Empirical Confirmation of the Square-Root Law of Market Impact in a U.S. Large-Cap Equity

We test the square-root law (SRL) of market impact on a single U.S. large-capitalisation equity, Apple Inc. (AAPL), using the full Nasdaq TotalView-ITCH market-by-order feed over 178 trading days (2 December 2024 -- 19 August 2025; ~0.5 billion events). Without broker-tagged parent orders, we reconstruct metaorders from the anonymous tape and calibrate impact as $I/\sigma_D = c\,(Q/V_D)^{1/2}$ with the exponent fixed at the universal value $1/2$. We find $c_{\rm raw} = 0.69$ (bias-corrected $c_{\rm eff} = 0.34$), conditional impact tracking $Q^{1/2}$, and a size-distribution tail exponent $\beta = 1.54 \pm 0.15$ -- both consistent with the worldwide cross-section. A direct model comparison decisively prefers the square-root form over linear ($\Delta{\rm AIC}=22$) and logarithmic impact, and the prefactor holds ($c_{\rm raw} \in [0.63, 0.77]$) across every reconstruction setting. Two structural tests confirm the impact is genuine: shuffling trade signs collapses directional impact to chance (86% to 51%); and scrambling event chronology destroys the SRL (0 of 80 calibrations remain viable). The underlying order flow is long-memory ($\gamma=0.66$) while the price stays diffusive (Hurst 0.49) -- the two ingredients of the universality theories. The prefactor is stable across 32 weekly walk-forward re-calibrations. To our knowledge this is the first confirmation of the square-root law on a U.S. equity derived purely from anonymous order flow, without broker-tagged parent orders.