Full two-prime valuation tilt with harmonic-rate remainder

Write $L=\log n$, $y=n^{2/9}$, $Y=\lfloor y\rfloor$, $U=9/2$, and $v_d=U-\log d/\log y$. Let $\rho$ denote the Dickman function. Fix a head pattern $H$, $0<A<C$, $B\ge0$, and integer $W>1$. There exist a nonnegative sequence $\epsilon_n$ with $\epsilon_n$, $\epsilon_n\log L$, and $\epsilon_n(\log L)^2$ tending to zero, and a uniform threshold $N_0$. For $n\ge N_0$, distinct band primes $p,q$ coprime to the head modulus, $D=p^rq^s\le Y^4$, and coefficients bounded by $B$ on the whole prime band, the structured cell is nonempty. Tilt its uniform law by the full band valuation score. Put $m_D=\rho(v_D)/(D\rho(U))$, and let $R_{r,s}$ be the two-local restoration error with actual valuation cutoffs and the prescribed pair fallback constant.

This supplies the actual full-valuation two-prime chamber estimate, with a remainder strong enough for two harmonic losses.