Family Relationships and Plain-Language Summary of Abdullah's Inheritance Equation

This paper derives the evolutionary distance L for five within-family relationship types from the main equation, and provides a plain-language explanation of the equation's two core parts. **Main Equation** G(t₂⟨σ⟩) = α_A⟨σ⟩ · G(t₁⟨σA⟩) + α_B⟨σ⟩ · G(t₁⟨σB⟩) + Λ⟨σ⟩ where Λ⟨σ⟩ = k⟨σ⟩ ∫ ε⟨σ⟩(t) dt (from t₁⟨σ⟩ to t₂⟨σ⟩) **Part 1: What you inherited by birth** α_A⟨σ⟩ · G(t₁⟨σA⟩) + α_B⟨σ⟩ · G(t₁⟨σB⟩) — the mixture from mother and father, with α_A + α_B = 1. **Part 2: What you gained by living** k⟨σ⟩ ∫ ε⟨σ⟩(t) dt — where ε(t) is the moment-by-moment rate of genetic change, the integral is the total accumulated genetic change over a lifetime, and k ∈ [0,1] is the fraction of that change passed on to a child (k=0: none passes; k=1: all passes). **Five Family Relationships Derived** 1. **Same individual:** ⟨σ₁⟩ = ⟨σ₂⟩ → L = 0 (identity axiom) 2. **Identical (monozygotic) twins:** α_A⟨0⟩ = α_A⟨0₂⟩, α_B⟨0⟩ = α_B⟨0₂⟩ (single fertilization, split once) → L = |Λ⟨0⟩ − Λ⟨0₂⟩| — the entire difference is epigenetic; heredity contributes zero. 3. **Fraternal (dizygotic) twins:** two separate eggs, same t₁ → L = |Δα_A·G_A + Δα_B·G_B + ΔΛ| — mathematically identical to an ordinary sibling. 4. **Ordinary sibling:** same mother and father, different t₁ → L = |Λ⟨0⟩ − Λ⟨0₂⟩| (ideal case, α's equal); general case adds the Δα term as in (3). 5. **Half-sibling:** one shared parent only (e.g. same mother, different fathers) → L = |α_B·ΔG_B + ΔΛ| — the hereditary term from the differing parent does NOT cancel, so this L is systematically larger than for an ordinary sibling. **Resulting natural ordering (derived, not assumed):** L(same individual) ≤ L(identical twin) ≤ L(ordinary sibling) ≤ L(half-sibling) This ordering matches known results from real twin studies. --- 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