Irregular Fixation: I. Fixed points and librating orbits of the Brown Hamiltonian

The paper derives the modified fixed point by setting ω1 = π/2 and solving the closed-form condition obtained from the total Hamiltonian H = Hsec + HB, yielding 1 − e_fix^2 = sqrt[5(1 + (9/8) εSA jz) / (3(1 − (9/8) εSA jz))] |jz| and cos ι_fix = sign(jz) (3/5)^{1/4} [(1 − (9/8) εSA jz)/(1 + (9/8) εSA jz)]^{1/4} |jz|^{1/2} (their Eqs. 14–16) . The candidate reconstructs the same result by explicitly differentiating H with respect to ω1, showing ∂H/∂ω1 ∝ e1^2 sin 2ω1 with a strictly positive coefficient for the allowed parameter range, hence ω1 = π/2, and then solving the resulting algebraic equation for e1^2; the final expressions coincide with the paper’s Eqs. 15–16 and reduce correctly to the quadrupole ZLK limit (Eqs. 17–18) when εSA → 0 . For the libration criterion, the candidate’s small‑e expansion establishes the sign of H(e_fix, π/2) − H_sep in terms of S(εSA, jz) = 9 − 15 jz^2 − (27/8) εSA jz (3 + 5 jz^2), whose zero set reproduces exactly the paper’s bifurcation locus obtained by setting e_fix = 0 (their Eq. 21) , using the paper’s separatrix energy (Eq. 19) . Thus, the derivations agree in substance and outcome; any differences (e.g., the candidate’s explicit ∂H/∂ω1 factorization) are stylistic, not substantive.