Jcost_G_eq_cosh_sub_one
plain-language theorem explainer
The declaration shows that the log-reparametrized J-cost satisfies G(t) = cosh(t) - 1 for every real t. Workers on T5 uniqueness proofs or d'Alembert identities cite it to connect the composition law to hyperbolic functions. The proof is a short tactic sequence that unfolds the definitions of G and Jcost then rewrites via exp negation and the cosh definition.
Claim. $G(J, t) = 2^{-1}(e^{t} + e^{-t}) - 1$ for all real $t$, where $G(F, t) := F(e^{t})$ and $J(x) := 2^{-1}(x + x^{-1}) - 1$.
background
Module Cost.FunctionalEquation supplies helper lemmas for the T5 cost uniqueness proof. G is the log reparametrization $G_F(t) = F(e^t)$. Jcost is the canonical reciprocal cost $J(x) = (x + x^{-1})/2 - 1$. Upstream results include the explicit form in Gravity.JCostInflaton where the same G(t) is defined as cosh(t) - 1, and the general G def in Constants.Codata.
proof idea
simp unfolds G and Jcost. A one-line lemma establishes (exp t)^{-1} = exp(-t). The final rw applies the library definition of cosh to reach the target identity.
why it matters
The identity is invoked by washburn_uniqueness, washburn_uniqueness_aczel, T5_uniqueness_complete and dAlembert_cosh_sum. It realizes the T5 J-uniqueness step of the forcing chain by exhibiting the explicit cosh form forced by the Recognition Composition Law. It also supplies the algebraic bridge used in the d'Alembert sum identity for thermodynamic instability results.
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