From Möbius Mollification to Screw Positivity

We connect the tapered Möbius mollifier V_N(s) = Σ_{n≤N} μ(n)(1 − log n/log N) n^{−s} with Suzuki's screw function Ψ of the Riemann zeta function by exact identities, and determine which kinds of estimates can and cannot turn this connection into information about the zeros. (i) For every Dirichlet series F with a(1)=1, the coefficients of the inversion residual 1 − F·V_{F,N} equal −Λ_F(m)/log N for 2 ≤ m ≤ N; higher tapers give Selberg's generalized von Mangoldt functions. For ζ this is classical; the universal form shows that the identity carries no ζ-specific information. (ii) Suzuki's triangular prime potential is exactly a value-plus-derivative jet of the truncated Möbius residual at s = 1/2, which gives a Möbius-side form of Suzuki's criterion Ψ ≥ 0 ⇔ RH. (iii) After removing the exact polar main term 4√N − 2 log N − 4, the centred remainder R^c(N) satisfies Ψ(log N) = κ log N + c_0 − R^c(N) + ε(N) with explicit constants κ = −0.68609…, c_0 = 0.29933…. RH is equivalent to a one-sided O(1) bound for R^c(N) − κ log N; under RH this quantity lies in an explicit interval of width < 0.094, although R^c is extracted from a prime sum of size 4√N. The growth exponent of R^c equals sup Re ρ − 1/2. (iv) In H² of the half-plane Re s > 1/2 the jet functional at distance ε from the critical line has the exact dual norm (((2εL−1)²+1)/(8ε³))^{1/2}, which diverges as ε → 0. (v) A raw coefficient Cauchy–Schwarz bound for the prime potential is too large by (log N)² relative to the potential and by √N (log N)² relative to the RH window. No proof of the Riemann hypothesis is claimed. All identities are proved; the embedded audit script reproduces every numerical value. Six open problems are formulated, including a Davenport–Heilbronn falsification benchmark.