omega_BIT_pos
The theorem establishes that the BIT carrier frequency is strictly positive in RS-native units. Quantum computing theorists modeling decoherence via the Bosonic Identity Theorem would cite this to anchor T2 ratio derivations. The proof is a one-line term reduction that unfolds the definition and closes the inequality with positivity of phi via linarith.
claim$0 < 5φ$ where $φ$ is the golden-ratio fixed point of the Recognition Composition Law and the BIT carrier is defined as $ω_{BIT} := 5φ$.
background
The module treats decoherence channels that arise when a qubit substrate couples to the Bosonic Identity Theorem carrier at frequency $ω_{BIT} = 5φ$. The carrier is identified with the dark-energy / superconductivity / consciousness-substrate frequency; its positivity is required before any ratio of $T_2$ times can be formed. Upstream, the Constants structure from LawOfExistence bundles the CPM constants and supplies $φ > 0$, while carrier definitions in the engineering modules simply set carrier := 5 * phi.
proof idea
Term-mode proof: unfold omega_BIT, obtain h_phi := Constants.phi_pos, then close the goal by linarith.
why it matters in Recognition Science
The result supplies the elementary positivity fact needed for the structural claim that $T_2$ ratios between qubit classes are integer powers of $φ$. It sits inside the algebraic layer of the Bosonic Identity Theorem (T5–T7 forcing chain) and precedes the band and ratio theorems listed as siblings. The module treats specific Z-rung assignments as open hypotheses while the $φ$-power structure itself is theorem-grade.
scope and limits
- Does not derive the numerical value 5φ from more primitive axioms.
- Does not assign Z-rungs to concrete qubit families.
- Does not produce numerical $T_2$ predictions.
formal statement (Lean)
56theorem omega_BIT_pos : 0 < omega_BIT := by
proof body
Term-mode proof.
57 unfold omega_BIT
58 have h_phi := Constants.phi_pos
59 linarith
60
61/-- The BIT carrier frequency is in the band `(7.5, 8.1)`. -/