Average n-Selmer Size Equals Divisor Sum σ(n) over F_q(t) — E8 Intelligence Research

FINDING: The average size of n-Selmer groups of elliptic curves over F_q(t) in the large-q limit equals the sum of divisors function σ(n) — a direct arithmetic-statistical law with deep lattice-theoretic content. | MATH: Let Sel_n(E/F_q(t)) denote the n-Selmer group. The geometric average (over a suitable moduli stack of elliptic curves, q→∞) satisfies: ⟨#Sel_n(E)⟩ = σ(n) = Σd|n d. For n=2: σ(2)=3 (classical Poonen–Rains result). For n=3: σ(3)=4. For n=4: σ(4)=7. This is not a heuristic — it is proven in the function-field large-q limit (Landesman, et al.). The proof uses the cohomology of the moduli space of elliptic curves, specifically the stable cohomology of the symplectic group Sp2g(Z) and the Grothendieck–Lefschetz trace formula over finite fields. | CONNECTION: **Root system Aₙ₋₁**: The sum of divisors σ(n) is precisely the number of positive roots in the root system Aₙ₋₁ counted with multiplicity by height? More precisely: σ(n) = Σd|n d = number of elements Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com