Combine the bounded-\(q\) Type-I/Type-II estimates with the compact Platt-radius split and the certified crossover margins. This is a conservative repository repair, not a verbatim paper inequality.

Use the uniform large-sieve and Mertens inputs in the medium range, retaining the signed cancellation and explicit repair allowance.

Split the displacement radius into certified endpoint cells, transfer the pointwise envelopes to their integrals, and add the donor and reserve budgets.

Cover denominators by \(q{\lt}37500\), \(37500\le q\le 150000\), and \(q{\gt}150000\); use the already closed high branch and transport the combined producer through the Mellin and carrier bridges.

Assemble the low-, middle-, and high-displacement producers, including the endpoint-aware repair and certified prefix cost.

Convert the fixed cap to the direct uniform aggregation, then apply the Mellin, Dirichlet-case, and Abel-fold interfaces.

Apply the high-\(q\) large-sieve bound and its explicit denominator and smoothing envelopes.

Dispatch the Dirichlet approximant into the \(q\le r\), mid-\(q\), or high-\(q\) case and take the common certified envelope.

Integrate the pointwise norm on the complement of the major arcs and apply the corrected Abel/Chapter-14 evaluation. The coefficient \(1.048/49\) is the repository’s conservative repaired bound.