MTT interpolation from the two signed moment measures

For every prime natural number $p$, natural numbers $N>0$ and $k\ge2$, and unital ring homomorphisms $\iota:\overline{\mathbb Q}\to\mathbb C$ and $\iota_p:\overline{\mathbb Q}\to\mathbb C_p$, the following implication holds. Here $\overline{\mathbb Q}$ is an algebraic closure of $\mathbb Q$. Suppose $f$ consists of a weight-$k$ cusp form $F$ for $\Gamma_1(N)$, a Dirichlet character $\varepsilon$ modulo $N$ with values in $\overline{\mathbb Q}$, and coefficients $a_n\in\overline{\mathbb Q}$ for every natural $n$, such that the coefficient of degree $n$ of the width-one $q$-expansion of $F$ is $\iota(a_n)$, $a_1=1$, and, for every $\gamma=\left(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\right)\in\Gamma_0(N)$ and every $z$ in the complex upper half-plane, $F(\gamma z)=\iota(\varepsilon(D))(Cz+D)^kF(z)$. Suppose also that for every prime natural $\ell$ and every such $z$, $\ell^{-1}\sum_{b=0}^{\ell-1}F((z+b)/\ell)+\iota(\varepsilon(\ell))\ell^{k-1}F(\ell z)=\iota(a_\ell)F(z)$, including primes dividing $N$. Put $I_t(r)=2\pi\int_0^\infty F(r+iu)(r+iu)^t\,du$ for natural $t$ and rational $r$. Suppose a period system $P$ supplies nonzero complex numbers $\Omega_s$ for $s\in\{\mathrm{true},\mathrm{false}\}$ and values $V_s(t,r)\in\overline{\mathbb Q}$ for every such $s,t,r$, with $\iota(V_s(t,r))=[I_t(r)+\sigma_s(-1)^tI_t(-r)]/(2\Omega_s)$ whenever $t\le k-2$, where $\sigma_{\mathrm{true}}=1$ and $\sigma_{\mathrm{false}}=-1$, and with the $\mathbb Z$-submodule generated by all $V_s(t,r)$ with $t\le k-2$ finitely generated. Suppose $\alpha\in\mathbb C_p$ satisfies $\lVert\alpha\rVert=1$ and $\alpha^2-\iota_p(a_p)\alpha+\iota_p(\varepsilon(p))p^{k-1}=0$. Let $U=\mathbb Z_p^\times$, let $x_c\in\mathbb C_p$ be the image of $x\in U$ under $\mathbb Z_p\subset\mathbb Q_p\to\mathbb C_p$, and suppose $\mu_s:C(U,\mathbb C_p)\to\mathbb C_p$ is a continuous $\mathbb C_p$-linear functional for each Boolean $s$ (these are the measures in the statement). Define $A_s(j,a,m)=\sum_{t=0}^j\binom jt m^t a^{j-t}V_s(t,-a/m)$ in $\overline{\mathbb Q}$ for rational $a,m$. Assume, for each $s$, every natural $n>0$, every integer $a$ relatively prime to $p$, and every natural $j\le k-2$, there exists a continuous function $g:U\to\mathbb C_p$ equal to $x_c^j$ when the reduction of $x$ modulo $p^n$ equals the reduction of $a$, and equal to zero otherwise, such that $\mu_s(g)=\alpha^{-n}\iota_p(A_s(j,a,p^n))-\iota_p(\varepsilon(p))p^{k-2}\alpha^{-(n+1)}\iota_p(A_s(j,a,p^{n-1}))$. Then for every natural $n\ge0$, every primitive $\overline{\mathbb Q}$-valued Dirichlet character $\chi$ modulo $m=p^n$, and every natural $j\le k-2$, there exist a continuous function $h:U\to\mathbb C_p$ and an algebraic number $v\in\overline{\mathbb Q}$ such that $h(x)=\iota_p(\chi(x\bmod m))x_c^j$ for all $x$, $\iota(v)=m^{j+1}j!\,L_{\chi,j}/[(-2\pi i)^jG(\chi^{-1})\Omega_s]$, and $(\mu_{\mathrm{true}}+\mu_{\mathrm{false}})(h)=\alpha^{-n}(1-\iota_p(\chi^{-1}(p))\iota_p(\varepsilon(p))p^{k-2-j}/\alpha)(1-\iota_p(\chi(p))p^j/\alpha)\iota_p(v)$. Here $s$ is true exactly when $\chi(-1)(-1)^j=1$ and is false otherwise; character arguments are residues at their displayed modulus; $G(\psi)=\sum_{a\in\mathbb Z/m\mathbb Z}\iota(\psi(a))\exp(2\pi i\widetilde a/m)$, with $\widetilde a$ the representative in $\{0,\ldots,m-1\}$; and $L_{\chi,j}=\frac{(2\pi)^{j+1}}{j!}\int_0^\infty [G(\chi)^{-1}\sum_{a\in\mathbb Z/m\mathbb Z}\iota(\chi(a))F(iu+\widetilde a/m)]u^j\,du$. All these integrals are the total Bochner integrals with respect to real Lebesgue measure restricted to $u>0$, which take value zero if the integrand is not integrable; field inversion and division are total, with inverse of zero equal to zero, so no separate nonvanishing hypothesis is imposed on a Gauss sum. The assumptions on moments only involve positive depths, while the conclusion includes $n=0$, hence modulus one, and includes $j=0$ and the case $k=2$, in which $j=0$ is the only allowed moment. Natural subtractions occurring here agree with ordinary subtraction under the stated bounds, and $\alpha\ne0$ follows from its norm. The values $V_s(t,r)$ for $t>k-2$ have no comparison requirement and do not enter these moment identities. No assumption that $p$ is relatively prime to $N$ is present, and the conclusion is conditional on the supplied period system and both measures, rather than asserting their existence or uniqueness.