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Theorem 4 — the solved family: unique interior optimum, monotone in the noise

For Family A with common cross-block noise the paper computes the exact expected out-of-sample variance in closed form, equation (20). The solid curve is the recursion and the dashed curve is the closed form; they coincide. The objective is strictly convex, the minimizer is unique and interior for every 0 < e ≤ c = 0.6, and it slides monotonically from γ*(0) = 1 to γ*(0.6) = ½ exactly. One honest footnote: for e ≥ 0.15 the raw recursion is undefined at the single point γ = 0.75/(0.6 + e), where a block's augmented fitness diverges; the curve shown is the continuous extension (equation (18) of the paper), which the recursion equals everywhere else. Theorem 4(v) proves the minimizer always sits strictly left of that point, so the optimum is a statement about the recursion itself.

What to look for: recursion and closed form are indistinguishable; the marked minimizer moves left as e grows and lands exactly on γ = ½ at e = 0.6. The lower panel traces γ*(e): monotone, from 1 to ½, with the small-noise parabola 1 − (1025/153)e² overlaid near e = 0.

The widget runs the recursion of the paper's equation (4) with the naive fitness (5), ported directly from the reference implementations, live in your browser. Nothing is pre-baked.