HBP | Inter | 5.1 • Riemann Hypothesis and the Multiplicative Choice

We present a compact, fully spelled–out conditional route from a specific, explicit choice of multiplication to the conclusion that all nontrivial zeros of ζ lie on the critical line. The link passes through three items that are part and parcel of that choice: (i) power–compatibility on the discrete side; (ii) the unitary Mellin transform on the midline ℜs = 1/2; and (iii) scale–neutral “blur” (Fejér/Paley–Wiener) on the log–line. The only additional hypothesis is a boundary positivity that is exactly what those multiplicative constraints naturally allow (Fejér–Mellin positivity).

We do not claim a stand–alone proof of RH here. Rather, we show that, under these axioms, RH (in the sense of location of zeros) follows formally and transparently; we also list the assumptions explicitly, explain why they are intrinsic to the multiplicative choice, and why they do not undermine the structural conclusion. We then explain in two voices—formal and plain–language— why simplicity of zeros is not forced by the basic multiplicative axioms alone, and how a natural “no hidden degeneracy” addendum (dilational irreducibility) aligns simplicity with the same choice. Finally, we explain how the Helson framework (guards, detector, four flows) is used bottom–to–top to derive the needed positivity and to address simplicity, should one seek unconditional rigor beyond this illustrative note.