critical_modes_specification
plain-language theorem explainer
For positive real t and g the critical number of modes equals -ln(t/τ₀)/(g ln φ), inverting the decoherence timescale formula. Modelers of quantum-to-classical transitions in Recognition Science frameworks cite this to locate the Gap-45 threshold for a target coherence time. The proof is a direct unfolding of the criticalModes definition followed by simplification under the positivity hypotheses.
Claim. For all real numbers $t>0$ and $g>0$, the critical number of environmental modes $N(t,g)$ satisfies $N(t,g)=-ln(t/τ_0)/(g ln φ)$, where $τ_0$ is the reference recognition tick and the formula inverts $τ_0 φ^{-N g}=t$.
background
The QFT.Decoherence module derives decoherence timescales from the Gap-45 threshold, the boundary at which a system exceeds ~10^45 operations and loses coherence to the environment. Coherence persists below this threshold; above it the system entangles with environmental modes and decoheres. The timescale formula is $τ≈τ_0 φ^{-N g}$, with $τ_0$ the fundamental tick imported from GravityBridge.tau0_seconds (≈7.3×10^{-15} s) and φ the golden ratio. criticalModes solves the inverse: given target time t and coupling g it returns the mode count N at which decoherence occurs.
proof idea
The term proof introduces t and g under the positivity hypotheses, unfolds the criticalModes definition (which contains an if-condition on positivity), and applies simp to drop the conditional and reduce directly to the explicit logarithmic expression.
why it matters
This result inverts the decoherence formula that implements the Gap-45 mechanism of QF-009, supplying the explicit mode count needed to locate the quantum-to-classical boundary for any chosen timescale. It rests on the upstream tau0_seconds reference and the criticalModes definition itself. Within the Recognition Science chain it supplies the concrete inversion step that lets the phi-ladder scaling predict decoherence thresholds without additional hypotheses.
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