Cyclic sums of comparative indices and their applications

The paper’s main theorem defines S_{1,2,...,m} as the full off-diagonal block matrix with entries wi,j = Y_i^T J Y_j for all pairs i ≠ j (equation (1.16)), and proves µ_c^±(Y_1,...,Y_m) = ind(±S_{1,2,...,m}) (equation (1.17)) . By contrast, the candidate solution explicitly replaces S_{1,2,...,m} with a different, banded “nearest-neighbor” (block-cyclic) matrix whose only nonzero off-diagonal blocks are w(Y_i,Y_{i+1}) and w(Y_{i+1},Y_i). This mismatch invalidates the subsequent pair-elimination argument, which assumes that the pivot “couples only to two neighbors,” an assumption that is false for the full matrix (1.16), where each block couples to all other blocks via wi,j . The paper’s proof proceeds instead by symplectic invariance of cyclic sums (Proposition 2.5) and a careful inertia analysis based on Tian’s block-inertia formula (equation (2.4)) to handle the full, dense off-diagonal structure, culminating in the identities (1.17) and the related representations in Corollary 3.3 and Theorem 3.4 . Although the model’s use of the Moore–Penrose pseudoinverse and the definitions of M, T, P match the paper’s comparative-index components (see (1.4)–(1.6)), applying a local LDL^T “pair-collapse” to the wrong S leads to an incorrect conclusion for the stated theorem. The m=2 sanity check aligns with (1.9), but the general case fails due to the structural error in S .