pairwise_RCL_balance
plain-language theorem explainer
The theorem establishes the exact algebraic identity for the net J-cost change under a paired stretching event x to lambda x together with x to x over lambda. Researchers modeling discrete fluid dynamics or verifying J-cost monotonicity in Recognition Science would cite this when confirming energy balances for paired events. The proof is a direct algebraic reduction that unfolds the definitions, clears denominators via field_simp, and finishes with ring.
Claim. For $x > 0$ and $lambda > 0$, the pairwise stretching change satisfies $J(x lambda) + J(x / lambda) - 2 J(x) = 2 J(x) J(lambda) + 2 J(lambda)$, where $J(y) = (y + y^{-1})/2 - 1$ is the canonical cost function.
background
The module isolates the algebraic core for paired stretching and compression events in discrete Navier-Stokes settings, remaining PDE-free so that concrete operators can reuse the same definitions. The central definition is pairwiseStretchingChange x lam := Jcost (x * lam) + Jcost (x / lam) - 2 * Jcost x, which records the net cost increment for the pair. Jcost is the canonical J-function from PhiForcingDerived, obeying the Recognition Composition Law J(xy) + J(x/y) = 2 J(x) J(y) + 2 J(x) + 2 J(y) and known to be strictly convex with minimum at 1 from PhysicsComplexityStructure.
proof idea
The proof unfolds pairwiseStretchingChange and Jcost to obtain the explicit expression. It introduces the facts that x and lam are nonzero, applies field_simp to clear denominators, and concludes with the ring tactic to verify the identity.
why it matters
This supplies the precise Recognition Composition Law balance required for paired stretching in the Navier-Stokes module. It is invoked directly by the downstream theorem pairwise_RCL_balance_factored, which rewrites the left-hand side as 2 (Jcost x + 1) Jcost lam. In the framework it closes the algebraic step for J-cost monotonicity programs, connecting to the eight-tick octave and phi-ladder without introducing continuous fields.
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