Mother Function — Al-Jabri Unified Theory

🏡 Back to The Oasis 🌐 العربية

Unified Cosmic Parent Function - Al-Jabri Unified Theory Z+C+A=1

Official Theory Poster

MAIN FUNCTION

Z(x) = x⁵ ln(x) · sin(2π/x) · exp(−x / x_p)

Domain: x > 0, x_p > 0 — characteristic scale parameter

Oscillatory behavior from sin(2π/x) produces decaying oscillations as x increases
Exponential term exp(−x/x_p) provides damping for large x
Prefactor x⁵ ln(x) modulates amplitude with logarithmic growth

NORMALIZATION IDENTITY

Z_x = Z + C + A · Z + C + A = 1

Conservation condition: the sum Z + C + A is normalized to unity, ensuring closure of the system. Z_x is the normalized total.

⚡ Multiscale Structure

Combines power-law growth x⁵, logarithmic factor ln(x), and periodic term sin(2π/x) to capture oscillations across scales.

🌀 Damping & Decay

Exponential damping exp(−x/x_p) ensures the function converges to zero as x → ∞, modeling stable attenuation.

🌌 Physical Role

Proposed as a spectral-type function for modeling structured oscillatory distributions in theoretical physics and cosmology.

🔗 Normalization

The condition Z + C + A = 1 provides a conserved unit sum, useful for probabilistic or structural interpretations.

📈 Illustrative Plot

Z(x) for x_p = 2.0 — Oscillatory Decay

📖 Research Context

This formulation explores a damped oscillatory function intended for modeling spectral-like distributions in theoretical cosmology and mathematical physics. The structure integrates power-law, logarithmic, and periodic components with exponential decay to represent realistic attenuation processes observed in natural oscillatory systems. The normalization constraint ensures probabilistic interpretation and total conservation across the defined domain.

📊 Equations and Graph

📸 Illustrative Materials for the Mother Function

English Explanation 1 - Mother Function Z(x)

English Explanation 2 - Mother Function Z(x)

English Explanation 3 - Mother Function Z(x)

FROM SANA'A TO THE COSMOS