Self-composition of Frobenius a real number of times, and some conjectures around Weil’s Riemann hypothesis

The paper explicitly proves: If Standard Conjecture D holds for X×X, then every polarized endomorphism f of X acts semisimply on cohomology (Result 1 in the abstract and Theorem 0.3). The proof combines (i) a linear-algebra lemma bounding sp(AA^τ) and (ii) an inequality from Hu–Truong–Xie (Lemma 2.5 and the proof of Theorem 1.1(2) there), to rule out nontrivial Jordan blocks for f* on each H^k(X) under D alone. See the abstract claim and the detailed statement and proof of Theorem 0.3, including the reliance on Lemma 2.5 of [HTX25] and the role of negative powers of Frobenius in the norm estimates. These appear verbatim in the uploaded PDF (Result 1: all polarized endomorphisms are semisimple under D; Theorem 0.3 and proof) . In particular, the paper notes that Theorem 1.1 of [HTX25] implies that, under D, if one polarized endomorphism is semisimple then all are, and then it supplies an argument (using the Frobenius-based inequality with s negative) that removes the need to assume Frobenius semisimplicity, directly proving semisimplicity for all polarized endomorphisms under D . By contrast, the candidate solution asserts the statement is still open under D alone and only equivalent to Frobenius semisimplicity. That directly contradicts the paper’s main theorem and proof, so the model is wrong relative to the provided paper.