J_mass
J_mass(r) supplies the recognition cost term for mass configurations at scale ratio r in the observability limits derivation. Researchers modeling stellar M/L ratios under flux and coherence constraints cite it when minimizing total J-cost. The definition is realized as a direct one-line alias to the core Jcost function.
claimThe J-cost for a mass configuration at scale ratio $r$ is $J_ {mass}(r) := J_{cost}(r)$, where $J_{cost}$ is the cost induced by the multiplicative recognizer on positive ratios.
background
The module develops Strategy 3 for recognition-bounded observability of stellar systems. Observable flux must satisfy $F ≥ E_{coh}/τ_0$ while mass assembly is limited by coherence volume $V ~ l_{rec}^3$. J_mass supplies the mass contribution to the total cost that is minimized to obtain M/L ratios in the set of phi powers.
proof idea
One-line wrapper that applies Cost.Jcost to the input scale ratio r.
why it matters in Recognition Science
J_mass supplies the mass term to J_total, whose minimization under observability constraints yields M/L ∈ {φ^n : n ∈ [0,3]} with typical value φ. It realizes the module's core insight connecting to the phi-ladder (T6) and Recognition Composition Law. The definition closes the mass side of the Strategy 3 derivation.
scope and limits
- Does not derive or expand the explicit form of Jcost.
- Does not enforce or check observability thresholds.
- Does not compute numerical M/L values for concrete systems.
- Does not address light or total cost terms.
Lean usage
noncomputable def J_total (r_m r_L : ℝ) : ℝ := J_mass r_m + J_light r_Lformal statement (Lean)
82noncomputable def J_mass (r : ℝ) : ℝ := Cost.Jcost r
proof body
Definition body.
83
84/-- The J-cost for luminosity configuration at scale ratio r -/