omega_BIT_band
Researchers in quantum computing decoherence cite this bound when anchoring the Bosonic Identity Theorem carrier to substrate T2 data. It asserts that the BIT frequency lies in the open interval (7.5, 8.1) in RS-native units. The proof is a direct algebraic reduction that applies the established bounds 1.5 < phi < 1.62 to the expression 5 phi via nlinarith.
claimLet omega_BIT = 5 phi. Then 7.5 < omega_BIT < 8.1.
background
The module establishes decoherence times T2(k) = T2_0 / phi^k for qubit substrates coupled to the BIT carrier at frequency omega_BIT = 5 phi. The golden ratio phi is bounded by the lemmas phi_gt_onePointFive (phi > 1.5) and phi_lt_onePointSixTwo (phi < 1.62). These place the carrier inside the stated band and feed the structural claims on T2 ratios being exact phi-powers.
proof idea
The term proof unfolds the definition of omega_BIT, obtains the two phi bounds from Constants, and applies nlinarith to each conjunct of the goal.
why it matters in Recognition Science
This theorem supplies the first field of the DecoherenceFromBITCert structure and thereby supports the one-statement theorem decoherence_from_BIT_one_statement. It verifies the carrier band under the phi-ladder model of the Recognition framework (T5 J-uniqueness and T6 self-similar fixed point).
scope and limits
- Does not assign specific Z-rungs to qubit classes such as transmon or fluxonium.
- Does not derive the BIT frequency from first principles beyond the 5 phi definition.
- Does not address experimental measurement precision of the band.
formal statement (Lean)
62theorem omega_BIT_band : (7.5 : ℝ) < omega_BIT ∧ omega_BIT < 8.1 := by
proof body
Term-mode proof.
63 unfold omega_BIT
64 have h1 := Constants.phi_gt_onePointFive -- φ > 1.5
65 have h2 := Constants.phi_lt_onePointSixTwo -- φ < 1.62
66 refine ⟨?_, ?_⟩
67 · nlinarith
68 · nlinarith
69
70/-! ## §2. Substrate Z-rungs and `T₂` from BIT coupling -/
71
72/-- The structural decoherence time at substrate Z-rung `k`,
73 in RS-native units: `T₂(k) = T₂_0 / φ^k`. Higher Z-rung →
74 stronger BIT coupling → faster decoherence. -/