gap45CrossoverModes
plain-language theorem explainer
The number of environmental modes at the Gap-45 crossover for a one-second biological timescale and unit coupling is defined by direct application of the criticalModes function. Researchers modeling quantum decoherence or quantum error correction would cite this value when estimating the mode count that drives entanglement with the environment. The definition is a one-line wrapper around criticalModes together with the logarithmic approximation N ≈ ln(τ_bio/τ₀)/ln(φ) that yields a value near 215.
Claim. The number of environmental modes $N$ at the Gap-45 crossover for target time scale $1$ s and unit coupling strength is given by the evaluation of criticalModes at these parameters, yielding the approximation $N ≈ 45 × ln(10)/ln(φ) ≈ 215$ where $φ$ is the golden ratio.
background
The QFT.Decoherence module derives decoherence timescales from the Gap-45 threshold. Gap-45 is the ratio of order $10^{45}$ separating the Planck tick from the biological scale of one second; systems whose environmental interactions exceed this ratio lose coherence and become entangled with the surroundings. The module states the decoherence time formula $τ_decoherence ≈ τ_0 × φ^{-N}$ with $N$ the number of coupled modes and $φ$ the golden ratio scaling factor.
proof idea
This is a one-line definition that applies criticalModes to tau_bio and unit coupling. The attached comment supplies the explicit approximation $N ≈ ln(τ_bio/τ_0)/ln(φ)$ using τ_0 ≈ 5.4 × 10^{-44} s, which evaluates to roughly 210 modes and is rounded in the doc-comment to 215.
why it matters
The definition supplies the concrete mode count required by the Gap-45 crossover mechanism in QF-009. It anchors the patent proposals for threshold-based quantum error correction and links to the Recognition Science forcing chain through phi scaling and the eight-tick octave. No downstream declarations are recorded, so the value functions as a module-level parameter rather than an input to further theorems.
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