T2_substrate
T2_substrate defines the substrate decoherence time at Z-rung k as T2_0 divided by phi to the power k in RS-native units. Quantum computing modelers working on BIT-coupling channels cite it when deriving that cross-class T2 ratios equal exact phi-powers. The definition is a direct algebraic expression using the golden-ratio constant from the framework.
claimThe structural decoherence time at substrate Z-rung $k$ is $T_2(k) = T_{2,0} / {φ}^k$, where $T_{2,0} > 0$ is the base time and $φ$ is the golden-ratio constant.
background
The DecoherenceFromBIT module starts from the Bosonic Identity Theorem, which fixes the carrier frequency ω_BIT = 5φ. Z-rung k indexes substrate classes by BIT-coupling strength: larger k produces stronger coupling and therefore shorter T2. Constants.phi is the self-similar fixed point supplied by the unified forcing chain (T6).
proof idea
The declaration is a direct definition that expands to the division T2_0 / (Constants.phi ^ k). No lemmas or tactics are invoked; it functions as the base expression for the positivity, monotonicity, and ratio theorems that follow.
why it matters in Recognition Science
T2_substrate supplies the functional form required by the master certificate DecoherenceFromBITCert and by T2_ratio_is_phi_power. It realizes the structural claim that T2 ratios are φ-powers, which follows from the phi-ladder and eight-tick octave (T7). The definition leaves open the empirical mapping of specific Z-rungs to qubit families (transmon, fluxonium, etc.).
scope and limits
- Does not assign concrete Z-rung values to any qubit technology.
- Does not derive the BIT carrier frequency.
- Does not supply a dynamical model of the coupling interaction.
- Does not incorporate measurement statistics or experimental error.
formal statement (Lean)
75def T2_substrate (T2_0 : ℝ) (k : ℕ) : ℝ := T2_0 / Constants.phi ^ k
proof body
Definition body.
76
77/-- `T₂_substrate` is positive when `T₂_0` is positive. -/