V_coherence
V_coherence defines the coherence volume as the cube of the recognition length l_rec. Astrophysicists working on Recognition Science observability bounds cite it when bounding maximum stellar assembly mass from density and cycle count. The definition is a direct algebraic extension of l_rec with no further reduction.
claimThe coherence volume is defined by $V = l_{rec}^3$, where $l_{rec}$ is the recognition length at the Planck scale.
background
In the Recognition-Bounded Observability module, stellar systems must satisfy photon flux above the recognition threshold $F_{threshold} ~ E_{coh}/τ_0$ while mass assembly is limited by coherence volume. The recognition length $l_{rec}$ is introduced as the fundamental scale $√(ℏG/(πc³))$, with the supplied definition using $√(1/π)$ as a normalized placeholder in RS units. The module derives M/L ratios as powers of φ from J-cost minimization under these geometric constraints.
proof idea
One-line definition that directly cubes the imported l_rec value.
why it matters in Recognition Science
This supplies the volume factor inside M_max(N, ρ) := ρ · V_coherence · N, completing the geometric step that yields M/L ∈ {φ^n : n ∈ [0,3]}. It anchors the observability derivation to the recognition length scale and feeds the J-total optimization that recovers the same φ-power ratios obtained from the forcing chain and RCL.
scope and limits
- Does not evaluate numerical mass or flux for concrete stellar objects.
- Does not incorporate time evolution or relativistic corrections.
- Does not derive l_rec from the underlying axioms.
- Does not address non-coherent or multi-component assembly.
formal statement (Lean)
74noncomputable def V_coherence : ℝ := l_rec ^ 3
proof body
Definition body.
75
76/-- Maximum mass from coherent assembly in N cycles -/