HBP | Main | 1.1 • Mittag–Leffler for Helson Zeta via Blur

We develop a six-part blur/packet program for a Helson-Mittag-Leffler theory built from fresh dyadic prime blocks, boundary guards, and packet exhaustion, under the standing no-small-primes convention
\[
\chi(p)=1 \qquad (p\le P_0).
\]

The final outcome is a constructive prescribed-divisor theorem for Helson zeta-functions above any fixed left boundary \(\sigma_-<1\). The special case \(\sigma_-=1/2\) is the one used later in the blur program.

Paper I (Blueprint)  
States the revised architecture of the program. The key finite step is the finite signed packet theorem; the global completion is nested packet exhaustion.

Paper II (Endpoints)  
Packages the short complex-analytic consequences: factorization, divisor readout, and the passage from a constructed Helson zeta to the final prescribed-divisor statement.

Paper III (Local controllability)  
Proves the one-window actuator: inside one far-out smooth log-window of primes, one can realize finite target data while keeping finitely many frozen constraints small.

Paper IV (Boundary transfer)  
Develops the geometric side: boundary blur, deblurring, logarithmic Rouché, and the transfer from boundary overlap to actual divisor control.

Paper V (Finite signed packets)  
Proves the local finite signed packet theorem in the thin-packet regime. Old-boundary constraints and packet-site constraints are handled simultaneously by one augmented finite-dimensional collocation problem. This yields one signed packet insertion, then finitely many by induction.

Paper VI (Global exhaustion)  
Passes from the finite theorem to a locally finite divisor theorem above \(\sigma_-<1\). Using disjoint packet discs, summable guard budgets, and a diagonal exhaustion, one obtains a meromorphic Helson zeta with exactly the prescribed divisor.

In short: the final program shows that the natural ambient for the Helson-Mittag-Leffler problem is the dyadic Euler side itself. Blur remains essential, but as a geometric control and read-out mechanism rather than as a replacement for the arithmetic engine.