To address the requirement of storing **continuums and floating-point values without approximation**, the RSIE (Relational State Information Encoding) system must be refined to ensure **lossless encoding and perfect reconstruction** of data. Below is a revised version of the **non-provisional patent application** focused on this objective, adhering to USPTO standards and avoiding reliance on speculative hypotheses. Key adjustments are highlighted to emphasize exact representation. --- # **Non-Provisional Patent Application** **Title of the Invention**: **“Lossless Relational State Information Encoding for Floating-Point Data Storage Using Multi-Dimensional Matrix Transformations”** --- ## **[0001] 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. --- ## **[0002] Background of the Invention** **(a) Field of the Invention** The present invention pertains to data storage and retrieval, specifically addressing the challenge of **lossless storage** of continuous and floating-point data while optimizing density and energy efficiency through multi-dimensional matrix transformations of relational dependencies [[Theme 1]]. **(b) Description of the Related Art** Current storage methods for floating-point data (e.g., IEEE 754 encoding) inherently introduce approximation errors due to fixed-bit representations [[Theme 1]]. While compression techniques (e.g., lossless algorithms like Huffman coding) reduce storage overhead, they still treat data as isolated values, failing to exploit relational patterns for density gains [[null]]. Matrix factorization methods like SVD are often used for dimensionality reduction, but they typically involve lossy compression. The present invention departs from these approaches by encoding **exact relational dependencies** within floating-point datasets using invertible matrix transformations, ensuring **zero-loss reconstruction**. Prior art includes: - Relational databases and graph databases, which manage relationships post-storage but do not encode dependencies directly at the storage level [[null]]. - Quantum computing techniques (e.g., qubit-based systems) focused on computation, not storage [[null]]. - Error correction codes like LDPC, which improve integrity but do not address relational encoding for density [[null]]. Thus, there remains an unmet need for a storage paradigm that encodes **exact relational dependencies** to achieve lossless compression and ultra-dense storage of continuous data. --- ## **[0003] Summary of the Invention** The invention provides a **lossless encoding system** for floating-point and continuous data by capturing relational dependencies in multi-dimensional matrices. Key features include: - **Exact Representation**: Floating-point values are stored as relational matrices without approximation errors. - **Lossless Compression**: Matrix transformations (e.g., invertible linear algebra operations) ensure perfect reconstruction. - **Energy Efficiency**: Reduced data volume and optimized matrix access patterns lower energy consumption. - **Scalability**: Compatible with classical and future quantum architectures via matrix operations [[notes/0.6/2025/02/6/6]]. --- ## **[0004] Detailed Description of the Invention** **(a) Technical Framework** RSIE encodes continuous data into **invertible matrices** where entries represent **exact relational dependencies** (e.g., differential relationships, functional dependencies). These matrices are losslessly transformed to reduce redundancy while preserving all information. **(b) Encoding Process** The method includes: 1. **Relational Dependency Analysis**: - For floating-point datasets (e.g., sensor readings, scientific measurements), identify **exact dependencies** like: - **Temporal correlations** (e.g., linear or nonlinear relationships between time-series points). - **Spatial gradients** (e.g., derivatives in spatially distributed data). - **Functional dependencies** (e.g., polynomial fits or differential equations governing the data). 2. **Matrix Construction**: - Build matrices where entries encode **exact relationships**. For example: - A **differential matrix** for time-series data, where entries represent precise differences between consecutive samples. - A **functional parameter matrix** for datasets governed by equations (e.g., coefficients of a polynomial fit). 3. **Lossless Matrix Transformation**: - Apply invertible transformations (e.g., **exact QR decomposition**, **lossless sparse encoding**) to compress matrices without data loss. - Example: For a 3D sensor dataset, construct a tensor of spatial gradients and apply invertible transforms to capture dependencies in a compact form. 4. **Storage**: - Store transformed matrices in **lossless formats** (e.g., compressed sparse row [CSR] for sparse matrices, or exact integer-encoded relational parameters). **(c) Decoding Process** - **Matrix Reconstruction**: Apply inverse transformations (e.g., QR inverse) to recover the original relational matrix. - **Data Reconstruction**: Use the relational matrix to **exactly reconstruct** floating-point values. For example: - Reconstruct time-series data by integrating differential values. - Reconstruct spatial data by applying functional parameters to a base dataset. **(d) System Architecture** - **Input Module**: Accepts raw floating-point data. - **Relational Analysis Engine**: - **Algorithm 1 (Temporal Dependency Extraction)**: Input: Time-series sensor data. Output: A **differential matrix** capturing exact differences between samples. - **Algorithm 2 (Functional Dependency Extraction)**: Input: Scientific measurements (e.g., temperature vs. pressure). Output: A **functional parameter matrix** (e.g., polynomial coefficients) describing the relationship between variables. - **Encoding Engine**: - Apply invertible transformations (e.g., lossless sparse compression [[notes/0.6/2025/02/6/6]] or exact matrix decomposition) to encode relational matrices. - **Storage Medium**: Optimized for matrix data (e.g., 3D NAND with matrix-friendly layouts). - **Decoding Engine**: - Reconstruct matrices via inverse operations and use them to **exactly regenerate** original floating-point values. --- ## **[0005] Claims** **Independent Claims**: 1. A method for losslessly encoding floating-point data for storage, comprising: - Analyzing, using a relational analysis engine, an input floating-point data stream to identify **exact relational dependencies** (e.g., temporal differentials, functional parameters); - 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; - Reconstructing the original floating-point 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 floating-point data (e.g., differential relationships, functional parameters); - An encoding engine configured to: - Construct matrices encoding these dependencies; - Apply **lossless transformations** (e.g., invertible matrix operations) to compress the matrices; - A non-transitory storage medium for the transformed matrices; - A decoding engine to invert transformations and regenerate data **exactly** using the stored dependencies. **Dependent Claims**: 3. The method of Claim 1, wherein the relational dependencies include temporal differentials, spatial gradients, or functional parameters. 4. The method of Claim 1, wherein the transformation uses **exact QR decomposition** or lossless sparse encoding (e.g., CSR for sparse dependency matrices). 5. The system of Claim 2, wherein the storage medium is 3D NAND configured to optimize access patterns for matrix retrieval. 6. The system of Claim 2, further comprising a decoding engine configured to reconstruct data via inverse matrix operations and dependency-based integration. --- ## **[0006] Theoretical Justification** **Exact Representation via Relational Encoding**: - Traditional floating-point storage (IEEE 754) approximates values due to finite bits [[Theme 1]]. RSIE instead stores **exact relational parameters**, eliminating approximation errors. For example: - A time-series dataset can be stored as an initial value plus a differential matrix. Reconstruction integrates differentials precisely. - A sensor array’s spatial data is encoded as base measurements plus a gradient tensor, enabling exact spatial interpolation. **Non-Obviousness**: - The inventive step lies in combining **exact dependency analysis** (e.g., differential extraction) with **invertible matrix transformations** to achieve lossless compression. While matrix operations are known, their application to **floating-point data storage** for **zero-loss encoding** is novel. --- ## **[0007] Abstract** Disclosed is a system and method for **lossless relational state information encoding (RSIE)** of floating-point and continuous data. The invention encodes datasets into multi-dimensional matrices capturing **exact relational dependencies** (e.g., temporal differentials, functional parameters), then applies invertible transformations (e.g., QR decomposition, lossless sparse encoding) to reduce storage density without approximation. The RSIE system 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. --- # **Key Adjustments** 1. **Focus on Floating-Point Data**: - Claims explicitly target **continuous and floating-point data**, contrasting with prior art that introduces approximation errors [[Theme 1]]. - Emphasize **exact dependency extraction** (e.g., differentials, gradients) and **invertible transformations** for lossless storage. 2. **Lossless Transformations**: - Replace “dimensionality reduction” with **invertible operations** (e.g., QR decomposition, lossless sparse encoding) to ensure no data loss. 3. **Reconstruction Accuracy**: - Specify that data is regenerated **without approximation**, using inverse algorithms tied to the exact relational parameters stored. 4. **Prior Art Differentiation**: - Highlight that RSIE avoids approximation errors (unlike IEEE 754) and focuses on **floating-point-specific relational encoding**, distinct from generic matrix compression [[notes/0.6/2025/02/6/6]]. --- # **Enforceability Risks & Mitigation** - **Enablement**: Provide pseudocode examples for exact dependency extraction (e.g., differential matrices for time-series data) and invertible transforms. - **Novelty**: Argue that no prior art combines **lossless relational encoding** with **floating-point data storage**. - **Non-Obviousness**: Stress the paradigm shift from fixed-bit approximations to **dependency-based exact representation**. --- # **Final Notes** This draft aligns with USPTO standards and emphasizes **lossless encoding** of floating-point data through relational matrices. It avoids speculative frameworks and grounds claims in established principles like invertible matrix operations and dependency analysis. Further refinements with a patent attorney are recommended to tighten claim language and address jurisdictional specifics. **Reference Sources**: [[Theme 1]] (2009-10-05): Highlights inefficiency of fixed-bit storage for continuous data. [[notes/0.6/2025/02/6/6]] (2024): Provides algorithmic examples for matrix-based encoding.