Flip Signatures

The paper’s Theorem D proves that under a D∞-conjugacy, the flip signatures have the same number of negative entries and equal leading signature, via a D∞-SSE decomposition (Proposition A) and careful control of J- and K-pairings on the eventual kernels through D∞ half–elementary equivalences (Proposition B, Proposition C, Lemma 5.1) . By contrast, the model asserts a stronger, incorrect conclusion (it claims full equality of the flip-signature tuples) and relies on an unsubstantiated “cohomology identity” S^T K S − J = A^T H − H A. The paper neither states nor uses such an identity, and, crucially, cycle lengths need not be preserved under a half–elementary step (they can shift by ±1), contrary to the model’s assumption (see (3.10)–(3.11) and (1.5)) .