buying_to_bundle_profit_intermediate_surrogate_gap_bound

First gap bound in the proof of Theorem 4.6 of Buying to Bundle: Optimal Sourcing from Monopolistic Sellers (App. C.2 pp. 35–36). Define the intermediate surrogate profit $\varpi'(x)=E_{\boldsymbol\mu}\big[Rev\big(\sum_i x(\mu_i)(\mu_i+\sigma Z_i)\big)\big]-N\,E_m[\varphi(m)x(m)]$ — the profit with the random inclusion pattern replaced by deterministic inclusion weights inside the buyer valuation (the payment term is written $N\int\varphi\,x\,d\Phi$, equal to $E_{\boldsymbol\mu}[\sum_i\varphi(\mu_i)x(\mu_i)]$ by symmetry — the same shape as eq. (1) in the root statement). Then there exists $K>0$ depending only on $(\sigma,\gamma,\xi,\mu_H)$ such that for every IC allocation rule $x$ and every $N\ge1$,

Paper proof: for every realization $(\boldsymbol\mu,\mathbf I)$, triangle inequality through the revenues at $Y=\sum_iI_i\mu_i$ and $\Upsilon=\sum_ix(\mu_i)\mu_i$; the outer terms are $O(N^{2/3})$ by Lemma 4.5 (monopoly_revenue_subexponential_dispersion_bound, applied with $C=Y,a_i=I_i$ and $C=\Upsilon,a_i=x(\mu_i)$, using $C\le N\mu_H$, $A\le N$), and the middle term integrates to $\le\mu_H\sqrt N$ by eq. (5) (buying_to_bundle_expected_bundle_quality_fluctuation_bound).