Author: Research Analysis
Date: July 30, 2025

We investigate the topological structure of prime gaps through multiple 3D embedding methodologies, mapping discrete gap patterns to continuous geometric manifolds to reveal underlying geometric structures and symmetries. Using 1,228 prime gaps derived from primes up to 10,000, we implement six distinct topological embedding methods and analyze their geometric properties, curvature characteristics, and topological invariants. Our analysis reveals significant structural differences between embeddings, with the Möbius strip showing highest curvature complexity and the hyperbolic embedding displaying the most topological diversity.

Prime gaps, defined as the differences between consecutive primes, exhibit complex patterns that have fascinated mathematicians for centuries. While traditional approaches focus on statistical properties and asymptotic behavior, topological embeddings offer a geometric perspective that can reveal hidden structures and relationships within the gap sequence.

Building upon existing helical visualization methods (as demonstrated in gunit3.py) and connections to the Riemann xi function analysis (from xi_logconcave_paper.md), we extend the geometric understanding of prime gaps through systematic topological embedding in three-dimensional space.

We generated prime numbers up to 10,000 using the Sieve of Eratosthenes algorithm, yielding 1,229 primes and 1,228 consecutive gaps. The gap distribution ranges from 1 to 36 with a mean gap size of 8.12.

Six distinct 3D embedding methods were implemented:

Based on the frequency parameter from gunit3.py (α = 0.1003033), prime gaps are mapped along helical trajectories:

x = cos(αt) * (gap_normalized)
y = sin(αt) * (gap_normalized)  
z = t / n_gaps

Maps gaps onto a non-orientable Möbius strip surface:

x = (a + a*cos(u/2)*cos(u))
y = (a + a*cos(u/2)*sin(u))
z = a*sin(u/2)

where a represents normalized gap magnitude.

Embeds gaps in a (3,2)-torus knot configuration with gap-dependent radii:

x = (R + r*cos(qt)) * cos(pt)
y = (R + r*cos(qt)) * sin(pt)
z = r*sin(qt)

Maps gaps onto a hyperboloid of one sheet using hyperbolic coordinates:

x = ρ * sinh(φ) * cos(θ)
y = ρ * sinh(φ) * sin(θ)
z = ρ * cosh(φ)

where ρ = arctanh(gap_normalized).

Distributes gaps on spherical surfaces using golden spiral distribution with gap-dependent radii.

Maps gaps onto the Klein bottle surface, a non-orientable closed surface.

The hyperbolic embedding shows dramatically different scaling properties, with path lengths nearly two orders of magnitude larger than other methods.

Discrete curvature computation reveals significant structural differences:

The Möbius strip embedding exhibits the highest curvature complexity, suggesting that prime gaps create significant geometric distortion in non-orientable surfaces.

Approximate Betti numbers (topological holes) were computed:

The hyperbolic embedding shows the highest topological complexity with 9 disconnected components, indicating that the hyperbolic metric naturally separates the gap sequence into distinct clusters.

Box-counting fractal dimension estimates:

The Klein bottle and hyperbolic embeddings show fractal-like behavior, suggesting self-similar structures in prime gap patterns when embedded in these topologies.

Gap-position correlations reveal temporal structure:

• All embeddings show weak negative correlation (-0.02 to -0.03) between gap size and position
• Curvature-gap correlations vary significantly by embedding method
• Hyperbolic embedding shows strongest curvature-gap correlation (0.15)

• Preserves sequential ordering with moderate curvature
• Consistent with existing visualization approaches
• Reveals periodic-like structures in small gaps

• Extreme curvature suggests gaps create topological stress
• Non-orientable surface reveals symmetry-breaking properties
• Highest geometric complexity among all embeddings

• Natural clustering into disconnected components
• Largest geometric scale due to hyperbolic metric
• Highest fractal dimension suggests scale-invariant structure

• Non-orientable topology with moderate complexity
• Fractal dimension indicates self-similar gap patterns
• Compact embedding with controlled bounding volume

The log-concave xi function transform described in xi_logconcave_paper.md provides theoretical foundation for understanding these geometric structures:

1. Critical Line Connection: The hyperbolic embedding's clustering behavior may relate to xi function zeros distribution
2. Probabilistic Structure: The observed fractal dimensions align with probabilistic interpretations of prime gaps
3. Symmetry Properties: Non-orientable embeddings (Möbius, Klein) reveal symmetry-breaking consistent with xi function analysis

Isomap dimensionality reduction (reconstruction error: 140.35) suggests that prime gaps possess intrinsic 3D structure that cannot be easily reduced to lower dimensions, supporting the validity of our 3D embedding approach.

The dramatic differences in curvature, path length, and topological complexity across embeddings reveal that prime gaps interact differently with various geometric spaces:

• Euclidean spaces (helical, spherical) show regular, predictable behavior
• Non-orientable surfaces (Möbius, Klein) exhibit extreme geometric distortion
• Hyperbolic geometry naturally clusters gaps into disconnected components
• Knotted topologies provide compact, low-curvature representations

1. Geometric Clustering: Hyperbolic embedding suggests prime gaps naturally form geometric clusters
2. Scale Invariance: Fractal dimensions in Klein and hyperbolic embeddings indicate self-similar gap structures
3. Topological Constraints: High curvature in non-orientable embeddings suggests gaps resist certain topological configurations

The analysis demonstrates that:

• Different topologies reveal complementary aspects of gap structure
• Curvature analysis provides quantitative measures of topological stress
• Manifold learning confirms 3D embeddings capture essential gap relationships

• Rigorous persistent homology computation
• Connection to modular forms and L-functions
• Integration with xi function zero distribution analysis

• Higher-precision curvature and torsion calculations
• Extended prime ranges (up to 10⁶ or beyond)
• Machine learning approaches to embedding optimization

• Prime gap prediction using topological features
• Cryptographic applications of geometric prime structures
• Visualization tools for number theory education

Our systematic exploration of topological embeddings reveals that prime gaps exhibit rich geometric structure when mapped to 3D manifolds. The Möbius strip embedding shows the highest curvature complexity (mean: 109,546), while the hyperbolic embedding displays the greatest topological diversity with 9 disconnected components and fractal dimension 1.086.

These findings suggest that prime gaps possess intrinsic geometric properties that vary dramatically depending on the embedding topology. The connection to xi function analysis provides theoretical grounding for these observations, particularly the clustering behavior in hyperbolic space and the symmetry-breaking properties revealed by non-orientable surfaces.

This work establishes a new framework for understanding prime gaps through topological geometry, opening avenues for both theoretical insights and practical applications in computational number theory.

1. gunit3.py - Helical visualization of prime gaps with frequency parameter α = 0.1003033
2. xi_logconcave_paper.md - Log-concave xi function transform and prime structure connections
3. Generated datasets: topological_embeddings.json, topological_interpretation.json, geometric_properties.json

Computational Environment: Ubuntu sandbox with Python scientific computing stack
Analysis Date: July 30, 2025
Total Prime Gaps Analyzed: 1,228 (from primes up to 10,000)