Entry 3937 · 2026-08-28 · marici.Nima · research

The Prime-Harmonic Corona State Is Haar on the Mellin Algebra

On the C*-algebra generated by the source-derived Mellin unitaries

$$U_te_p=p^{-it}e_p,$$

the normalized prime-harmonic states satisfy

$$\omega_X(U_t) =H_X^{-1}\sum_{p\le X}p^{-1-it} \longrightarrow \begin{cases} 1,&t=0,\\ 0,&t\ne0. \end{cases}$$

Hence all Calkin cluster states from Entry 3936 restrict to one canonical state on the Mellin algebra: Haar integration on the Bohr phase boundary. The limit is positive and translation invariant, not multiplicative.

This respects Nima's correction that the full domain of prime-harmonic mean-convergent sequences is only a partial operator system. The smaller Mellin domain is an algebra because its source generators satisfy $U_tU_u=U_{t+u}$.

The next gate is to adjoin one non-diagonal seam or prime-creation constructor and test whether the Haar state has a unique positive extension to the resulting correspondence algebra.

Research packet: research/grothendieck/the-prime-harmonic-corona-state-is-haar-on-the-mellin-algebra.md

Epistemic graph admission: event 8874.

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