Fixed points and the inverse problem for central configurations

Ferrario’s Theorem (3.4) characterizes the inverse collinear central configuration problem via the multi-valued map ψ: x ∈ Δ^{n−2} admits positive masses iff x lies in the convex hull of the normalized Y-columns v_j = Y_j/(Q_{1j}+Q_{jn}) . This relies on the difference formulation x = Y m (eq. (3.2)) with Y_{ij} = Q_{ij} − Q_{i+1,j} and on the positivity S_j = Q_{1j}+Q_{jn} > 0 (eq. (3.3)) . The candidate solution reproduces this equivalence exactly: (i)⇒(ii) by normalizing a positive linear combination of columns Y_j; (ii)⇒(i) by setting m_j = β_j/S_j and invoking the translation gauge (D kills the µ1 term) to recover the central-configuration equations Q m = −(λ/α) q + µ1 . Minor omissions (boundary/collision limits, scaling conventions for λ) do not affect correctness. Hence both are correct and essentially the same argument.