Generate an advanced python script that envelops this idea with GUI:

```
**Title:** **Numerical ASE Point Puzzle Algorithm**

---

### **Algorithm Overview: Numerical ASE Point**

**Objective:**  
Numerical ASE Point is an advanced puzzle algorithm designed for multi-layer encryption-inspired problem-solving tasks. It combines elements of numerical manipulation, encoding techniques, and sequence alignment inspired by Morse Code, AES encryption principles, and C++ structure. This puzzle is structured for a challenging level of complexity suitable for expert users, leveraging the following parameters:

- **Puzzle Layering**: 4x Layers
- **Advanced Functional Design**: 10x Advanced Mechanisms
- **Difficulty Scaling**: 8x Difficulty Integration

---

### **Algorithm Framework**

#### **1. Puzzle Layer Structure**

The puzzle layers (L1, L2, L3, L4) contain interlinked data encoding and decoding operations to achieve multiple stages of logical encryption within a numerical sequence. Each layer progressively increases complexity through transformations in the numerical and symbolic domains.

1. **Layer 1 (L1): Base Sequence Initialization**
   - Initialize a prime-number sequence (`P[]`) mapped to ASCII values for alphanumeric and symbolic representation.
   - Encode starting input as Morse-like code but in numerical terms. Use a binary approach (1’s and 0’s) similar to Morse’s dot and dash.
  
2. **Layer 2 (L2): Transformation Encoding**
   - Implement an Advanced Shift Encoding (ASE) function. This function emulates AES rounds, using bitwise operations and modular shifts.
   - Use a key-derived sequence (`K[]`) with permutations applied based on modular arithmetic with a base in prime numbers.

3. **Layer 3 (L3): Numerical Inversion and Mapping**
   - Invert the sequence from L2 using a complex modulus inversion formula with shifting offsets.
   - Apply symbolic mapping through the Fibonacci sequence to establish dependencies on prior puzzle states.
   
4. **Layer 4 (L4): Verification Checkpoint**
   - Final layer encodes verification checkpoints to ensure that decoded values match the expected result.
   - Utilize C++-like conditional operators for verification checkpoints, combining encoded layers into a cumulative state.

#### **2. Advanced Functional Design**

- **Encoding Functions**:
  - **ASE Encoding**: Advanced Shift Encoding combines rotational encoding and bitwise AND/OR operations.  
    - `ASE(x) = ((x << s) & K1) ^ ((x >> s) | K2) % mod(P[x])`
    - Input sequence (`x[]`) shifts according to `s`, a dynamically derived shift value.
  - **Numerical Morse Mapping**:
    - Map each number to a pseudo-Morse code: 1 as dot and 2 as dash, creating a binary equivalent for alphanumeric encryption.

- **Decoding Functions**:
  - **ASE Decoding**: Reverse-engineer ASE encoding using key reversal techniques based on bitwise XOR operations and shifts.
  - **Symbolic Mapping Decoder**:
    - Reverse Fibonacci mapping to restore numeric inputs from symbols.

#### **3. Complexity and Mapping Design**

Each layer integrates increasing complexity through specialized mappings:

- **Prime Mapping**: Maps ASCII and special characters to distinct prime numbers, ensuring unique encoding.
- **Modular & Bitwise Complexity**: Use of modular arithmetic combined with bitwise rotations that increase with each layer.
- **Fibonacci-Based Inversion**: Symbolic dependency on Fibonacci values adds an interdependency layer across the puzzle's encoded outputs.

### **Example Execution**

1. **Initialize Prime Sequence (`P[]`)**: {2, 3, 5, 7, 11, 13, …}  
2. **Layer 1 Encoding**: Map input characters to Morse-like binary equivalents, transforming each into the base `P[]`.
3. **Layer 2 Transformation (ASE)**: Apply advanced encoding using a shifted key-based sequence.
4. **Layer 3 Mapping Inversion**: Map back using Fibonacci to align symbols and numbers.
5. **Layer 4 Verification**: Confirm through conditional C++-like operators, concluding if layers align to expected value.

---

### **Pseudo-Code Outline**

```cpp
// Initialize Sequence and Keys
int P[] = {2, 3, 5, 7, 11, ...};
int K1, K2; // Derived keys

// Layer 1: Numerical Morse Encoding
for (char c : input) {
    numSequence.push_back(mapToMorse(c, P[]));
}

// Layer 2: ASE Encoding
for (int i = 0; i < numSequence.size(); i++) {
    numSequence[i] = ASE(numSequence[i], K1, K2, i);
}

// Layer 3: Fibonacci-Based Inversion
for (int i = 0; i < numSequence.size(); i++) {
    numSequence[i] = invertFibonacci(numSequence[i], i);
}

// Layer 4: Verification
bool isValid = verifyLayer(numSequence, expectedOutput);
if (isValid) {
    outputResult(numSequence);
}
```

### **Conclusion**

The Numerical ASE Point Puzzle Algorithm effectively combines multi-layered numerical encoding, prime mappings, and symbolic dependencies to create a challenging, advanced puzzle. Each layer builds upon the previous, requiring understanding of bitwise operations, modular arithmetic, and numerical inversions.