The Josephson Realization of the ΩF 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\[E_C⁽ⁿ⁾=E_/8p^n,_J⁽ⁿ⁾={E_β^2}{2p²ⁿ},{E_J⁽ⁿ⁾}{E_C⁽ⁿ⁾}=4β^2p⁻ⁿ,\]so the Mathieu parameter of the toy model is precisely Qₙ=-EJ⁽ⁿ⁾/(2EC⁽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-EJ/EC operating regime of a transmon qubit. A direct charge-basis derivation is then given. IfH=4ECN̂²-EJcosφ̂ at offset charge ng=0, the free-rotor-normalized, ground-referenced free-energy response satisfies\[-k_B Tlog{Tr e^{-H^{}(E_J)/(k_B T)}}{Tr e^{-H^{}(0)/(k_B T)}}=E_J^2/2E_CF\!(4E_C/π k_B T)+O\!(E_J^4/E_C^3),\]where\[F(u)={∑m≥12m^2/4m^2-1e-π um^2}{∑ₘe-π um^2}.\]Thus F is the dimensionless excess quadratic Josephson response of the ideal cosine rotor. For the geometric circuit family, the response strength scales as (EJ⁽ⁿ⁾)²/EC⁽ⁿ⁾∝ 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\[{Ω_F=∫_0^∞ u^2F(u)\,d u=c_0(p)log p=0.0359811440770352697…}\]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 -U³/6. Extensions to nonzero offset charge, odd Josephson harmonics, finite towers, and a capacitively coupled two-rung model are derived. Finally, ΩF 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.