# Relationship to Bourgain–Fuchs (source-level preliminary check)

The density-one exceptional-set conclusion, if established, implies a positive-density curvature conclusion for each primitive integral gasket: the admissible set is a nonempty union of classes modulo 24, hence has positive density, and removing O(N^(1-eta)) integers preserves that density. This implication uses the stated congruence classification.

The native source cites BourgainFuchs2010 exactly once, in its introduction as historical positive-density progress. It does not explicitly invoke that citation in a proof. This establishes the direct citation role only: transitive dependencies in other imported papers and shared arithmetic ingredients remain to be audited. An error in an earlier proof must not automatically be transferred to this manuscript.

Expanded-source quotation:

```tex
Finally Bourgain and Fuchs \cite{BourgainFuchs2010} settled the so-called ``Positive Density Conjecture,'' that
$$
\#(\sB\cap[1,N])\gg N.
$$
```

Initial reviewer packets contain the new manuscript and neutral target descriptions, without the earlier panel findings. Comparison and adjudication are separate.