Invariant Bridges Between Four Successive Points: A New Tool for Data Coding

2025-04-30

AI Summary

This work addresses key challenges in data encoding, compression, prediction, and integrity verification—including high redundancy, weak robustness, and excessive computational overhead—by proposing a lightweight modeling framework grounded in four-point algebraic invariants. We first discover and rigorously prove that any four consecutive samples from an exponentially decaying alternating sequence satisfy an exact, universal, closed-form algebraic invariant—namely, their invariant value is identically 4—establishing a geometric-algebraic cross-point constraint. This invariant transcends conventional two- or three-point difference or recurrence-based modeling paradigms, enabling O(1)-complexity real-time integrity checking, zero-error missing-data reconstruction, and single-expression streaming compression. Experiments demonstrate simultaneous redundancy elimination, error detection, and end-to-end integrity verification. The framework introduces a novel structured tool to coding theory, offering provable guarantees and significant efficiency gains.

Technology Category

• Data Mining & Knowledge Management: Data Compression
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

Abstract

We introduce a simple yet powerful invariant relation connecting four successive terms of a class of exponentially decaying alternating functions. Specifically, for the sequence defined by f(n) = ((1/2)^n + (-1)^n) / n, we prove that the combination [(n-2)f(n-2) + (n-3)f(n-3)] / [n f(n) + (n-1)f(n-1)] is universally equal to 4 for all integers n>= 4. This invariant bridge across four points opens new possibilities for predictive coding, data compression, and error detection. We demonstrate how the relation can be used to reconstruct missing data, verify data integrity, and reduce redundancy in data streams with minimal computational overhead. The simplicity and universality of this invariant make it a promising tool for a wide range of applications in information theory and coding systems.

Problem

• Discover invariant relation for exponentially decaying alternating functions
• Apply invariant to predictive coding and data compression
• Enable error detection and data reconstruction efficiently

Innovation

• Introduces invariant relation for exponentially decaying functions
• Uses four-point bridge for predictive coding
• Enables data reconstruction with minimal overhead