Sharp bounds on the central moments of the individual causal effect from several marginal central moments: two open cells settled by exact dual certificates

the finitely supported form of the problem and its LP-duality machinery: mom, mix, mu as Finset sums over ℝ, dual_bound (a pointwise nonnegative integrand has nonnegative weighted sum), expand (a general element of the dual cone – constant, u¹..u⁴, v¹..v⁴, uv, uv², u² v – expands into the moment data), and muₜwo, muₜhree (μ⁽²⁾ = σ₁⁽²⁾ + σ₀⁽²⁾ - 2E[UV], μ⁽³⁾ = σ₁⁽³⁾ - σ₀⁽³⁾ + E[3UV² - 3U²V])

(m, K) = (2, 2,3,4), the first cell the source leaves open for the variance: at σ₁ = (2, 6, 26) and σ₀ = (3, 0, 15) for k = 2, 3, 4, every finitely supported joint law of the two centred potential outcomes has μ⁽²⁾ ≥ 1 (mu2_geₒne), and the three-atom law on (-1,-2), (0,1), (4,3) with weights (2/5, 1/2, 1/10) realises the instance and attains μ⁽²⁾ = 1 (attainsₒne) – so 1 is the SHARP lower bound. Certificate: the quartic F(u,v) = 43/144 + u + 35/36 u² - 11/36 u³ + 5/144 u⁴ - v/2 + v²/8 + v³/18 + v⁴/48 - uv is an exact rational sum of three squares, its zero set is the three atoms, and it integrates to 2 - E[UV]

the other endpoint of the same cell: at the same instance every finitely supported joint law has μ⁽²⁾ ≤ 9 (mu2ₗeₙine), and the reflected three-atom law on (-1,2), (0,-1), (4,-3) with the same weights realises the same instance and attains μ⁽²⁾ = 9 (attainsₙine). So the sharp interval is exactly [1, 9]. Because σ₀⁽³⁾ = 0 the reflection v ↦ -v fixes the instance, so the single certificate F gives both endpoints

the source's Theorem 7(4) interval is strictly wider at both ends at this instance: (√2 - √3)² < 1 and 9 < (√2 + √3)², i.e. 5 - 2√6 < 1 ≤ μ⁽²⁾ ≤ 9 < 5 + 2√6 (paperᵢntervalₛtrictly_wider)

(m, K) = (3, 2,3,4), the first open cell of the odd-order table: at σ₁ = (3, -6, 36) and σ₀ = (7, -12, 76) for k = 2, 3, 4, every finitely supported joint law has μ⁽³⁾ ≤ 48 (mu3ₗe), and the three-atom law on (-4,-2), (0,2), (2,-4) with weights (1/8, 5/8, 1/4) realises the instance and attains μ⁽³⁾ = 48 (attains₄8) – so 48 is the SHARP upper bound. Certificate: the quartic F(u,v) = 112/11 + 12u + 112/11 u² + 29/11 u³ + 4/11 u⁴ - 36/11 v - 32/11 v² + 3/11 v³ + 4/11 v⁴ - 3uv² + 3u²v is an exact rational sum of three squares vanishing exactly at the three atoms, and integrates to 42 - E[3UV² - 3U²V]

the source's Theorem 11(2) bound at that instance, characterised without radicals: if p, q > 0 with p⁴ = 36, q⁴ = 76 and B > 0 with 27 B⁴ = 4 (p + q)¹² – i.e. B = (√2/⁴√27)(⁴√σ₁⁽⁴⁾ + ⁴√σ₀⁽⁴⁾)³ – then 48 < B (paper_bound_gt)

non-degeneracy of both instances: no monic quadratic annihilates either marginal. For the m = 2 instance E[(U² + αU + β)²] ≥ 4 and E[(V² + αV + β)²] ≥ 6 for all real α, β; for the m = 3 instance ≥ 15 and ≥ 45/7. Equivalently all four order-4 Hankel matrices [[1,0,σ⁽²⁾],[0,σ⁽²⁾,σ⁽³⁾],[σ⁽²⁾,σ⁽³⁾,σ⁽⁴⁾]] are positive definite (determinants 8, 18, 45, 45; Schur complements 4, 6, 15, 45/7). (no_quadraticₖillsᵤ, no_quadraticₖillsᵥ in both Mu2 and Mu3)