T2_transmon_to_trapped_ion_ratio
plain-language theorem explainer
The ratio of decoherence times between transmon and trapped-ion qubits equals phi cubed when their Z-rungs differ by three under BIT coupling. Quantum hardware modelers working on substrate-dependent T2 would cite this to obtain exact phi-power predictions from the Recognition Science ladder. The proof is a direct instantiation of the general T2_ratio_is_phi_power theorem on the rung pair for these classes, followed by unfolding and omega discharge of the integer difference.
Claim. $ T_2(k_t) / T_2(k_i) = phi^3 $ where $ T_2(k) := T_{2,0} / phi^k $ for base time $ T_{2,0} > 0 $, with $ k_t $ the Z-rung of the transmon substrate and $ k_i $ the Z-rung of the trapped-ion substrate.
background
The module treats decoherence as arising from coupling between the BIT carrier at frequency 5 phi and a qubit substrate whose Z-rung k sets the coupling strength. The structural definition T2_substrate(T2_0, k) = T2_0 / phi^k therefore encodes faster decoherence at higher rung. The general theorem T2_ratio_is_phi_power then forces any cross-class ratio to equal phi raised to the rung difference, with the specific integer difference supplied by the rung assignments for each hardware family.
proof idea
The proof applies T2_ratio_is_phi_power to the concrete pair z_rung_transmon and z_rung_trapped_ion. The required integer difference of three is discharged by unfolding both rung definitions and invoking omega. The final simpa step substitutes the unfolded rung values to obtain the phi^3 equality directly.
why it matters
This supplies the transmon-to-trapped-ion entry in the five-field master certificate for Decoherence from BIT. It instantiates the phi-power ratio structure that follows from the Bosonic Identity Theorem and the phi-ladder forced by the T5-T8 chain. The result closes one concrete case in the algebraic classification of substrate-dependent decoherence channels while leaving the empirical rung assignments as open hypotheses.
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