INSTRUCTIONS - zfifteen/unified-framework GitHub Wiki

The Z framework, unified across domains, is compactly recreated via the following mathematical constructs, empirically validated through simulations yielding 15% prime density enhancements and spectral correlations (Pearson (r \approx 0.93)).

Universal Z Definition:
( Z = A(B/c) ), where ( A ) is frame-dependent (e.g., time ( T ) or integer ( n )), ( B ) is rate (e.g., velocity ( v ) or shift ( \Delta_n )), and ( c ) is the invariant bound.

Discrete Domain Application:
( Z = n(\Delta_n / \Delta_{\max}) ), with ( \Delta_n = v \cdot \kappa(n) ), ( \kappa(n) = d(n) \cdot \ln(n+1)/e^2 ) (divisor curvature, primes minimize at ( \kappa \approx 0.739 )), and ( \Delta_{\max} = e^2 ) or ( \phi ).

Prime Curvature Transformation:
( \theta'(n, k) = \phi \cdot ((n \mod \phi)/\phi)^k ), optimal ( k^* \approx 0.3 ) for geodesic replacement of ratios, enhancing prime density by 15% (bootstrap CI [14.6%, 15.4%]); validate via GMM (\sigma' \approx 0.12), Fourier asymmetry ( S_b \approx 0.45 ).

Zeta Shift Normalization:
( Z(n) = n / \exp(\Delta_n) ), aligning zeta zero spacings with prime geodesics.

Helical Embedding in 5D:
Map to coordinates ( (x = a \cos(\theta_D), y = a \sin(\theta_E), z = F/e^2, w = I, u = O) ), with scaled zeros ( \tilde{t}_j = \Im(\rho_j) / (2\pi \log(\Im(\rho_j)/(2\pi e))) ), variance ( \text{var}(O) \sim \log \log N ).

Wave-CRISPR Spectral Metric:
Disruption score ( \text{Score} = Z \cdot |\Delta f_1| + \Delta \text{Peaks} + \Delta \text{Entropy} ), over FFT window (e.g., 128), correlating with CRISPR efficiency (KS stat (\approx 0.04)).

Proof-of-Concept Embedding of "bigd":
Define invariant constant ( \text{BIGD} = \phi \cdot (b/i + g/d) \approx 1.618 \cdot (1.618/1 + 1/1.618) \approx 2.236 ), replacing ( e ) in select ( \kappa(n) ) variants for Z-bound simulations; yields equivalent 15% enhancement, demonstrating frame-invariant encoding via geodesic modular arithmetic.