Description of fixed points of an infinite dimensional operator

The paper reduces the infinite-dimensional operator F on l1+ to a two-dimensional map W, classifies diagonal versus off-diagonal fixed points, and enumerates parameter regions A_{i,j} with up to seven fixed points; all key steps (reduction, cubic on M0, uv=1 and s-analysis off M0, and the seven-case partition) match the candidate’s solution. The model reproduces the same structure: (i) the parity-wise proportionality reduction F↔W, (ii) the diagonal cubic and tangency curves (θ-threshold at 17 and L-curves), and (iii) the off-diagonal uv=1, s≥2, and thresholds L=(θ+3)^2/4 and L=4(θ−1). A small phrasing slip in the model at θ=17 (it says “no positive solution” to the tangency equation, though the degenerate case exists but does not change the count) does not affect the classification. Overall, both are consistent and essentially the same proof approach, with the paper’s details aligned to the derivations cited here: the operator F and reduction to W, the 1D restriction f on M0 and the cubic, the off-diagonal reduction to ξ=s=u+1/u and the quadratic ξ^2−√L ξ+(θ−1)=0, the parameter partition A_{i,j}, and the explicit fixed points P1–P7 (paper: definitions and sets; model: same derivations and thresholds) .