From Dream to Definition: Bridging Intuition and Formalism in Nested Number Sequences
By Igor Chomko
Abstract:
This article formalizes the intuitive framework presented in the blog post Unbounded Nested Number Sequences and introduces a reproducible structure known as Nested Modulus Sequences. We explore its arithmetic consistency, define key terms, and extend the concept to include intersection analysis, integer-preserving behavior, and cryptographic applications.
1. Introduction
The original blog post presents a personal discovery of arithmetic patterns inspired by a dream. Though informal, the structure it proposes—centered around a formula involving modulus and nesting—exhibits consistent behavior across examples. This paper aims to:
- Translate the blog’s intuitive framework into formal mathematical language
- Compare its claims with reproducible results
- Extend the concept into a generalized, tool-supported framework
2. Original Framework: Intuition and Terminology
2.1 Definitions (from the blog)
- Nest (N): The base integer around which a sequence is built
- Modulus (M): The position index within the sequence
- Index Number (IN): Additive step for full sequence
- Integer Index (InI): Additive step for integer-only positions
- Formula:
(M × N) + (M / N) + (M - N) + (M + N)
2.2 Observations
- IN is constant for a given nest and used to generate terms
- Integer values appear at regular intervals (e.g., every N terms)
- Multiplying IN by M yields the term value directly
- Integer-preserving behavior across nests (e.g., 88 = 81 ∈ 77)
3. Formalization: Nested Modulus Sequences
3.1 Definitions
Let SN be a sequence defined by a nest N, with:
IN(N) = SN(1)
SN(M) = IN(N) × M
Integer positions occur at modulus values M = kN, for integer k.
3.2 Generalized Formula
SN(M) = (M × N) + (M / N) + (M - N) + (M + N)
This yields:
- IN(N) = SN(1)
- Integer-preserving positions: SN(kN) ∈ SN−1
4. Comparative Analysis
| Feature | Blog Framework | Formalized Framework |
|---|---|---|
| Origin | Dream-inspired, anecdotal | Empirical, reproducible |
| Terminology | Modular arithmetic terminology | Formal definitions |
| Proofs | None | Empirical validation, visualizations |
| Integer-preserving behavior | Observed in examples | Formalized via modulus-based indexing |
| Applications | Pattern observation | Cryptography, clustering, visualization |
5. Empirical Simulation of Cross-Nest Intersections
We simulated all nested sequences for Nest ∈ [1,49] and Modulus ∈ [1,49] using the formula:
(M × N) + (M / N) + (M - N) + (M + N)
We filtered all results below a threshold of 1000 and recorded values that appeared in multiple nests.
5.1 Sample Intersections
| Modulus M | Nest N | Value |
|---|---|---|
| 1 | 1 | 4.0 |
| 2 | 1 | 8.0 |
| 3 | 1 | 12.0 |
| 4 | 1 | 16.0 |
| 5 | 1 | 20.0 |
5.2 Integer-Preserving Behavior
We observed that values like 81 appear in multiple nests:
- 88 = 81 → 81 ∈ 77
- 99 = 81 → 81 ∈ 88
This confirms the recursive nesting property: TN(N) ∈ SN−1
5.3 Heatmap Visualization
The diagonal and vertical bands suggest modular layering and integer-preserving zones across nests.
Interactive Heatmap:
6. Conclusion
While the original blog post is anecdotal in tone, its structural consistency invites formalization. By translating its intuitive framework into a reproducible system, we uncover deeper mathematical properties and open pathways for application in modular analysis and cryptography.
Empirical simulations confirm that:
- Integer-preserving behaviour across nests is consistent and predictable
- The formula allows direct computation of terms without full sequence generation
- Shared values across nests suggest modular clustering and layered structure
These findings support the transition from intuitive insight to formal framework, demonstrating that even dream-inspired arithmetic can yield reproducible and extensible mathematical structures.