ON FLOER-TYPE NUMERICAL INVARIANTS, GIT QUOTIENTS, AND ORBIT BIFURCATIONS OF REAL-LIFE PLANETARY SYSTEMS (WITH NUMERICAL EXPLORATIONS OF THE JUPITER-EUROPA AND SATURN-ENCELADUS SYSTEMS)

The uploaded paper states the exact iteration formula for the Hörmander index of a nondegenerate symmetric periodic orbit x^k in terms of Chebyshev polynomials, namely s(x^k) = (1/2)·sign((Id − T_k(A)) U_{k−1}(A)^{-1} C^{-1}), with the one-iterate case s(x) = (1/2)·sign((Id − A) C^{-1}) and with the reduced monodromy in Wonenburger form M = [A B; C A^T] satisfying B = B^T, C = C^T, AB = BA^T, A^T C = C A, A^2 − BC = I. It also records the needed nondegeneracy condition and Chebyshev definitions, and cites [12] for the result, but does not present a full proof in the text. The candidate model provides a detailed proof outline: (i) deriving the Chebyshev iterate identities A_k = T_k(A), B_k = U_{k−1}(A)B, C_k = C U_{k−1}(A); (ii) explaining why nondegeneracy implies C and C_k are invertible; and (iii) computing the Hörmander index as a signature of a symmetric form depending only on the endpoint and extending this to iterates. These steps match the stated theorem and standard arguments. Thus the statement in the paper is correct, and the model’s proof is correct and compatible, though not verbatim the same as the paper’s presentation, which cites the result rather than proving it. Key elements and assumptions used by both are explicitly present in the PDF, including the block relations, the nondegeneracy definition, and the theorem statement with the Chebyshev formula (Theorem 3.1) .