On a perturbed Hofstadter $Q$-recursion

The Hofstadter Q-sequence is a prominent example of nested recurrence. Despite decades of study, it is not even known whether Q(n) is defined for all n. Mantovanelli introduced a parity-perturbed variant $\tilde{Q}$, obtained by adding $(-1{)}^n$ to the recursion, which surprisingly replaces the chaotic behaviour of Q by an exact dyadic self-similarity. In this paper we prove that $\tilde{Q}$ is well-defined for all n and satisfies $\mathrm{\mid }\tilde{Q}(n)\mathrm{/}n-1\mathrm{/}2\mathrm{\mid }=O(1\mathrm{/}\sqrt{\log n})$. The proof exploits the self-similar structure of the sequence, where alternating arches arise whose frequency combinatorics are governed by the Catalan numbers. A complementary analysis of the arch amplitudes, conditional on two minimal conjectural properties, refines the asymptotic formula to ${\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}\mathrm{\mid }\tilde{Q}(n)\mathrm{/}n-1\mathrm{/}2\mathrm{\mid }\sqrt{{\log }_{2}n}=1\mathrm{/}(3\sqrt{2\pi })$. Numerical experiments suggest the conjecture $Q(n)-\tilde{Q}(n)=O(n\mathrm{/}\sqrt{\log n})$, indicating that $\tilde{Q}$ may serve as a tractable proxy for Q. This experimental direction will be investigated elsewhere.