The Josephson Realization of the $\Omega_F$ Tower: Cooper-Pair-Box Response, the Josephson-Array Phase Boundary, and a Discrete-Scale-Invariant Mellin Invariant

Our circle construction of a discrete-scale-invariant Schrödinger-operator tower, introduced previously, is shown to be, after restoration of one energy unit and removal of an additive scalar, exactly a geometric family of zero-offset Cooper-pair-box Hamiltonians. At level $n$ the circuit parameters are\[E_C^{(n)}=\frac{E_\star}{8p^n},\qquadE_J^{(n)}=\frac{E_\star\beta^2}{2p^{2n}},\qquad\frac{E_J^{(n)}}{E_C^{(n)}}=4\beta^2p^{-n},\]so the Mathieu parameter of the toy model is precisely $Q_n=-E_J^{(n)}/(2E_C^{(n)})$. This identifies the same Cooper-pair-box Hamiltonian family that underlies the transmon, while locating the tower in the charge-dominated perturbative regime rather than in the usual large-$E_J/E_C$ operating regime of a transmon qubit. A direct charge-basis derivation is then given. If$H=4E_C\hat{N}^2-E_J\cos\hat{\phi}$ at offset charge $n_g=0$, the free-rotor-normalized, ground-referenced free-energy response satisfies\[-k_B T\log\frac{\mathrm{Tr} e^{-H^{\flat}(E_J)/(k_B T)}}{\mathrm{Tr} e^{-H^{\flat}(0)/(k_B T)}}=\frac{E_J^2}{2E_C}F\!\left(\frac{4E_C}{\pi k_B T}\right)+O\!\left(\frac{E_J^4}{E_C^3}\right),\]where\[F(u)=\frac{\displaystyle\sum_{m\geq1}\frac{2m^2}{4m^2-1}e^{-\pi um^2}}{\displaystyle\sum_{m\in\mathbb{Z}}e^{-\pi um^2}}.\]Thus $F$ is the dimensionless excess quadratic Josephson response of the ideal cosine rotor. For the geometric circuit family, the response strength scales as $(E_J^{(n)})^2/E_C^{(n)}\propto p^{-3n}$, which is exactly the weight of the original tower. Its coefficient functional obeys a Mahler-type dilation equation and has a log-periodic amplitude whose mean gives\[\boxed{\Omega_F=\int_0^\infty u^2F(u)\,\mathrm{d} u=c_0(p)\log p=0.0359811440770352697\ldots}\]for every $p>1$. The ground reference is shown to be essential: the corresponding raw coefficient is $R(u)=2F(u)-1/2$, and its third moment diverges as $-U^3/6$. Extensions to nonzero offset charge, odd Josephson harmonics, finite towers, and a capacitively coupled two-rung model are derived. Finally, $\Omega_F$ is formulated as a universal excess-response sum rule under explicit idealizing assumptions. Its most direct prospective use is a scale-quotiented Hamiltonian checksum and deformation diagnosticfor Josephson circuits and engineered discrete-scale-invariant analog simulators.