K = 12 Lattice Saturation | Constructive Verification in SSM

Constructive Veriﬁcation of K = 12 Lattice Saturation

Exploring Kinematic Consistency in the Selection-Stitch Model

Raghu Kulkarni

Independent Researcher

January 18, 2026

Abstract

The Selection-Stitch Model (SSM) proposes a view of the vacuum not as a pre-existing back-

ground, but as a constructed geometry built from a simple ”Stitch” operator (L). However,

discrete models like this often face a diﬃcult hurdle: simple local rules frequently lead to

”jammed” or disordered structures rather than smooth space. In this study, we test whether

the SSM can overcome this geometric challenge. Using a constructive simulation that enforces

the model’s strict ”Lift” operator (

2/3L), we observe that the network evolves without

jamming, naturally settling into a saturated lattice with a coordination number of K = 12.

We identify this local geometry as the Cuboctahedron, which corresponds to

Face-Centered Cubic (FCC) packing. While further thermodynamic analysis is needed,

these results suggest that the SSM’s axioms are geometrically consistent, oﬀering a viable

kinematic framework for a discrete vacuum.

1 Introduction

Physics currently faces deep questions about the nature of space-time. The Selection-Stitch

Model (SSM) oﬀers a potential path forward by treating the vacuum as a simplicial complex

built from discrete operations [1]. The core of this idea is the Stitch Operator, which binds

nodes at a ﬁxed, invariant distance L.

But building space from scratch is not easy. A major risk for any such theory is ”geometric

frustration”—the tendency for local building rules to create gaps or overlaps that prevent a

coherent, dense structure from forming. For the SSM to be a serious candidate, it needs to

demonstrate that it doesn’t just create a messy ”glass” of nodes, but rather a saturated lattice,

speciﬁcally reaching a coordination number of K = 12 [1]. This paper is an attempt to check

that basic consistency. We ask a simple question: If we strictly follow the Unitary Stitch rule,

does the geometry hold together, or does it fall apart?

2 Methodology

To explore this, we built a custom simulation to model how the Stitch Operator propagates in 3D

space. The code is open-source and available for review at: https://github.com/raghu91302/

ssmtheory/blob/main/ssm_simulation.py [4].

2.1 The ”Lift” Operator

Unlike simulations that shake a box of spheres until they settle (energy minimization), the SSM

requires us to build geometry constructively. We have to get it right the ﬁrst time. When adding

a new node to an existing triangle so that all connections remain length L, geometry gives us a

speciﬁc constraint. The height of the new node must be:

h =

− recircumradius

− (L/

√

2/3L ≈ 0.816L (1)

Our simulation uses this Lift Operator as a strict rule. It allows us to test whether applying

this rigid constraint recursively preserves the structure of the lattice or leads to conﬂicts.

2.2 How the Simulation Builds

The universe in our model grows step-by-step:

1. Genesis: We begin with a single tetrahedron.

2. Surface Scan: We ﬁnd all the triangular faces exposed on the surface.

3. Recursive Lift: We try to ”lift” a new node from these faces at the speciﬁc height

h =

2/3L.

4. Exclusion Check: This is the critical test. A new node is only accepted if it doesn’t

crowd its neighbors (enforcing a ”Hard Shell” radius of 0.95L).

Justiﬁcation: We deﬁne the Exclusion Radius (R

) at 0.95L rather than the strict 1.0L.

Physically, this tolerance models the ’Metric Jitter’ or quantum uncertainty inherent in

the stitch. In a purely rigid classical simulation, ﬂoating-point precision errors or perfect

geometric frustration can arrest growth artiﬁcially. The 5% tolerance acts as a ’thermal

lubricant,’ allowing the lattice to settle into the Cuboctahedral minimum without getting

trapped in local jamming states.

5. Stitch: If the node ﬁts, it locks in and becomes part of the lattice.

2.3 Modeling Evolutionary Phases

To capture the timeline suggested by the SSM, we use a dynamic probability function. Instead of

a static rule, the likelihood of creating volume (Lift) vs. expanding the surface (Stitch) changes

as the network grows:

• Phase I: We start by suppressing volumetric growth to simulate an initial ”ﬂat” sheet.

• Phase II: We switch to aggressive volumetric growth, mimicking an inﬂationary expan-

sion.

• Phase III: Finally, we let the system relax to equilibrium to see if it crystallizes.

3 Results

We ran the simulation up to N = 5, 000 nodes. As shown in Figure 1, the network successfully

propagated a continuous, dense manifold without fragmenting.

Figure 1: Visualization of Lattice Consistency. The simulation shows that the Lift Operator

can successfully tile 3D space. The yellow core represents nodes that have reached the saturation

limit of K = 12, indicating a consistent internal structure.

3.1 Veriﬁcation: No Jamming

Perhaps the most signiﬁcant result is a negative one: The simulation did not jam. In

constructive packing, it is common for shells to overlap in ways that stop growth. The fact

that we could apply the Lift Operator 5,000 times without violating the exclusion radius is

encouraging. It indicates that the K = 12 lattice is a valid solution for these operators.

3.2 Topological Statistics

The resulting structure showed clear patterns:

• Maximum Degree: The connectivity saturated at exactly 12. No node exceeded this

limit. This local geometry corresponds to the Cuboctahedron, the signature of

Face-Centered Cubic (FCC) packing.

• Bulk Average: The average connectivity was 9.66. This is slightly below 12, reﬂecting

the natural vacancies and ”gaps” we might expect in a dynamic vacuum rather than a

frozen perfect crystal.

4 Discussion

4.1 Geometric Consistency

It is important to interpret this carefully. The Cuboctahedral geometry arises naturally from

the math of the Lift Operator. This simulation doesn’t prove that a random gas of spheres must

freeze into this shape (a thermodynamic claim), but it does prove Geometric Consistency. It

conﬁrms that if the vacuum follows the SSM rules, the resulting structure works—it ﬁts together

mathematically. This distinguishes the SSM from discrete models that might break down into

non-manifold geometries.

4.2 Future Work

Our current model uses a simple hard exclusion radius (0.95L). The next logical step would be to

introduce force-based relaxation (like Lennard-Jones potentials) to see if this structure represents

a true energy minimum compared to amorphous glass. For now, however, the kinematic viability

of the model stands on ﬁrmer ground.

5 Conclusion

We have computationally veriﬁed that the Selection-Stitch Model’s local operators are consistent

with a saturated 3D lattice. By enforcing the unitary bond length L, the system naturally evolves

into a Cuboctahedral (K = 12) state without jamming. While this result is constructive

rather than thermodynamic, it demonstrates that the SSM framework describes a geometrically

viable architecture for the vacuum.

References

1. R. Kulkarni, THE SELECTION-STITCH MODEL (SSM): Space-Time Emergence via

Evolutionary Nucleation in a Polycrystalline Tensor Network, Zenodo (2026). DOI: 10.5281/zen-

odo.18138227

2. R. Kulkarni, Lattice Pressure Multipliers and Cosmic Acceleration, Zenodo (2026). DOI:

10.5281/zenodo.18238511

3. R. Kulkarni, The Geometric Origin of Mass Hierarchies, Zenodo (2026). DOI: 10.5281/zen-

odo.18253327

4. R. Kulkarni, SSM Theory Simulation Code, GitHub (2026). https://github.com/raghu91302/

ssmtheory/blob/main/ssm_simulation.py