Cubic Supercongruences for Coefficients of Powers of Euler Products

Let $P_d(q)=\prod_{m\ge1}(1-q^m)^{-m^d}$ and let $F_{\boldsymbol c}(q)=P_2(q)^{c_1}P_2(q^2)^{c_2}P_2(q^4)^{c_4}$, where $\boldsymbol c\in\mathbb Z^3$. We establish cubic congruence towers for the diagonal coefficients $[q^{\alpha n}]F_{\boldsymbol c}(q)^{\beta n}$ for all primes $p\ge7$, and also for $p=5$ when $c_1+2c_2+4c_4\equiv0\pmod5$. The parallel conclusion for $P_4$ holds at all primes $p\ge5$. The parameters $\alpha,n,r$ are positive integers, $\beta$ is any integer, and the congruence modulus is $p^{3r}$ at height $r$, without a coprimality restriction on $n$. The proof combines a formal square-Cartier criterion, integral Katz cusp coordinates and their exact two-radius Gauss norms, Coleman's positive-weight Frobenius trace, $p$-adic Eisenstein interpolation, and separate treatments of the exceptional primes. For the quartic product at $p=5$, a finite twelve-block calculation is paired with an infinite analytic tail estimate. We also establish an unconditional quadratic tower for more general integral Euler products, characterize a sharp $5$-adic parameter obstruction, and give exact coefficient counterexamples to three literal formulations associated with OEIS A380582 and A380583. These counterexamples do not invalidate the corresponding properly powered cubic towers. Licensing note: The manuscript is licensed under CC BY 4.0. The OEIS b-file snapshots included in the supplementary archive retain their original CC BY-SA 4.0 license and attribution. The independently authored verification code is made available under CC BY 4.0.