This is an additional proposed refinement from the cap reviewer. It is not a dependency of RESULTS.md or the cutoff-selection algorithm. Its explicit constant has not received the finite numerical checks or fresh proof review used for the main results, so it is retained as a candidate argument rather than promoted by the main run's PASS status.
Let delta(q) be any fixed nonnegative profile bounded by D. Put
\[ I_\delta=\sum_{q\ge1}\delta(q)(q^{-1/2}-(q+1)^{-1/2}), \qquad J_\delta=\int_0^1\delta(\lfloor t^{-2}\rfloor)\log t\,dt. \]
Both are absolutely convergent, with 0<=I<=D and -D<=J<=0. The proposed explicit bound, for integer N>=8, is
\[ \left|S_N-\tfrac12\sqrt N\log N I_\delta-\sqrt N J_\delta\right| \le12D N^{1/3}\log N. \tag{P} \]
Argument supplied: extract the square sum as in RESULTS.md equation (2). Write x=sqrt N, Q=floor(N^(1/3)), and z=x/sqrt(Q+1)<=N^(1/3). For each of the first Q quotient cells, replace L(x/sqrt q)-L(x/sqrt(q+1)) by the integral of log v over the same interval. The factorial remainder gives total error at most 2DQ(1+log x). The remaining square sum and integral have absolute total at most 2Dz log z+2D. These terms together are at most 4D N^(1/3) log N for N>=8. Adding the nonsquare remainder bound with constant eight gives (P). The substitution v=xt evaluates the whole integral as x log x I_delta+x J_delta.
If this argument is retained after its separate checks, every fixed bounded profile would satisfy S_N/(sqrt N log N) -> I_delta/2. The finite bound would also apply to varying bounded profiles with their corresponding I_delta_N,J_delta_N; it would not turn those changing quantities into a fixed leading coefficient. When only the attained quotient cells are known, one may extend that finite profile by zero outside them for that N.
This does not assert that arbitrary coefficient families have bounded deficits, or that the new balanced-prefix family has a fixed deficit profile.