The Floor of an Inverse-Sqrt Kickmix Circuit: A Full-Domain Optimality Analysis

The Google Quantum AI "kickmix" challenge scores a submitted quantum circuit for computing 2/√x by the product Q·C, where Q is the declared qubit count and C the per-shot executed cost (Toffoli count plus one quarter of the rotation T-charge). The public competition frontier converged, across a 22-step lineage of independent submissions, to a single point: a 62-qubit, 97,747.215-cost circuit implementing a Newton–Raphson fixed-point pipeline. This report presents a complete, measured optimality analysis of that circuit across seven axes: op-level editability (1,475 toffoli-like gates scanned — zero constant controls, zero canceling pairs, zero degenerate gates), the rotation floor (information-bounded at ~32 bits; 41 measured rotations are individually live and accuracy-locked), the algorithm configuration space (156 candidate configurations; exactly one feasible shape), register liveness (all 40 scratch slots genuinely live across a 16,366-line occupancy plateau), erase-and-recompute economics (a set-cover formulation over 381 feasible replay windows; break-even 25.8 cost-centers per qubit against a measured recompute cost of 52 — uneconomic by 2×), value-fact edits (three full-domain-verified arithmetic identities with no gate-level referent in the fused synthesis), and block-level exact tightness (full-domain ablation: 204/204 gates of the decoded multiplier block load-bearing). Every verdict is backed by full-domain exact computation over all 3,133,452 in-domain inputs, not sampling. The conclusion is that 62q/97,747.215 is the floor of the Newton–Raphson design family: reaching the #1 leaderboard position (~60,300 implied cost) requires a new algorithm family, not circuit surgery. The durable contribution is methodological: a reusable toolchain — full-domain ablation plus whole-circuit equivalence proving — that closes design spaces with proofs rather than conjectures, and that generalizes to any kickmix-style or verifiable-circuit competition.