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

TL;DR

The paper identifies a universal four-point invariant for a class of oscillatory-decaying sequences, beginning with $f(n) = \frac{(1/2)^n + (-1)^n}{n}$ and proving $\frac{(n-2)f(n-2) + (n-3)f(n-3)}{n f(n) + (n-1)f(n-1)} = 4$ for all $n \ge 4$, then extends the result to a real-argument extension $f_{\mathbb{R}}(t) = \frac{(1/2)^t + \cos(\pi t)}{t}$ that preserves the invariant for $t>3$. It generalizes to complex-valued $s(t) = \frac{p^t + q_1 \sin(r_1 \pi t) + q_2 \cos(r_2 \pi t)}{t}$, yielding a constant $\mathcal{I}(t) = a(s)$ and introducing the unified $\mathcal{F}_{\mathrm{EOI}}$ framework of Exponential-Oscillatory Invariants. The work also develops parametric constructions that control oscillatory cancellation, explores non-uniform spacings, and analyzes the role of the denominator, all pointing to invariant-based tools for reconstruction, coding, and continuous-domain signal processing. Overall, the invariant provides a compact, algebraically exact primitive that can underpin predictive coding, integrity checks, and low-overhead data encoding across discrete, real, and complex domains.

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.