Boolean Walsh Eta Units and Eisenstein Bases For Squarefree Levels

Let $M>1$ be squarefree and let $D(M)$ be its Boolean divisor cube. To each Boolean character ${\chi }_T$ we attach the eta-quotient $$R_T^{(M)}(\tau )=\prod _{d\mid M}\eta (d\tau {)}^{{\chi }_T(d)},{\chi }_T(d)=(-1{)}^{\mathrm{\mid }T\cap \text{supp}(d)\mathrm{\mid }}\mathrm{.}$$ At squarefree level, the finite Fourier transform on the divisor cube simultaneously diagonalizes the squarefree Ligozat cusp-order matrix, Fricke complementation, Atkin–Lehner action on cusp labels, and the constant-term map for logarithmic Eisenstein series. In particular, for $T\mathrm{\ne }\mathrm{\varnothing }$, $${\text{ord}}_{1\mathrm{/}c}{R}_T^{(M)}=\frac{{\mathrm{\Lambda }}_T^{(M)}}{24}{\chi }_T(c),{\mathrm{\Lambda }}_T^{(M)}=\prod _{p\in T}(p-1)\prod _{\begin{array}{c}p\mid M\\ p\mathrm{\notin }T\end{array}}(p+1),$$ and the forms $D\log R_T^{(M)}$ form a Walsh basis of the Eisenstein subspace of $M_2({\mathrm{\Gamma }}_0(M))$. The structural theorem also determines explicit Fricke constants, Atkin–Lehner eigenvalues, good-prime Hecke eigenvalues, local $U_p$ triangular blocks, a simultaneous bad-prime eigenbasis, and the indices of two explicit principal cuspidal divisor sublattices inside the formal degree-zero cusp-divisor lattice. As an application we specialize to the Heegner prime product $$N=2\cdot 3\cdot 7\cdot 11\cdot 19\cdot 43\cdot 67\cdot 163.$$ The first Boolean boundary gives the eta-normalized Heegner-coloured partition product, while the top Walsh character gives the Möbius eta-unit identity $$D\log R_{\mathcal{P}}^{(N)}(\tau )=40415760-\sum _{n\ge 1}{\sigma }_1(n^{\perp })q^n\mathrm{.}$$ The same application gives algebraic modular-unit relations for the reciprocal partition product and exact Fricke-fixed logarithmic derivative identities for $1\mathrm{/}\pi$, interpreted through the modular completion of $E_2$ and through accelerated paired products.