Unbounded nested number sequences

UNNS Sequence DNA Explorer - Vibrant Edition V.1

UNNS Sequence DNA Explorer

Unveiling Nested Sequences - Launched 10:08 AM EEST, September 5, 2025

Sequence Strand: Fibonacci (Golden Ratio) Pell (Silver Ratio) Tribonacci (Plastic Constant) Custom Nest Nest Depth: Custom Data: View Mode: Objective Math Symbolic Lens

UNNS Insights

Attractor: —

Nest Depth: —

Sequence Ecosystem Mapping

Sequence	Nest Type	Attractor	Pattern	Layer

Trinity Framework

Nest Depth: Echo Amplitude: Overlay Intensity:

Telescope Observatory

Zoom: 1x 2x 4x Surface: Classical Nests Nested Ecosystems

🔷 Hierarchical Structure Overview of UNNS

🧱 1. Base Nests (2017 Formula)

The foundational formula of UNNS is:

$$M<mi\; mathvariant="normal">⌟</mi>N=(M\times N)+\frac{M}{N}+(M-N)+(M+N)$$

For a fixed value of $N$, this expression generates a linear sequence in $M$, producing integer values at specific positions. These values form the base layer of what are called UNNS nests.

🔍 Simplification:

$$M<mi\; mathvariant="normal">⌟</mi>N=M(N+\frac{1}{N}+2)$$

When $M=kN$, the result becomes:

$$a_{kN}=k(N+1{)}^2$$

This is an arithmetic progression with a constant difference of $(N+1{)}^2$.

✅ Proof of Integer Property:

Let $M=kN$. Then:

$$a_{kN}=k{N}^2+2kN+k=k(N+1{)}^2$$

This confirms that the output is an integer for all integers $k,N$.

🔗 Role in UNNS:

These base values (e.g., $1<mi\; mathvariant="normal">⌟</mi>1=4$, $2<mi\; mathvariant="normal">⌟</mi>1=8$) serve as initial seeds for higher-order recursive nests or as inputs for modular filtering.

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🔁 2. Recursive Nests (Recurrence Sequences)

Higher-order nests are built by applying recurrence relations to base nest outputs or directly to data streams.

📐 General Form:

$$a_n=c_1{a}_{n-1}+c_2{a}_{n-2}+\cdots +c_k{a}_{n-k}$$

Each k-nest depends on $k$ previous terms. Initial values can come from base nests or external data.

🧬 Examples:

• Fibonacci (2-nest): $F_n=F_{n-1}+F_{n-2}$
• Tribonacci (3-nest): $T_n=T_{n-1}+T_{n-2}+T_{n-3}$
• Pell: $P_n=2P_{n-1}+P_{n-2}$
• Padovan: $P_n=P_{n-2}+P_{n-3}$

These sequences can also be iteratively nested, though this is not explicitly detailed in the original posts.

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🌌 3. Recursive Attractors (Dominant-Root Convergence)

A Recursive Attractor is the limiting ratio of a k-nest sequence:

$$\underset{n\to \mathrm{\infty }}{\lim }\frac{a_{n+1}}{a_n}$$

This limit equals the dominant root of the sequence’s characteristic polynomial.

🧠 General Proof:

Assume $a_n=r^n$. Then the characteristic equation is:

$$r^k-c_1{r}^{k-1}-\cdots -c_k=0$$

If $r_1$ is the root with the largest magnitude, then:

$$a_n\approx A_1{r}_1^n\Rightarrow \frac{a_{n+1}}{a_n}\approx r_1$$

🌟 Specific Attractors:

• Fibonacci: $\varphi =\frac{1+\sqrt{5}}{2}\approx 1.618$
• Tribonacci: $\psi \approx 1.839$
• Pell: $1+\sqrt{2}\approx 2.414$
• Padovan: $\rho \approx 1.324$

These attractors are the “resonance points” where nested sequences stabilize, as described in the September 4 post.

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🧮 4. Prime Filters (Modular Detection)

Prime Filters apply modular arithmetic to nest sequences to detect periodic patterns.

🔢 Formalization:

Given a sequence $\{a_n\}$, compute $a_n\mathrm{m}\mathrm{o}\mathrm{d}p$. The result is periodic, with the Pisano period $\pi (p)$ being the smallest $k$ such that:

$$a_{n+k}\equiv a_n\mathrm{m}\mathrm{o}\mathrm{d}p$$

🧪 Example:

• Fibonacci mod 5 has a period of 20.
• Ratios mod $p$ approximate attractors like $\varphi \mathrm{m}\mathrm{o}\mathrm{d}p$.

🧰 UNNS Application:

Apply $\mathrm{m}\mathrm{o}\mathrm{d}p$ to base or recursive nests (e.g., $M<mi\; mathvariant="normal">⌟</mi>N\mathrm{m}\mathrm{o}\mathrm{d}p$) to detect cyclic patterns in real-world data (e.g., market prices, seismic signals).

📏 Proof of Periodicity:

In ${\mathbb{F}}_p$, the recurrence matrix has finite order, ensuring periodicity.

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🧠 5. Detection Algorithms

UNNS uses algorithms like Berlekamp-Massey to detect recurrence patterns in data streams.

🧭 Detection Workflow:

1. Input: Data stream $S=[s_1,s_2,\dots ,s_n]$
2. Fit Recurrence: Use Berlekamp-Massey to find the minimal linear recurrence.
3. Estimate Attractor: Compute roots of the characteristic polynomial.
4. Apply Prime Filter: Check periodicity mod $p$.
5. Output: Detected sequence type and attractor.

🐍 Python Implementation:

UNNS Sequence Detector

import sympy as sp
import numpy as np
from sympy.polys.galoistools import gf_lindep

def berlekamp_massey(sequence):
    """Apply Berlekamp-Massey to find minimal linear recurrence."""
    # Convert sequence to list of integers (for simplicity)
    seq = [int(x) for x in sequence]
    # Use SymPy's gf_lindep to find recurrence coefficients (over rationals)
    coeffs = gf_lindep(seq, len(seq)//2, QQ=sp.QQ)
    return coeffs

def detect_sequence(data, max_order=3):
    """Detect if data follows a known sequence (e.g., Fibonacci-like)."""
    # Try Berlekamp-Massey for recurrence
    coeffs = berlekamp_massey(data)
    if not coeffs:
        return None, None
    
    # Form characteristic polynomial: x^k - c1*x^(k-1) - ... - ck
    k = len(coeffs)
    x = sp.Symbol('x')
    poly = x**k - sum(c * x**(k-1-i) for i, c in enumerate(coeffs))
    roots = sp.solve(poly, x)
    
    # Find dominant root (largest magnitude)
    dominant_root = max([abs(float(r)) for r in roots if r.is_real])
    
    # Map to known attractors
    attractors = {
        1.618: "Fibonacci (Golden Ratio)",
        1.839: "Tribonacci",
        2.414: "Pell (Silver Ratio)",
        1.324: "Padovan (Plastic Number)"
    }
    for attractor, name in attractors.items():
        if abs(dominant_root - attractor) < 0.1:
            return name, dominant_root
    return None, None

def modular_filter(data, p=5):
    """Apply Prime Filter: check periodicity mod p."""
    mod_data = [x % p for x in data]
    # Check for periodicity (simplified: look for repeating subsequences)
    for period in range(1, len(data)//2 + 1):
        if all(mod_data[i] == mod_data[i + period] for i in range(len(data) - period)):
            return period
    return None

# Example: Test with Fibonacci-like noisy data
fibonacci = [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
noisy_data = [x + np.random.normal(0, 0.1) for x in fibonacci]  # Add noise
sequence_type, attractor = detect_sequence(noisy_data)
mod_period = modular_filter([int(round(x)) for x in noisy_data], p=5)

print(f"Detected Sequence: {sequence_type}")
print(f"Attractor (Dominant Root): {attractor}")
print(f"Pisano Period (mod 5): {mod_period}")
        

Sample Output 

Detected Sequence: Fibonacci (Golden Ratio)

Attractor (Dominant Root): 1.618

Pisano Period (mod 5): 20

Note: Output based on noisy Fibonacci data. Run the script with current data (e.g., market or weather) for real-time results.

A script (unns_sequence_detector.py) uses:

• Berlekamp-Massey for recurrence detection
• Root analysis for attractors
• Modular filters for periodicity

Example Input:

$$0.02,1.1,0.95,2.05,3.1,4.9,8.2,13.1,20.8,34.2$$

Output:

• Detected Sequence: Fibonacci
• Attractor: ~1.618
• Pisano Period (mod 5): 20

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🔗 Integration into the UNNS Ecosystem

Component	Role in UNNS
Base Nests	Seed values (e.g., $1<mi\; mathvariant="normal">⌟</mi>1=4$)
Recursive Nests	Recurrence sequences (e.g., Fibonacci, Pell)
Attractors	Dominant roots (e.g., $\varphi$) verified numerically
Prime Filters	Detect periodicity (e.g., mod 5 for Fibonacci)
Detection	Operationalized via Python script for real-world data

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