T2_ratio_is_phi_power
plain-language theorem explainer
The theorem establishes that under the BIT-coupling decoherence model the ratio of substrate T2 times at Z-rungs k_a and k_b equals phi to the power of their difference when k_b is at least k_a. Quantum computing modelers working with the Bosonic Identity Theorem would cite this algebraic identity to relate decoherence times across qubit families. The proof is a short tactic sequence that unfolds the substrate definition, applies field simplification to the quotient, and invokes the power-subtraction lemma.
Claim. Let $T_{2,0}>0$ and let $k_a,k_b$ be natural numbers satisfying $k_a≤k_b$. Then $T_2(k_a)/T_2(k_b)=φ^{k_b-k_a}$, where the substrate decoherence time is defined by $T_2(k):=T_{2,0}/φ^k$.
background
The DecoherenceFromBIT module develops the structural consequences of coupling the Bosonic Identity Theorem carrier (at frequency 5φ in RS-native units) to qubit substrates. The referenced definition T2_substrate states that the decoherence time at Z-rung k equals T2_0 divided by phi to the power k, with higher rungs producing stronger coupling and therefore shorter times. The module treats specific rung assignments to qubit classes as hypothesis-grade inputs while proving the ratio relation algebraically.
proof idea
The tactic proof first unfolds T2_substrate on both sides. It records positivity of phi and non-zero powers, then rewrites the left-hand side quotient via field_simp to obtain phi^{k_b} over phi^{k_a}. The final step rewrites division as multiplication by the inverse and applies the lemma pow_sub₀ (under the assumption k_a ≤ k_b) to reach phi to the difference.
why it matters
This supplies the T2_ratio_phi_power component of the master certificate decoherenceFromBITCert and is invoked by the one-statement theorem decoherence_from_BIT_one_statement together with the concrete ratio theorems for transmon-fluxonium and transmon-trapped-ion pairs. It completes the algebraic structure of the φ-power ratio required by the BIT-coupling model, aligning with the Recognition Science forcing of phi as the self-similar fixed point. The module leaves the empirical calibration of individual Z-rungs as an open hypothesis.
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