A strict comparison becomes an equality
Where the issue occurs
Section 9, proof of Proposition 9.1, unnumbered product-variation display immediately before Equation (9.10) · p. 54
Q TV[q,∞)(h₋/u) ≤ Q(0.024/q + 0.012/q) < 0.036 min(1, Q/7.5).
The final strict inequality in this chain is the error; it becomes an equality when Q ≥ ζ. Equation (9.10) is a subsequent bound, not this display.
Issue & evidence
The final strict comparison in the quoted display is false when Q≥ζ: with q=max(Q,ζ) and ζ>7.5, both Q(0.024/q+0.012/q) and 0.036 min(1,Q/7.5) equal 0.036. This occurs within the stated first-atom domain, for example at b=4/5 and x=1/10, where Q=8>ζ. The error concerns the intermediate comparison; the intended strict total-variation bound remains valid.
Proposed repair by OSVerify
Replace the chain in the quoted display by Q TV_[q,∞)(h_-/u)≤3mQ/q<0.036 min(1,Q/7.5), retaining the exact maximum m defined earlier in this proof. The product-variation inequality gives the first bound because TV_[q,∞)(h_-)≤2m, sup h_-≤m, and TV_[q,∞)(1/u)=sup(1/u)=1/q. The strict inequality follows from m<0.012 and Q/q≤min(1,Q/7.5). This preserves the subsequent absolute-variation estimates in Equations (9.10) and (9.12) and their contributions to the edge-deficit bounds.
Related context
Background Section 9, proof of Proposition 9.1, first-atom domain
Shows that Q ≥ ζ can occur within the stated parameter range.
Background Section 9, same proof, definitions of ζ and the exact negative minimum m
The retained strict bound m < 0.012 supplies the proposed repair.
Downstream use Section 9, Equations (9.10) and (9.12)
These later variation bounds are preserved by the proposed correction; neither is identified as a separate error.