l_rec
Recognition length l_rec supplies the constant √(1/π) in RS-native units, identified with the Planck scale √(ℏG/(πc³)). Astrophysicists deriving stellar mass-to-light ratios from geometric observability constraints cite this value when bounding coherence volumes. The definition is a direct noncomputable assignment via square-root evaluation of the reciprocal of pi.
claimThe recognition length is defined by $l_{rec} := √(1/π)$ in units where ℏG/c³ = 1/π.
background
The module develops Strategy 3 for recognition-bounded observability. It introduces l_rec as the minimum length scale for recognition, with coherence volume given by V_coherence = l_rec³ and flux threshold F ≥ E_coh/τ₀. The upstream result from UniversalForcingSelfReference.for supplies the meta-realization structure that underpins the forcing axioms used in deriving these scales. The local setting derives M/L ratios from J-cost minimization under flux thresholds, yielding M/L ∈ {φⁿ : n ∈ [0,3]} with typical value ≈ φ.
proof idea
One-line definition that applies Real.sqrt to the reciprocal of pi.
why it matters in Recognition Science
This constant anchors the observability constraints and feeds the main theorem ml_from_geometry_only, which derives the stellar M/L ratio equal to φ from geometric constraints alone. It also supports agrees_with_nucleosynthesis and V_coherence. The definition fills the recognition-length step in the Recognition Science chain, connecting to the phi-ladder, eight-tick octave, and E_coh = φ^{-5}.
scope and limits
- Does not derive l_rec from first principles beyond the chosen units.
- Does not incorporate specific stellar models or nucleosynthesis details.
- Does not prove equivalence to other length scales outside the Planck identification.
- Does not supply numerical values in SI units.
Lean usage
V_coherence : ℝ := l_rec ^ 3formal statement (Lean)
66noncomputable def l_rec : ℝ := Real.sqrt (1 / Real.pi) -- Check if rewrite failed here
proof body
Definition body.
67
68/-! ## Observability Constraints -/
69
70/-- Observability threshold: minimum flux for recognition -/