MTT interpolation at conductor one from depth-one moments

Let $p$ be prime, let $N>0$ and $k\ge 2$, and let $f$ be a normalized algebraic cuspidal Hecke eigenform of level $N$ and weight $k$. Fix compatible complex and $p$-adic embeddings, a period system $P$, an ordinary root $\alpha$, and two signed bounded measures $\mu^+$ and $\mu^-$ on $\mathbf Z_p^\times$. Assume each signed measure realizes the prescribed critical polynomial moment on every positive-depth residue disk.

For the primitive character of modulus $p^0=1$ and every $0\le j\le k-2$, there are a continuous function $g$ and an algebraic number $v$ satisfying the conductor-one interpolation identity

with $g(x)=\iota_p(\chi(x))x^j$ and $\iota(v)$ equal to the corresponding period-normalized critical value. Both Euler factors in $E_p$ are retained. This is the boundary case recovered by partitioning $\mathbf Z_p^\times$ into residue disks of depth one and applying the moment relations there.