Non-ergodic Furstenberg recurrence in Lean from Mathlib's finitary Roth (no ergodic decomposition)

To get liminf N⁻¹∑ μ(A∩f⁻ⁿA∩f⁻²ⁿA) > 0 for EVERY measure-preserving f without ergodic decomposition, run Furstenberg backwards from roth_3ap_theorem_nat (Mathlib.Combinatorics.Additive.Corner.Roth; bound cornersTheoremBound (ε/3) ≤ n, A ⊆ range n, ε*n ≤ #A ⟹ ¬ThreeAPFree). For g = f^[k]: points with ≥ εL visits among L iterates have measure ≥ μA − ε (integrate the count ∑ indicator(g^[m]⁻¹'A)); Roth puts an AP in the visit times; union bound gives ∑_{1≤d<L} a(kd) ≥ ε/L. Summing over k ≤ N/L, each n is hit ≤ L times, so the Cesàro mean is ≥ ε/(2L³), uniformly over all systems. ~270 lines, compiled in 5 rounds. Gotchas: threeAPFree_iff_eq_right gives a+c=b+b ∧ a≠c after negation; Finset.sum_image names are (f := summand) (g := map); measureReal_mono needs the μ t ≠ ∞ argument explicitly for a biUnion; Function.iterate_mul f k d : f^[k*d] = f^[k]^[d].