# CSM_RH Paper 41 v0.1

Campaign 40 audits the actual Heath-Brown arithmetic factors behind the Type-II major-arc problem.

Main results:
- the 2026 Type-II grouping is based on dyadic lengths, not on preservation of a Möbius carrier;
- a legal smooth-only Type-II component exists by taking all Möbius variables equal to 1 and three smooth variables of scale X^(1/3);
- a legal pure-Möbius-core Type-II component also exists by taking the top Heath-Brown level, fixing the log variable at 2 and the remaining smooth variables at 1, while the short Möbius variables carry the product scale;
- therefore neither a universal Möbius-carrier rule nor a universal complementary-smooth-factor anti-resonance rule is valid;
- the hard surviving arithmetic object is a product of several genuine short Möbius Dirichlet polynomials in a translated high-frequency window;
- this high-frequency problem is distinct from the t=0 fixed-power Mertens lock, although no lower-strength theorem is proved;
- cross-j cancellation in the exact Heath-Brown identity remains a logically possible alternative and is not ruled out;
- Campaign 41 attacks the pure-Möbius high-frequency core.

No proof or disproof of RH is claimed.