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How the NPF System Works

• 0 is represented as ∅ (empty set, symbolizing absence).
• 1 is represented as [1] (a singleton leaf).
• Negative integers are prefixed with a minus sign (-) applied to the representation of their absolute value (e.g., -n is -[representation of n]).

• Start with the canonical prime factorization of n = p1^α1 * p2^α2 * ... * pk^αk, where p1 < p2 < ... < pk are primes and αi > 0.
• Represent n as a list [R(α1), R(α2), ..., R(αk)], where R(αi) is the recursive representation of the exponent αi.
• Recurse on each exponent αi until reaching base cases (1 or primes, which are leaves).
• Primes are represented as themselves in leaves, but if an exponent is prime, it is left as is unless further decomposed (primes stop recursion).

• If an exponent α is 1, it is [1].
• If α is prime, it is the prime value (e.g., 2, 3, 5).
• If α is composite, apply the NPF recursively to α.

Why Revolutionary?

• Hierarchical Insight: The nested structure mirrors the "building blocks" of numbers, making it intuitive for educational purposes, number theory analysis, and AI applications (e.g., input to graph neural networks for predicting properties like divisibility).
• Compactness for Large Numbers: High exponents are nested deeply rather than repeated, reducing representation length compared to linear factorization strings. For example, 2^{64} would be a deeply nested [ [...] ] rather than "2^64".
• Operational Efficiency: Arithmetic can be performed recursively—multiplication concatenates or nests lists, addition requires "merging" trees with carries resolved via recursion. This could accelerate computations in symbolic math software or quantum simulation, where nested structures align with qubit entanglement models.
• Extension to Other Domains: Easily adapts to rationals (numerator/denominator as two trees) or complexes, and supports error-resilient encoding by adding redundancy in nests.

Examples

• 0: ∅
• 1: [1]
• -1: -[1]
• 2 (prime): 2
• 3 (prime): 3
• 4 = 2^2: [2] (since exponent 2 is prime)
• -4: -[2]
• 6 = 2^1 * 3^1: [ [1], [1] ]
• 8 = 2^3: [3] (exponent 3 is prime)
• 12 = 2^2 * 3^1: [2, [1]]
• 16 = 2^4 = 2^(2^2): [[2]] (recursive on exponent 4 = 2^2)
• 1200 = 2^4 * 3^1 * 5^2: [ [2], [1], 2 ] (exponent 4 = [2], 1 = [1], 2 = 2)
• 65536 = 2^16 = 2^(2^4): [[[2]]] (deep nesting for 16 = 2^4, 4 = 2^2, 2 = 2)
• -65536: -[[[2]]]

Comparison with Traditional Systems

| Number | Binary      | Decimal | Factorial     | NPF (Lists) | Notes on NPF Advantage 
|--------|-------------|---------|---------------|-------------|------------------------
| 1      | 1           | 1       | 1!0           | [1]         | Minimal nesting for unity.
| 4      | 100         | 4       | 2!0           | [2]         | Reveals prime exponent directly.
| 8      | 1000        | 8       | 2!1!0         | [3]         | Compact for power-of-2. 
| 12     | 1100        | 12      | 2!2!0         | [2,[1]]     | Shows factors separately.
| 1200   | 10010110000 | 1200    | 4!3!2!1!0!0   | [[2],[1],2] | Nested for exponents, shorter than linear factorization (2^4*3^1*5^2 = 15 chars).
| 65536  | (17 bits)   | 65536   | Long sequence | [[[2]]]     | Highly compact for exponential towers; binary requires 17 chars, NPF uses 7.