MokraBela's Spectral Laws: Complete Discovery Package – Conceptual Summary & Reference Sheet

Author: Ahmed Mokrane | ORCID: 0009-0005-2319-6098Affiliation: Independent Researcher, Sidi Bel Abbes, Algeria The MokraBela Discovery: Prime Numbers as a Wave System This package contains two complementary documents that together present a fundamental breakthrough in number theory. File 1: The Great Discovery "Understanding Prime Numbers Without Tracking Them" The Vision: Answers the two great questions of prime distribution. The Mechanism: Explains the MokraBela Spectral Transformation. Quick Read: 3 pages – conceptual summary of the discovery. File 2: MokraBela's Spectral Law "The Complete Empirical Framework" The Evidence: Complete table of 7 massive-scale experiments (N = 70,000,000). The Constants: Formal definition of all discovered laws and invariants. Quick Reference: 2 pages – reference sheet for researchers. The Discovery in One Sentence "Prime numbers are not isolated points — they are a deterministic wave system that can be understood collectively." The Five MokraBela Spectral Laws Spectral Power Law: |A_k| ~ gamma_k^alpha (alpha = -0.933) Mokrane Constant: M / sqrt(N) = 52.5 Golden Ratio Identity: NM x N_alpha = 2756.25 --> alpha = -1 Energy Stability Law: E(K) = O(1) ≈ 0.0474 Convergence Law: alpha(N) = -1 + 1.02 / log N Statistical Validation GUE Statistics: D = 0.018 (vs Poisson D = 0.548) Selberg's Alignment: ||R||^2 / K = 0.0474 (within 2.4% margin) von Koch Limit: alpha -> -1 (Asymptotic convergence confirmed) [العربية] الاكتشاف الاكتشاف في جملة واحدة الأعداد الأولية ليست نقاطاً معزولة – بل هي نظام موجي يمكن فهمه بشكل جماعي. القوانين الطيفية الخمسة المكتشفة 1. قانون القوة الطيفية: |A_k| ∼ γ_k^α (α = -0.933 ± 0.015) 2. ثابت مكرانة: M/√N = 52.5 3. متطابقة النسبة الذهبية: π(N) × N_α = 2756.25 → α = -1 4. قانون استقرار الطاقة: E(K) = O(1) ≈ 0.045-0.054 5. قانون التقارب: α(N) = -1 + 1.02/log N التحقق الإحصائي - GUE: D = 0.018 (مقابل بواسون D = 0.548) - ثابت سيلبرغ: ‖R‖²/K = 0.0474 (بفارق 2.4%) - حد فون كوخ: α = -0.933 (بفارق 6.7%) [Français] la Découverte La Découverte en Une Phrase Les nombres premiers ne sont pas des points isolés — ils sont un système d'ondes qui peut être compris collectivement. Les Cinq Lois Spectrales de MokraBela Découvertes 1. Loi de Puissance Spectrale : |A_k| ∼ γ_k^α (α = -0.933 ± 0.015) 2. Constante de Mokrane : M/√N = 52.5 3. Identité du Nombre d'Or : π(N) × N_α = 2756.25 → α = -1 4. Loi de Stabilité Énergétique : E(K) = O(1) ≈ 0.045-0.054 5. Loi de Convergence : α(N) = -1 + 1.02/log N Validation Statistique - Statistiques GUE : D = 0.018 (vs Poisson D = 0.548) - Constante de Selberg : ‖R‖²/K = 0.0474 (à 2.4%) - Limite de von Koch : α = -0.933 (à 6.7%) How to Cite Mokrane, A. (2026). MokraBela's Spectral Laws: Complete Discovery Package. Zenodo. DOI: 10.5281/zenodo.19884833.