tau19
The declaration defines τ₁₉ as τ₀ multiplied by φ raised to the 19th power, locating the rung at -19 on the phi-ladder for biological timescales near 68 ps. Researchers deriving time hierarchies from Planck units to macroscopic scales would cite this rung. The definition is a direct scaling assignment using the imported base tick and self-similar factor.
claimDefine τ₁₉ := τ₀ ⋅ φ¹⁹, where τ₀ denotes the fundamental tick duration and φ is the golden-ratio fixed point of the Recognition Science scaling.
background
Recognition Science organizes time scales on a phi-ladder in which each rung multiplies the prior duration by φ. The base unit τ₀ is the fundamental tick duration, defined in Constants as the duration of one tick and derived in Derivation from Planck constants in RS-native units. The module on Planck Scale from φ relates these to l_P = c × τ₀ × φ^(-n) for suitable n, with upstream scale defined as φ^k.
proof idea
This is a direct definition that sets tau19 equal to tau0 multiplied by phi to the power 19. It applies the imported constants for tau0 and the phi scaling without invoking further lemmas or tactics.
why it matters in Recognition Science
The definition supplies rung -19 on the phi-ladder, extending the hierarchy from Planck time to biological timescales as stated in the module documentation. It supports the claim that the full ladder spans from t_P to cosmological times and aligns with the self-similar fixed point φ in the forcing chain. No downstream theorems are listed.
scope and limits
- Does not derive or prove any relation between τ₁₉ and other physical constants beyond the scaling.
- Does not compute or verify the numerical approximation ≈ 68 ps.
- Does not address empirical matching to biological processes.
- Does not extend the ladder to higher or lower rungs.
formal statement (Lean)
129noncomputable def tau19 : ℝ := tau0 * phi^19
proof body
Definition body.
130
131/-! ## Quantum Gravity Predictions -/
132
133/-- RS predictions for quantum gravity:
134
135 1. **Minimum length = l_voxel**, not l_P
136 - Below l_voxel, spacetime is discrete
137 - l_P may be inaccessible
138
139 2. **φ-quantized energies** near Planck scale
140 - Energies at φ^n × E_P
141
142 3. **No singularities**
143 - Voxel structure prevents infinite densities
144
145 4. **Modified dispersion relations**
146 - At high energy, E² = p²c² + m²c⁴ + corrections -/