(1) symmetry. We rewrite the interaction term in complex
variables as −
iΩ
2
(ψ
∗
˙
ψ − ψ
˙
ψ
∗
). The associated conserved Noether charge density j
0
is:
j
0
= Im(ψ
∗
˙
ψ) + Ω|ψ|
2
(18)
As in standard Klein-Gordon theory, this relativistic charge is not strictly positive-definite for all
modes. However, the physical probability interpretation emerges in the non-relativistic limit. As
derived in Section IV, the dynamics reduce to the Schr¨odinger equation, where the conserved density
is strictly |ϕ|
2
, guaranteeing positive-definite probability conservation for stable matter.
IV. EVOLUTION AND THE SCHR
¨
ODINGER LIMIT
To recover standard mechanics, we transform to the Larmor Frame (co-rotating with the lattice
precession) using the ansatz ψ(x, t) = Φ(x, t)e
−iΩt
. We calculate the time derivatives:
˙
ψ = (
˙
Φ − iΩΦ)e
−iΩt
(19)
¨
ψ = (
¨
Φ − 2iΩ
˙
Φ − Ω
2
Φ)e
−iΩt
(20)
5
Substituting these into the complex wave equation:
(
¨
Φ − 2iΩ
˙
Φ − Ω
2
Φ) + 2iΩ(
˙
Φ − iΩΦ) − c
2
∇
2
Φ + ω
2
0
Φ = 0 (21)
The cross terms −2iΩ
˙
Φ and +2iΩ
˙
Φ cancel exactly. The terms −Ω
2
Φ and +2Ω
2
Φ combine to +Ω
2
Φ.
The result is the exact Klein-Gordon equation:
∂
2
Φ
∂t
2
− c
2
∇
2
Φ + (ω
2
0
+ Ω
2
)Φ = 0 (22)
We identify the effective rest mass m via:
m
2
c
4
ℏ
2
= ω
2
0
+ Ω
2
=⇒ m =
ℏ
c
2
q
ω
2
0
+ Ω
2
(23)
Finally, applying the non-relativistic envelope approximation (Φ(x, t) = ϕ(x, t)e
−imc
2
t/ℏ
and ne-
glecting
¨
ϕ), we recover the free Schr¨odinger equation:
−
2im
ℏ
˙
ϕ − ∇
2
ϕ = 0 =⇒ iℏ
∂ϕ
∂t
= −
ℏ
2
2m
∇
2
ϕ (24)
V. DISCUSSION AND PREDICTION
A. Limitations of Current Derivation
This derivation establishes a rigorously defined path from a chiral vacuum lattice to single-particle
quantum mechanics. We acknowledge the following boundaries of the current scope:
• Lagrangian Origin: While motivated by the Berry connection of the K = 13 trefoil de-
fect, the coefficients in Eq. (4) require further derivation from first-principles inter-atomic
potentials.
• Entanglement: The lattice naturally supports non-local correlations through the shared
elastic medium (the bulk connects all defects), but a rigorous derivation of Bell-type correla-
tions remains an open challenge.
B. Derivation of the Geometric Decoherence Limit
We now derive the specific mass scale for this breakdown based on the resolution limit of the
vacuum lattice. The reduced Compton wavelength λ
c
defines the quantum resolution scale of a
particle. For a wavefunction to be consistently represented on the discrete lattice, this wavelength
must exceed the lattice spacing:
λ
c
=
ℏ
mc
≥ a (25)
This condition defines the maximum mass a coherent quantum state can possess before the lattice
discreteness induces decoherence. Using the geometrically renormalized lattice spacing a ≈ 0.77l
P
derived in our previous work [7]:
m
max
=
ℏ
ac
≈
ℏ
(0.77l
P
)c
=
m
P
0.77
≈ 28µg (26)
6
Conclusion: This limit arises from the finite granularity of the K = 12 vacuum geometry. Any ob-
ject exceeding this mass limit (28µg) possesses a quantum wavelength smaller than the fundamental
resolution of the vacuum (a). Such a mode cannot propagate as a coherent wave; it effectively falls
through the lattice mesh, collapsing into a classical geometric deformation.
C. Comparison with Penrose Objective Reduction
It is notable that our derived saturation limit of 28µg aligns closely with the predictions of the
Di´osi-Penrose model of Objective Reduction (OR) [8]. Penrose posits that quantum state reduction
is a gravitational phenomenon occurring near the Planck Mass (≈ 21.7µg) due to the instability of
spacetime superposition (E
G
≈ ℏ/t
collapse
).
However, while Penrose attributes this collapse to a fundamental conflict between General Rela-
tivity and Quantum Linearity, the SSM identifies a distinct, mechanical origin: the elastic saturation
of the vacuum lattice. The convergence of these two distinct approaches—one relativistic and one
geometric—upon nearly the same mass scale (20 − 30µg) strongly suggests this is a fundamental
physical boundary, not an artifact of approximation.
VI. CONCLUSION
The Selection-Stitch Model provides a rigorous derivation of Quantum Mechanics from the me-
chanics of a Chiral Cosserat Vacuum. By identifying the origin of the complex unit, probability
conservation, and the Schr¨odinger equation in lattice gyroscopics, we offer a testable geometric
foundation for physical reality.
[1] G. ’t Hooft, The Cellular Automaton Interpretation of Quantum Mechanics (Springer, 2016).
[2] G. E. Volovik, The Universe in a Helium Droplet (Oxford University Press, 2003).
[3] H. Kleinert, Multivalued Fields in Condensed Matter, Electromagnetism, and Gravitation (World Scien-
tific, 2008).
[4] R. Kulkarni, ”The Selection-Stitch Model (SSM): Space-Time Emergence via Evolutionary Nucleation
in a Polycrystalline Tensor Network,” Zenodo, doi:10.5281/zenodo.18138227 (2026).
[5] R. Kulkarni, ”Thermodynamic Emergence: Deriving the Cuboctahedral Vacuum from Information En-
tropy,” Zenodo, doi:10.5281/zenodo.18334374 (2026).
[6] R. Kulkarni, ”The Geometric Origin of Mass: A Topological Derivation of the Proton-Electron Ratio
using Selection-Stitch Model (SSM),” Zenodo, doi:10.5281/zenodo.18253326 (2026).
[7] R. Kulkarni, ”Geometric Renormalization of the Speed of Light and the Origin of the Planck Scale in a
Saturation-Stitch Vacuum,” Zenodo, doi:10.5281/zenodo.18447672 (2026).
[8] R. Penrose, ”On gravity’s role in quantum state reduction,” Gen. Relativ. Gravit. 28, 581–600 (1996).
