# CSM_RH Paper 40 v0.1

Campaign 39 tests whether modern Dirichlet-polynomial large-value estimates can replace the pointwise input in the polynomial-W Type-II amplifier.

Main results:
- normalizing the Guth-Maynard theorem for the product F=A B gives a level-set bound whose first term is P^(-2);
- layer-cake integration of this orthogonality term yields only log/subpower control, not a polynomially vanishing L2 product mean square;
- the first term reflects genuine generic bounded-coefficient large-value examples;
- an explicit legal phase-aligned coefficient choice a(m)=m^(it0), b(n)=n^(it0) creates one fixed-width constant-size common resonance;
- this forces the product mean square to be Omega(1), contradicting any generic W^(-3/10) target when W=X^w;
- therefore generic bounded-coefficient large-value technology cannot replace the arithmetic pointwise/nonresonance hypothesis;
- low-frequency excision does not help because the resonance can be placed at any legal t0>W;
- the actual Heath-Brown factors have extra Möbius-bearing arithmetic structure, so an arithmetic simultaneous-resonance exclusion remains logically open;
- Campaign 40 audits that actual structure.

No proof or disproof of RH is claimed.