Dimension preserving set-valued approximation and decomposition via metric sum

The paper proves the exact statement the model judged “likely open.” Specifically, Theorem 4.19 shows that for any f ∈ C(I, Kc(R)) with dim_H G*(f) ≥ β > 1, there exist g*, h* with dim_H G*(g*) = dim_H G*(h*) = β and f = g* ⊕ h* . The proof uses: (i) D1 = {dim_H = 1} is Gδ-dense and every f decomposes as f = g ⊕ h with g, h ∈ D1 (Theorem 4.15) , via the translation lemma for dense Gδ-sets under ⊕ (Lemma 4.14) ; (ii) associativity on Kc(R) (Proposition 2.3) and the scalar-linearity of metric linear combination (Note 2.1) to algebraically construct g*, h* with g* ⊕ h* = g ⊕ h = f ; and (iii) the bi-Lipschitz invariance of dim_H under ⊕ with a Lipschitz map (Theorem 4.8) to control the dimension outside a compact set A where dim_H G*(f|A) = β . Minor exposition gaps (e.g., Lemma 4.14’s density step and the implicit use of Note 2.1 in Theorem 4.19) are fixable, but the argument is coherent. Hence the paper’s decomposition result stands, while the model’s “likely open” verdict is incorrect.