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

This paper introduces Abdullah's Evolutionary System, a mathematical framework for heredity and evolution. 1. Genetic Value GEvery individual is assigned a real number G(t), interpreted as a genetic position that moves through time — analogous to position x(t) in physics. 2. Evolution and Evolutionary SpeedΔG = G(t₂) - G(t₁)E = dG/dt 3. Evolutionary DistanceL = |G₁ - G₂| (instantaneous)dL/dt = |E₁ - E₂| (only in the monotone case; in general L ≤ Ξ where Ξ = ∫|E₁-E₂|dt is the cumulative path of divergence) 4. Lineage Path NotationEach individual is identified by a string σ over {A, B}, where A = mother's line, B = father's line:⟨0⟩ = self⟨A⟩ = mother⟨B⟩ = father⟨A²⟩ = ⟨AA⟩ = maternal grandmother⟨AB⟩ = maternal grandfather⟨A²B⟩ = ⟨AAB⟩ = maternal grandmother's father⟨AABBA⟩ = mother's mother's father's father's motherString algebra:⟨XX⟩ = ⟨X²⟩, ⟨XXX⟩ = ⟨X³⟩⟨0σ⟩ = ⟨σ⟩ (neutral element)|⟨σβ⟩| = |σ| + |β| 5. Two Time Points Per Individual ⟨σ⟩t₁^⟨σ⟩ = parental mating timet₂^⟨σ⟩ = any moment of life (covers birth and after)Connection rule: t₂^⟨σA⟩ = t₁^⟨σ⟩ and t₂^⟨σB⟩ = t₁^⟨σ⟩ 6. Main Axiom — Abdullah's Inheritance EquationG(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⟩ (simple form when ε is constant: Λ_⟨σ⟩ = k_⟨σ⟩ · ε_⟨σ⟩ · Δt^⟨σ⟩)α_A^⟨σ⟩ + α_B^⟨σ⟩ = 1 (recombination fractions), G ∈ [0,∞)k_⟨σ⟩ ∈ [0,1] (epigenetic transmission constant)ε_⟨σ⟩ = dG/dt (individual evolutionary speed)Δt^⟨σ⟩ = t₂^⟨σ⟩ - t₁^⟨σ⟩ (lifetime interval)Ideal system: α_A = α_B = ½, k = 0 7. Closed Box PrincipleOnly parental values at mating time enter the equation. Prior history is irrelevant:G(t₁^⟨σA⟩), G(t₁^⟨σB⟩) → G(t₂^⟨σ⟩) 8. Classical Theories as Special CasesMendel: α_A = α_B = ½, k = 0Hardy-Weinberg: α = ½, k = 0, ε = 0Wright r = (½)ⁿ: emerges as w^⟨σ⟩ = ∏α = (½)ⁿWeismann barrier: k = 0Lamarck: k = 1Epigenetics: 0 < k < 1Recombination deviation: α_A ≠ α_B 9. Connection to the Dinosaur and LUCAThrough recursive expansion as n → ∞, the weight of one specific ancestor:w^⟨single dinosaur⟩ = (½)^3,000,000 ≈ 0w^⟨LUCA⟩ = (½)^160,000,000 ≈ 0Important: this does not mean the total contribution vanishes. At every generation there are 2ⁿ simultaneous ancestors, and the sum of all their weights is always exactly 1. So while any one ancestor's individual weight is negligible, the mathematical connection to every generation, including the dinosaur and LUCA, never fully disconnects. --- 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