criticalModes
plain-language theorem explainer
The critical modes definition computes the number of environmental modes at which decoherence reaches a target timescale for given coupling. Researchers deriving Gap-45 crossovers in Recognition Science cite it for quantum-to-classical transitions. The definition solves the decoherence equation algebraically for the mode count using logarithmic inversion of the phi-powered scaling.
Claim. The number of modes $N$ at which decoherence time equals target time $t$ for coupling strength $g$ satisfies $N = -{ln(t/τ_0)}/(g ln φ)$ for positive $t$ and $g$, and equals zero otherwise.
background
The QFT.Decoherence module develops decoherence timescales from the Gap-45 threshold, which separates quantum coherence (information preserved) from classical behavior (entanglement with environment). Decoherence time follows $τ_decoherence ≈ τ_0 φ^{-N}$, with $τ_0$ the reference tick (7.3e-15 s) and $φ$ the golden ratio. This rests on the truncated modes construction from the Galerkin2D module and the reference tick definition.
proof idea
The definition directly encodes the solved formula for critical modes by applying the logarithm to invert the exponential scaling in the decoherence expression. It checks positivity of target time and coupling before computing -log(targetTime / tau0) / (coupling * log phi), defaulting to zero otherwise. No external lemmas beyond the imported constants are invoked.
why it matters
This supplies the mode count for the critical_modes_specification theorem that verifies the inversion and for the gap45CrossoverModes computation at human timescales. It advances the QF-009 derivation of decoherence from the Gap-45 boundary and the phi scaling in the eight-tick octave. It leaves open how mode coupling interacts with the full Recognition Composition Law at higher rungs.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.