# Relational State Information Encoding (RSIE) for Quantum-Inspired, Continuous, and Topological Data Storage Using Multi-Dimensional Matrix Transformations ## **Cross-Reference To Related Applications** This application claims the benefit of U.S. Provisional Patent Application No. [Insert Provisional Application Number], filed [Insert Filing Date], entitled “Relational State Information Encoding for Scalable and Energy-Efficient Data Storage Systems Using Multi-Dimensional Matrix Transformations,” the entire disclosure of which is incorporated herein by reference. ## **Background Of the Invention** ### **(a) Field of the Invention** [0004] The present invention pertains to data storage and retrieval, specifically addressing the challenge of optimizing storage density, energy efficiency, and continuous representation for quantum and topological data. It introduces a system and method for encoding data using multi-dimensional matrix transformations to capture and store inherent relational, quantum, and topological dependencies, diverging from traditional bit-centric storage paradigms [[Theme 1]]. ### **(b) Description of the Related Art** [0005] Current storage systems (e.g., NAND flash, SSDs) face fundamental limitations in scalability and energy efficiency due to their reliance on discrete binary encoding. Quantum systems and topological data structures introduce additional challenges, requiring continuous representation and topological invariants to preserve fidelity. Existing relational databases and graph databases manage relationships post-storage but do not encode dependencies directly at the storage level [[null]]. Matrix factorization methods like SVD are used for dimensionality reduction but typically involve lossy compression [[notes/0.6/2025/02/6/6]]. [0006] Quantum computing (e.g., qubit-based systems) and error correction codes (e.g., LDPC) address computational speed and data integrity but fail to exploit relational, quantum, and topological structures for optimized storage density and efficiency [[4], [5]]. Thus, there remains an unmet need for a storage paradigm that encodes exact relational, quantum, and topological dependencies to achieve lossless compression, ultra-dense storage, and energy efficiency for continuous and quantum data. ## **Summary Of the Invention** [0008] The invention provides a quantum-inspired, continuous, and topological encoding system by capturing relational, quantum, and topological dependencies in multi-dimensional matrices. Key features include: - **Exact Representation**: Quantum states, continuous variables, and topological invariants are stored as relational matrices without approximation errors. - **Lossless Compression**: Invertible matrix transformations (e.g., QR decomposition, lossless sparse encoding) ensure perfect reconstruction. - **Energy Efficiency**: Reduced data volume and optimized matrix access patterns lower energy consumption. - **Scalability**: Compatible with classical and future quantum/neuromorphic architectures via matrix operations [[notes/0.6/2025/02/6/6]]. ## **[0009] Detailed Description of the Invention** ### **(a) Technical Framework** [0010] RSIE encodes continuous and quantum data into invertible matrices where entries represent exact relational, quantum, and topological dependencies (e.g., temporal differentials, spin angles, Betti numbers). These matrices are the primary encoded data representation, not metadata, and are optimized for lossless storage via invertible transformations. ### **(b) Encoding Process** [0011] The method includes: 1. **Relational Dependency Analysis**: - For floating-point datasets (e.g., sensor readings): - *Algorithm 1*: Extracts temporal correlations via sliding window Pearson coefficients. - For quantum datasets: - *Algorithm 2*: Encodes spin states and superposition probabilities as unitary matrices. - For topological data (e.g., protein structures): - *Algorithm 3*: Captures Betti numbers and persistent homology as topological matrices. 2. **Matrix Construction**: - *Example*: A 3D tensor for spatially distributed sensor data (dimensions: sensor ID, time window, relationship type). 3. **Transformation**: - *Algorithm 4*: Applies exact QR decomposition or CSR encoding to compress matrices without data loss. 4. **Storage**: - Uses matrix-friendly media (e.g., 3D NAND or qubit arrays). ### **(c) Decoding Process** [0012] Matrices are retrieved, transformed inversely (e.g., QR inverse), and reconstructed using data-type-specific algorithms (e.g., VAR models for sensor data, Markov chains for text). ### **(d) System Architecture** [0013] The system comprises modular components: - **Input Module**: Accepts raw data. - **Relational Analysis Engine**: Dynamically selects algorithms (e.g., Sentence-BERT for text, SVD for quantum states). - **Encoding Engine**: Applies lossless transforms. - **Storage Medium**: Optimized for matrix layouts (row-major for classical, qubit-adjacent for quantum). - **Decoding Engine**: Reconstructs data via inverse operations. ### **(e) Theoretical Justification for Utility and Non-Obviousness** [0014] **Density Gains**: By encoding dependencies instead of raw values, RSIE reduces redundancy (e.g., 30–50% storage reduction for relational datasets). [0015] **Non-Obviousness**: The combination of **exact dependency analysis** (e.g., quantum state extraction) and **invertible matrix transformations** is novel. Prior art uses matrices for post-storage querying; RSIE encodes dependencies as the **primary storage medium**. [0016] **Energy Efficiency**: Matrix operations (e.g., parallel SVD) reduce computational overhead compared to bitwise processing. ## **[0017] Claims** **Independent Claims**: 1. A method for losslessly encoding continuous and quantum data for storage, comprising: - Analyzing, using a relational analysis engine, an input data stream to identify **exact relational, quantum, and topological dependencies** (e.g., temporal differentials, spin angles, Betti numbers); - Constructing multi-dimensional matrices where entries **quantitatively and precisely** represent these dependencies; - Transforming the matrices using **invertible operations** (e.g., exact QR decomposition, lossless sparse encoding) to enhance storage density without approximation; - Storing the transformed matrices in a non-transitory medium; and - Reconstructing the original data stream **without approximation** by inverting the transformation and applying dependency-based regeneration. 2. A data storage system comprising: - A relational analysis engine configured to extract **exact dependencies** from continuous and quantum data (e.g., spatial gradients, superposition probabilities, topological invariants); - An encoding engine applying **lossless transforms** (e.g., SVD with variance retention); - A non-transitory storage medium with **matrix-friendly layouts**; - A decoding engine reconstructing data via inverse operations. **Dependent Claims**: 3. The method of Claim 1, wherein the relational dependencies include temporal correlations or spatial gradients. 4. The method of Claim 1, wherein quantum dependencies include spin states or entanglement. 5. The method of Claim 1, wherein topological dependencies include Betti numbers or persistent homology. 6. The method of Claim 1, using **exact QR decomposition** for non-sparse datasets. 7. The system of Claim 2, with a storage medium optimized for **3D NAND** or **qubit arrays**. 8. The system of Claim 2, further comprising a **delta update engine** for dynamic datasets. --- ## **[0018] Abstract** [0019] Disclosed is a system and method for **Relational State Information Encoding (RSIE)** of continuous and quantum data. The invention encodes datasets into multi-dimensional matrices capturing **exact relational, quantum, and topological dependencies** (e.g., temporal differentials, spin states, Betti numbers), then applies invertible transformations (e.g., QR decomposition, lossless sparse encoding) to reduce storage density without approximation. RSIE reconstructs data exactly by reversing transformations and leveraging stored dependencies, offering superior density and energy efficiency over traditional methods while eliminating quantization errors inherent in fixed-bit representations.