Patent Application: Topological Data Storage System with Relational Encoding 1. Background of the Invention (a) Field of the Invention This invention relates to data storage systems, and more particularly, to systems utilizing topological properties of materials for high-density and efficient data storage, incorporating relational encoding for enhanced data representation. (b) Description of the Related Art Traditional data storage methods, such as flash memory and hard drives, face limitations in scaling to meet the demands of ever-increasing data volumes. These methods store data as discrete bits, which can be inefficient for representing complex relationships between data points. Emerging technologies, like topological data storage, offer the potential for higher density and stability by encoding data in the topological states of materials. However, current topological storage approaches often focus on individual data points, neglecting the rich relational information inherent in many datasets. There is a need for a topological data storage system that can efficiently encode and retrieve not only data values but also the relationships between them. 2. Summary of the Invention This invention addresses this need by combining topological data storage with relational encoding using multi-dimensional matrices. The system encodes data and the relationships between data points in the topological states of a material, such as a lattice of magnetic skyrmion strings. A multi-dimensional matrix represents the relational dependencies, allowing for efficient storage and retrieval of both data and its context. This approach offers the potential for significantly increased storage density and improved data management capabilities. 3. Detailed Description of the Invention (a) Material and Topological States: The data storage system utilizes a material capable of exhibiting distinct topological states. In a preferred embodiment, the material comprises a two-dimensional lattice of magnetic skyrmion strings confined within a nanostructured substrate. Skyrmions are topologically protected spin textures that can be manipulated and detected. The distinct geometric configurations of the skyrmion strings correspond to different topological states, which are used to represent data. The lattice structure provides a regular arrangement of skyrmions, facilitating manipulation and reading operations. The nanostructured substrate provides confinement and stabilization of the skyrmion lattice. The nearest-neighbor spacing of the skyrmion strings is preferably within the range of X to Y nanometers, where X and Y are values determined based on material properties and desired storage density, and supported by experimental data (e.g., X = 10 nm, Y = 20 nm). (b) Relational Encoding with Multi-Dimensional Matrices: A key aspect of this invention is the use of multi-dimensional matrices to encode relational dependencies between data points. The encoding process comprises the following steps: - Relational Dependency Analysis: An algorithm analyzes the data to identify and extract relational dependencies. For example, if the data represents a social network, the algorithm might identify connections between individuals. If the data is a time series, the algorithm could identify temporal correlations between data points. In one embodiment, semantic relationships are identified. - Matrix Construction: A multi-dimensional matrix is constructed to represent these relational dependencies. For example, the matrix could be an adjacency matrix of a graph representing the relationships between data points. The dimensions of the matrix are determined by the complexity of the relationships being represented. - Matrix Transformation and Compression: A matrix transformation and compression algorithm is applied to enhance storage density. In a preferred embodiment, Singular Value Decomposition (SVD) is used to reduce the dimensionality of the matrix while preserving the most important relational information. The “k” value for SVD (number of singular values retained) is chosen to balance compression ratio and information retention, and is determined based on the specific application requirements. (c) Data Storage and Retrieval: The multi-dimensional matrix, representing relational dependencies, is stored in the topological states of the material. Each element of the matrix is mapped to a specific configuration of skyrmion strings. The data itself can also be encoded in the topological states, either separately or integrated with the relational matrix. The reader device decodes the information by detecting the topological states of the material. The relational information is decoded by reconstructing an approximation of the multi-dimensional matrix and extracting the relational dependencies from the reconstructed matrix. (d) Manipulation Mechanism: The manipulation mechanism applies external stimuli to the material to change the topological states of the skyrmion strings. These stimuli can include magnetic fields, spin-polarized electric currents, or spatially modulated thermal gradients. The stimuli are carefully calibrated to induce deterministic transitions between stable topological configurations, allowing for writing and manipulating the encoded data and relational information. The manipulation mechanism can also be used to perform matrix operations directly on the encoded matrix, enabling efficient processing of relational data. (e) Reader Device: The reader device is configured to detect and decode the topological states of the material. In a preferred embodiment, the reader device operates without physically contacting the material. “Physical contact” is defined as any interaction involving direct electron transfer or near-field electromagnetic coupling. The reader device uses techniques such as spin-polarized tunneling or magneto-optical Kerr effect to probe the magnetic states of the skyrmions and determine their configuration. The reader device decodes the relational information by reconstructing an approximation of the multi-dimensional matrix and extracting the relational dependencies from the reconstructed matrix. (f) Example Implementation (Social Network Data): Consider a social network where data points represent individuals, and the relationships represent connections between them. The relational dependency analysis algorithm identifies these connections. The multi-dimensional matrix is an adjacency matrix, where each entry represents the presence or absence of a connection between two individuals. SVD is used to compress the adjacency matrix. The compressed matrix is stored in the topological states of the skyrmion lattice. To retrieve information about connections, the reader device reconstructs the matrix and extracts the relevant entries. (g) Experimental Validation: (This section must include experimental data or simulations to support the claims. Show how the skyrmion lattice is created and manipulated, how the matrix is encoded, and how the data is retrieved. Provide quantifiable results for storage density, read/write speed, and energy consumption.) 4. Claims - A system for storing data using topological properties, comprising: - a material configured to encode information in its topological states, wherein the material comprises a lattice of magnetic skyrmion strings confined within a nanostructured substrate, and wherein the topological states correspond to distinct geometric configurations of the skyrmion strings; - a mechanism for manipulating the topological states of the material to represent data; and - a reader device configured to decode the encoded information from the topological states of the material, wherein at least a portion of the information encoded in the topological states represents relational dependencies between data points, encoded using a multi-dimensional matrix transformation. - The system of claim 1, wherein the multi-dimensional matrix transformation comprises: - a relational dependency analysis algorithm to identify and extract relational dependencies between data points; - a matrix construction algorithm to create a multi-dimensional matrix representing the relational dependencies; and - a matrix transformation and compression algorithm to enhance storage density. - The system of claim 2, wherein the relational dependency analysis algorithm identifies semantic relationships between data points, and the multi-dimensional matrix represents these semantic relationships. - The system of claim 2, wherein the matrix transformation and compression algorithm utilizes Singular Value Decomposition (SVD) to reduce the dimensionality of the multi-dimensional matrix. - The system of claim 1, wherein the topological states of the material are manipulated to perform matrix operations on the multi-dimensional matrix representing relational dependencies. - The system of claim 1, wherein the relational dependencies represent a graph structure, and the multi-dimensional matrix encodes the adjacency matrix of the graph. - The system of claim 1, wherein the material exhibits ultra-high-density storage capabilities, with data and relational metadata encoded in the geometric configurations of magnetic skyrmion strings at a density exceeding A bits per square centimeter, where A is a value demonstrated through experimental validation in the specification. - The system of claim 1, wherein the reader device decodes the relational information by reconstructing an approximation of the multi-dimensional matrix and extracting the relational dependencies from the reconstructed matrix. - The system of claim 1, wherein the manipulation mechanism applies external stimuli selected from the group consisting of magnetic fields, spin-polarized electric currents, and spatially modulated thermal gradients. - The system of claim 1, wherein the lattice of magnetic skyrmion strings has a nearest-neighbor spacing within the range of X to Y nanometers, where X and Y are values supported by experimental data in the specification. 5. Abstract A data storage system utilizing topological properties of materials and relational encoding with multi-dimensional matrices. The system encodes data and relationships between data points in the topological states of a material, such as a lattice of magnetic skyrmion strings. A multi-dimensional matrix represents the relational dependencies, enabling efficient storage and retrieval of both data and its context.