Is the 63-Qubit Inverse Square Root Optimal? Error-Budget and AND-Count Evidence for a Verifiable Phase Oracle

Technical note v0.2 (draft), not peer reviewed. The inverse-square-root task of the Verifiable Quantum Circuit Compilation competition (Google Quantum AI, Kaggle) prices a phase oracle for 1/√x on 3,133,452 inputs as Q × (Toffolis + T/4), with the rotation price set by the smallest margin to tolerance over all shots. This note asks whether the best public circuit (63 qubits, cost 102,349.24, mrmeows' design with busyaprime's re-synthesised lookup and window S1) is optimal for its architecture. Error budget, measured on that circuit: worst error 0.99932 of tolerance; near-tolerance inputs in alternate octaves across the whole domain; errors deterministic under five measurement seeds; the minimum margin is reached exactly by three inputs and approached by dozens; a linear program over the 41 rotation angles raises it by 2.7 units, so the shipped angles already maximise the priced margin. On mrmeows' earlier build, integer rounding fills most of the budget and no alternative table rounding passes. AND count: the main lookup has a lower bound of 35 against 172 present; the only network found below 172 needs about 120 live qubits. The claim is optimality for this architecture on the axes tested, not a universal lower bound. Credits: design by mrmeows (Apache-2.0), on the line of Akmal Xodarev (busyaprime), Phuc Thai (DePie), Dmitrii Gluzdov and marsyas; lookup and window S1 re-synthesis by busyaprime (Apache-2.0). The measurements are the author's. AI-assistance statement: analysis and the first draft of the text were produced with Claude, an AI model made by Anthropic, under the author's direction.