Jcost_hyperbolic_ODE
plain-language theorem explainer
The log-lift of the J-cost satisfies the hyperbolic ODE G'' = G + 1. Researchers closing the curvature gate in the Recognition Science forcing chain cite this step when confirming that interaction and entanglement force the hyperbolic branch. The proof is a one-line wrapper that invokes the direct derivative verification already established for the explicit cosh definition.
Claim. The function $G(t) = 2^{-1}(e^{t} + e^{-t}) - 1$ satisfies the ODE $G''(t) = G(t) + 1$ for all real $t$.
background
The curvature gate in the four-gate triangulated proof requires that the log-lift G of the cost function obey the second-order ODE G'' = G + 1. This ODE is satisfied by the explicit solution Gcosh(t) := cosh(t) - 1, which is the log-lift of the J-cost J(x) = (x + x^{-1})/2 - 1. The module assembles interaction, entanglement, curvature, and d'Alembert gates to reach the unconditional result that F equals J and P equals the RCL combiner.
proof idea
The proof is a one-line wrapper that applies the upstream theorem Gcosh_satisfies_hyperbolic. That theorem computes the first derivative as sinh and the second as cosh, which equals Gcosh(t) + 1 by the definition of Gcosh.
why it matters
This declaration supplies the curvature-gate step inside the triangulated proof of full inevitability. It confirms that the J-cost selects the hyperbolic branch once the flat solution is excluded, aligning with T5 J-uniqueness and the RCL identity. The result is invoked by the final inevitability theorem that derives F = J and P = RCL from the four gates.
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