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A quantitative non-sharpness certificate for Example 2.9

theorem Papers.AnsariRockel2026RhoFootrule.correlationRatio_uniform_improvement (C : ProbabilityTheory.Copula 2) : 21 / 20 * copulaCorrelationRatio C ≤ 2 * C.chatterjeeXi + 3 / 250

A universal improvement using the endpoint restrictions of conditional CDFs.

theorem Papers.AnsariRockel2026RhoFootrule.quarter_xi_uniform_gap (C : ProbabilityTheory.Copula 2) (h : C.chatterjeeXi = 1 / 4) : copulaCorrelationRatio C ≤ 256 / 525

The whole quarter-xi slice stays a positive distance below eta=1/2.

theorem Papers.AnsariRockel2026RhoFootrule.quarter_xi_uniform_separation : ∃ (e : ℝ), 0 < e ∧ ∀ (C : ProbabilityTheory.Copula 2), C.chatterjeeXi = 1 / 4 → copulaCorrelationRatio C ≤ 1 / 2 - e

Non-sharpness holds even for the supremum; no compactness premise is required.