Hall-Petch limit: σ_y(d=1)=σ_0+k_HP; σ_y(d→∞)=σ_0

Sympy-exact witness of the Hall-Petch limiting behaviour. Setup: σ_y = σ_0 + k_HP/√d. Identity 1 (d=1 reference-grain-size anchor): σ_y(1) = σ_0 + k_HP — the characteristic grain-refinement contribution. Identity 2 (d → ∞,…