Abdullah's Evolutionary System: Lineage Path Notation and a Unified Framework for Heredity

This paper derives a special result from Abdullah's Inheritance Equation: the evolutionary distance between two siblings coming from the same parents reduces entirely to the epigenetic/individual term. **1. Starting Point — The Main Axiom** Λ⟨σ⟩ = k⟨σ⟩ ∫ ε⟨σ⟩(t) dt (from t₁⟨σ⟩ to t₂⟨σ⟩) G(t₂⟨σ⟩) = α_A⟨σ⟩ · G(t₁⟨σA⟩) + α_B⟨σ⟩ · G(t₁⟨σB⟩) + Λ⟨σ⟩ where ⟨σ⟩ is the lineage path of any individual; ⟨σA⟩ is the mother, ⟨σB⟩ is the father. **2. Notation for Siblings** In the standard system, ⟨0⟩ is a unique reference reserved for "oneself" — two different individuals cannot both be ⟨0⟩. For a sibling we therefore write ⟨0₂⟩, not ⟨0'⟩: you are ⟨0⟩, your sibling is ⟨0₂⟩. Both share the same first step of the lineage path: G(t₁⟨0A⟩) = G(t₁⟨0₂A⟩) = G(t₁⟨A⟩)G(t₁⟨0B⟩) = G(t₁⟨0₂B⟩) = G(t₁⟨B⟩) **3. The Equations for Two Siblings** G(t₂⟨0⟩) = α_A G(t₁⟨A⟩) + α_B G(t₁⟨B⟩) + Λ⟨0⟩ G(t₂⟨0₂⟩) = α_A G(t₁⟨A⟩) + α_B G(t₁⟨B⟩) + Λ⟨0₂⟩ **4. Derivation — Taking the Difference** L = |G(t₂⟨0⟩) − G(t₂⟨0₂⟩)| = |(α_A G_A + α_B G_B + Λ⟨0⟩) − (α_A G_A + α_B G_B + Λ⟨0₂⟩)| The common parental terms (α_A G_A + α_B G_B) cancel completely, since both siblings use the same parental values: L = |Λ⟨0⟩ − Λ⟨0₂⟩| **5. Critical Warning — Same Symbol, Different Value** Λ⟨0⟩ = k⟨0⟩ ∫ ε⟨0⟩(t) dt (from t₁⟨0⟩ to t₂⟨0⟩)Λ⟨0₂⟩ = k⟨0₂⟩ ∫ ε⟨0₂⟩(t) dt (from t₁⟨0₂⟩ to t₂⟨0₂⟩) These two terms can differ for three reasons:- k⟨0⟩ ≠ k⟨0₂⟩ — each individual's epigenetic transmission constant can differ- ε⟨0⟩(t) ≠ ε⟨0₂⟩(t) — each individual undergoes a different life experience- t₂⟨0⟩ ≠ t₂⟨0₂⟩ — the moment of measurement can differ, so the lifetime interval Δt differs The symbol Λ is written with the same letter for both siblings, but because the subscript ⟨σ⟩ differs, the two numerical values are generally not equal. **6. Two Special Cases** Ideal system (k⟨0⟩ = k⟨0₂⟩ = 0):Λ⟨0⟩ = 0, Λ⟨0₂⟩ = 0, so L = 0. The siblings are mathematically identical. Real system (k ≠ 0, general case):Λ⟨0⟩ ≠ Λ⟨0₂⟩ generally, so L ≠ 0. Even with identical inheritance, the entire difference between siblings comes from lived experience. **7. Generalization — Any Two Individuals Sharing a Common Point** L(σ, σ₂) = |Λ⟨σ⟩ − Λ⟨σ₂⟩| condition: ⟨σ⟩ and ⟨σ₂⟩ come from the same parental pair (⟨σA⟩ and ⟨σB⟩ terms are equal). In this form, L is completely purified of the hereditary (genetic) component — only the difference accumulated over each individual's own life remains. This is a clean mathematical statement that evolutionary distance comes not only from birth, but also from the life that is lived. --- All ideas, definitions, axioms, and formulas belong to Abdullah Baran. Claude served only as scribe — typesetting and mathematical verification only. [e-mail] | x.com/realABaran | ORCID: 0000-0003-2935-1835