The Kac formula and Poincaré recurrence theorem in Riesz spaces

The paper’s Kac formula states LS n_p = P_{LS p} e under T strictly positive, E T-universally complete, and S surjective, and proves it by (i) expressing n_p via the backward-recurrence construction (Lemma 4.1) and showing LS n_p = LS(∨_{k≥0} S^{-k}p) (Lemma 4.2), then (ii) identifying w := ∨_{k≥0} S^{-k}p with P_{LS p} e using S-invariance and band-projection arguments in Theorem 4.4. This argument is internally coherent; the only subtle step is the use of Lemma 2.4 with the operator LS, but LS is indeed strictly positive because TLS = T and T is strictly positive, so LS f = 0 with f ≥ 0 implies 0 = T(LS f) = T f, hence f = 0. Therefore the application P_{LS p} e ≥ P_p e = p is justified, and the proof closes correctly (Theorem 4.4; supporting Lemmas 4.1–4.2 and Theorem 2.5) . By contrast, the candidate solution relies on several incorrect or unproven claims: (a) it assumes LS f ∈ R(T) for all f, which contradicts the paper’s identification R(LS) = IS with only R(T) ⊂ R(LS) (Cor. 2.7) ; (b) it claims monotonicity of a forward-stack sequence a_n without proof (and the claim is generally false); and (c) it uses commutation/centrality properties that require elements in R(T), which it asserts for LS p but does not have. These gaps break the model’s argument, even though the final identity matches the paper’s conclusion.