tau_bio
The declaration fixes the biological timescale at one second for use in decoherence calculations. Physicists modeling the transition from quantum coherence to classical behavior via the Gap-45 threshold would reference this value when estimating environmental coupling modes. It is introduced as a direct numerical assignment without further derivation.
claimLet $τ_{bio}$ denote the biological or classical timescale in seconds. Then $τ_{bio} := 1$.
background
In the QFT.Decoherence module the Gap-45 threshold marks the boundary where quantum systems decohere due to environmental entanglement. Gap-45 is the approximate ratio of $10^{45}$ between the Planck timescale $τ_P ≈ 5.4 × 10^{-44}$ s and the human scale $τ_{bio} ≈ 1$ s. The module derives decoherence times as $τ_{decoherence} ≈ τ_0 × φ^{-N}$, where $N$ counts coupled modes and $φ$ is the golden ratio scaling factor.
proof idea
This is a one-line definition that directly assigns the real number 1.0 to represent the biological timescale in seconds.
why it matters in Recognition Science
This constant anchors the Gap-45 crossover computation in gap45CrossoverModes, which estimates the number of modes $N ≈ 215$ needed to reach classical behavior at human scales. It implements the core insight of QF-009 that decoherence occurs when interactions exceed the Gap-45 threshold. In the Recognition framework it connects the phi-ladder scaling to observable timescales, closing the loop from Planck to biological regimes.
scope and limits
- Does not derive the value from the forcing chain or J-cost axioms.
- Does not specify units beyond the implicit seconds convention.
- Does not address variations across different biological or classical systems.
Lean usage
noncomputable def crossoverModes := gap45CrossoverModesformal statement (Lean)
67noncomputable def tau_bio : ℝ := 1.0
proof body
Definition body.
68
69/-- The logarithmic gap between biological and Planck timescales.
70 log₁₀(tau_bio / tau_planck) ≈ log₁₀(1 / 5.4e-44) ≈ 43.3
71
72 We prove this is approximately 43-44 orders of magnitude. -/