Jcost_has_RCL_combiner
The declaration shows that the J-cost function obeys the Recognition Composition Law identity for positive reals x and y. Physicists and mathematicians working within Recognition Science would reference this result when verifying that the cost function satisfies the d'Alembert-derived combiner. The proof proceeds as a direct one-line application of the upstream d'Alembert identity lemma followed by linear arithmetic.
claimFor all positive real numbers $x$ and $y$, $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$, where $J$ denotes the J-cost function.
background
The AnalyticBridge module proves that structural axioms plus interaction force the d'Alembert functional equation on the combiner. The J-cost function, introduced in the Cost module, is the explicit form satisfying the required properties. The upstream dalembert_identity theorem states that J(xy) + J(x/y) = 2J(x) + 2J(y) + 2J(x)J(y) for positive x and y.
proof idea
The proof is a one-line wrapper that applies the dalembert_identity lemma from the Cost module to the positivity hypotheses on x and y, then uses linarith to match the target equality.
why it matters in Recognition Science
This result supports the parent theorem P_forced_to_RCL, which shows that under structural axioms and interaction the combiner P is forced to the RCL form. It fills the verification step of the Bridge Theorem in the AnalyticBridge module, confirming that Jcost satisfies the required functional equation and aligning with the Recognition Composition Law used to derive constants such as the alpha band.
scope and limits
- Does not derive the d'Alembert identity from first principles.
- Restricted to positive real arguments.
- Does not cover zero or negative values.
- Serves only as a verification step rather than an independent result.
formal statement (Lean)
798theorem Jcost_has_RCL_combiner (x y : ℝ) (hx : 0 < x) (hy : 0 < y) :
799 Jcost (x * y) + Jcost (x / y) = 2 * Jcost x * Jcost y + 2 * Jcost x + 2 * Jcost y := by
proof body
Term-mode proof.
800 have h := dalembert_identity hx hy
801 linarith
802
803end AnalyticBridge
804end DAlembert
805end Foundation
806end IndisputableMonolith