J
J(x) supplies the recognition cost for scale ratio x via the explicit formula ½(x + 1/x) - 1. Stellar modelers cite it when partitioning photon emission against bound mass to obtain M/L on the phi-ladder. The definition is a direct noncomputable alias to Cost.Jcost and carries no additional proof obligations.
claim$J(x) = ½(x + 1/x) - 1$ for real $x > 0$, where this is the recognition cost attached to a scale ratio.
background
The StellarAssembly module treats stellar collapse as a recognition-cost minimization between photon emission events (cost J(r_emit)) and mass-storage events (cost J(r_store)). Equilibrium occurs when the differential Δδ equals an integer multiple of J_bit, forcing M/L onto the phi-ladder. J itself is the convex cost function introduced by the Recognition Composition Law and realized as the Quantity Cost in RSNative.Core.
proof idea
One-line alias that redirects every call to Cost.Jcost; no tactics or lemmas are applied inside the definition.
why it matters in Recognition Science
The definition supplies the cost kernel for the module's derivation that M/L lies in {φ^n : n ∈ [0,3]}. It inherits the T5 J-uniqueness property and the eight-tick octave structure from the forcing chain, enabling the equilibrium condition stated in the module doc-comment. No downstream uses are recorded yet.
scope and limits
- Does not compute numerical M/L for specific stellar populations.
- Does not derive the closed form of J from the composition law.
- Does not extend the cost model beyond stellar collapse.
formal statement (Lean)
62noncomputable def J (x : ℝ) : ℝ := Cost.Jcost x
proof body
Definition body.
63
64/-- J is minimized at x = 1 with J(1) = 0 -/