J_total
J_total sums the recognition costs of mass and light configurations at independent scale ratios for a stellar system. Astrophysicists working on observability limits cite it when minimizing total cost subject to flux thresholds and coherence volumes. The definition is a direct sum of two separate Cost.Jcost applications with no interaction term.
claimThe total recognition cost for a stellar configuration is defined by $J(r_m, r_L) := J_m(r_m) + J_L(r_L)$, where $J_m$ is the J-cost applied to the mass scale ratio and $J_L$ is the J-cost applied to the luminosity scale ratio.
background
The Astrophysics.ObservabilityLimits module derives mass-to-light ratios from recognition constraints: photon flux must exceed the coherence energy threshold and mass must assemble inside volumes set by the recognition length. J-cost is the base recognition penalty function imported from Cost, applied separately to mass and luminosity. J_mass(r) delegates directly to this cost at the mass scale ratio r; J_light(r) does the same for luminosity.
proof idea
One-line wrapper that adds the results of the two sibling definitions J_mass and J_light, each of which is itself a direct call to Cost.Jcost.
why it matters in Recognition Science
J_total supplies the objective minimized by OptimalConfig under observability constraints, producing M/L ratios that are integer powers of phi. It supports the module claim that observable configurations satisfy M/L in {phi^n : n in [0,3]} and feeds downstream additive lemmas for independent components such as J_additive_for_independent. The construction aligns with the Recognition Science forcing chain where J-uniqueness and the eight-tick octave fix the allowed geometric scalings.
scope and limits
- Does not incorporate interaction terms between mass and light.
- Does not enforce flux or volume constraints.
- Does not select or optimize the input scale ratios.
- Does not evaluate to a numerical value.
formal statement (Lean)
88noncomputable def J_total (r_m r_L : ℝ) : ℝ := J_mass r_m + J_light r_L
proof body
Definition body.
89
90/-- Optimal configurations minimize J_total subject to observability -/