planckTime_rs
RS-native Planck time is the fundamental tick scaled by the square root of the derived G over c cubed with c fixed at 1. Researchers building Planck-scale quantities inside the Recognition Science unit system cite this to anchor time to ledger primitives. The definition reduces to a direct algebraic expression using the upstream G and tick constants.
claimIn RS-native units the Planck time is given by $τ_P = τ_0 √(G / c^3)$ where $τ_0$ is the fundamental tick, $G$ is the RS-derived gravitational constant, and $c = 1$.
background
The module defines an RS-native measurement system whose base units are the tick (one discrete ledger posting interval, set to 1) and the voxel (distance light travels in one tick). Derived quanta are the coherence energy φ^{-5} and the action ħ = E_coh · τ_0. All quantities sit on the φ-ladder and c is fixed at unity; SI conversion is optional and external.
proof idea
This is a definition that multiplies the tick primitive by the square root of the upstream G divided by c cubed. It relies on the imported definitions of G (λ_rec² c³ / (π ħ)) and tick together with the constant c set to 1. The implementation is a one-line algebraic expression.
why it matters in Recognition Science
The definition supplies the Planck time that is multiplied by c to obtain the Planck length in the same module. It places the fundamental tick at the scale where quantum and gravitational effects meet, consistent with the φ-ladder and eight-tick octave structure. It supports downstream mass and length formulas without external anchors.
scope and limits
- Does not derive G or the tick from the functional equation.
- Does not supply numerical values in SI units.
- Does not include relativistic or quantum-gravity corrections.
- Does not address conversion to external calibration constants.
Lean usage
noncomputable def planckLength_rs : Length := c * planckTime_rsformal statement (Lean)
243noncomputable def planckTime_rs : Time :=
proof body
Definition body.
244 -- Using the gate identity: τ_P = τ₀ · √(G/c³) where G,c are in RS units
245 -- Since c = 1, and G is derived, this is pure φ-structure
246 tick * Real.sqrt (Constants.G / (c ^ 3))
247
248/-- Planck length in RS units: ℓ_P = c · τ_P. -/