gap45CrossoverApprox
plain-language theorem explainer
The definition assigns the rational 210 to the approximate mode count at the Gap-45 crossover in decoherence models. Researchers deriving quantum-to-classical transitions from Recognition Science timescales cite this value when estimating environmental mode thresholds. It arises as a direct numerical evaluation of the logarithmic ratio ln(τ_bio/τ_0)/ln(φ) using Planck and biological scales.
Claim. The approximate number of environmental modes at the Gap-45 crossover is the rational $210$.
background
In the QF-009 module, decoherence timescales derive from the Gap-45 threshold, defined as the ratio of biological timescale τ_bio ≈ 1 s to Planck time τ_0 ≈ 5.4 × 10^{-44} s, yielding approximately 10^{45}. Quantum coherence persists below this threshold while entanglement with the environment induces decoherence above it. The formula τ_decoherence ≈ τ_0 × φ^{-N} governs the transition, with N the number of coupled modes. The upstream modes definition supplies the finite set of truncated modes in the 2D Galerkin projection used for mode counting.
proof idea
The definition performs a direct assignment of the rational 210, computed from the ratio of natural logs of the timescale ratio divided by ln(φ) ≈ 0.48, giving approximately 210.
why it matters
This supplies the numerical anchor for the theorem gap45_crossover_range establishing the interval (100, 300). It realizes the Gap-45 mechanism in QF-009, separating the quantum regime of preserved coherence from the classical regime of environmental entanglement. Within Recognition Science this aligns with phi-ladder scaling and the eight-tick octave structure underlying discrete time.
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