K_rs
K_rs defines the K-gate ratio as π/(4 ln φ) inside the RS-native unit system. Researchers modeling coherence quanta or φ-ladder scalings in Recognition Science reference this constant for all dimensionless ratios. The definition is a direct one-line alias to the K_gate_ratio computation imported from KDisplayCore.
claim$K_{rs} := K = {π}/(4 ln φ)$
background
The RS-native units module sets up a measurement system with tick τ₀ as the discrete ledger time quantum and voxel ℓ₀ as the causal spatial step satisfying c = 1. Derived quantities are the coherence energy E_coh = φ^{-5} and action ħ = E_coh · τ₀, with all ratios fixed by φ on the φ-ladder for masses, energies, times, and lengths. This definition draws on the RSUnits structure (tau0, ell0, c with c · tau0 = ell0) and the K_gate_ratio computation π/(4 log φ) from KDisplayCore, plus the abstract Constants bundle containing Knet and related terms.
proof idea
The definition is a one-line alias that directly references the K_gate_ratio evaluation Real.pi / (4 * Real.log phi) from the imported KDisplayCore module. No lemmas or reductions are applied beyond the import and alias.
why it matters in Recognition Science
K_rs supplies the fixed K-gate ratio for RS-native constants, supporting coherence scaling E_coh = φ^{-5} and the φ-ladder organization in the Recognition framework. It aligns with the eight-tick octave and D = 3 spatial dimensions by providing a φ-derived dimensionless parameter without external anchors. No downstream uses appear in the graph, leaving its role as a foundational constant definition.
scope and limits
- Does not perform numerical evaluation or SI conversion.
- Does not depend on specific tau0 or ell0 values beyond the RSUnits structure.
- Does not address physical interpretations or derivations of K.
formal statement (Lean)
269noncomputable def K_rs : ℝ := Constants.RSUnits.K_gate_ratio
proof body
Definition body.
270
271/-- Coherence scaling: E_coh = φ⁻⁵. -/