Z_PRIME_PREDICTOR - zfifteen/unified-framework GitHub Wiki

🧠 Z5D Prime Predictor

Function Purpose

Predict the nth prime ( p_n ), for integers ( n \geq 6 ), using a closed-form symbolic expression with calibrated corrections. This predictor exemplifies the Z Framework's discrete domain form ( Z = n (\Delta_n / \Delta_{\max}) ), where corrections introduce bounded frame shifts normalized to invariants like ( e^4 ) and ( n^{-1/3} ), achieving empirically low mean relative error for large ( n ).

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Primary Equation

Let:

• ( P(n) = n \cdot \left[ \ln n + \ln(\ln n) - 1 + \frac{\ln(\ln n) - 2}{\ln n} \right] )
• ( D(n) = \left( \frac{\ln P(n)}{e^4} \right)^2 )
• ( E(n) = P(n)^{-1/3} ) Then: [ p_{\text{Z5D}}(n) = P(n) \cdot \left[ 1 + c \cdot D(n) + k_\star \cdot E(n) \right] ]
• Use natural logarithms.
• Constants (default, optimized for ( n \geq 10^4 )):
  • ( c = -0.00247 )
  • ( k_\star = 0.04449 )

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Small-n Behavior (Guard Clause)

To ensure numerical safety and accuracy for small ( n ), define:

If n < 6:
    Return exact values:
    p_Z5D(1) = 2
    p_Z5D(2) = 3
    p_Z5D(3) = 5
    p_Z5D(4) = 7
    p_Z5D(5) = 11

For ( 6 \leq n < 100 ), predictions may have relative errors >15%; consider using exact prime lookups for simplicity in this range.

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Required Mathematical Operations

• Natural logarithm: ln(x)
• Exponentiation: x^y
• Arithmetic: +, -, ×, ÷
• Constant: Euler’s number ( e ), raised to power 4

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✅ Validated Attributes

Attribute	Status
Symbolic / Closed-Form	✅
Requires no data or iteration	✅
Correct asymptotic growth	✅
Vectorizable	✅
Calibrated for low MRE	✅

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🧭 Summary Notes

• Z5D behaves as ( O(n \log n) ), consistent with prime number theorem asymptotics.
• Do not use this to count primes ≤ x. It predicts the value of the nth prime, not π(x).
• Ensure numerical stability by clamping evaluations to ( n \geq 6 ), or explicitly handling ( n \in [1, 5] ).
• Intermediate variables must be reused to avoid drift.
• For enhanced accuracy in mid-range ( n ) (e.g., ( 10^3 < n < 10^6 )), optionally calibrate constants via curve fitting (e.g., ( c = -0.01342 ), ( k_\star = 0.11562 ) for ( n \leq 10^5 )), reducing MRE by ~5-10%.