dAlembert_cosh_sum
plain-language theorem explainer
The d'Alembert sum identity for hyperbolic cosines follows from the Recognition Composition Law applied to the J-cost function. Researchers modeling extraction costs or optimality in recognition frameworks would cite it to decompose sums of cosh terms. The proof invokes Jcost_cosh_add_identity, substitutes the explicit form from Jcost_G_eq_cosh_sub_one, and closes via ring algebra plus linarith.
Claim. For all real numbers $a$ and $b$, $cosh(a + b) + cosh(a - b) = 2 cosh a cosh b$.
background
The J-cost function satisfies Jcost(exp t) = cosh t - 1, so its associated G obeys G(Jcost, t) = cosh t - 1 by the upstream result Jcost_G_eq_cosh_sub_one. The Recognition Composition Law is realized as the cosh-add identity Jcost_cosh_add_identity, which encodes the functional equation J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y) in exponential coordinates. This theorem lives in the Ethics.ThermodynamicInstabilityOfExtraction module, where cost identities derived from the RCL are applied to extraction systems and optimality conditions.
proof idea
The tactic proof opens the Cost.FunctionalEquation namespace and obtains rcl := Jcost_cosh_add_identity a b. It rewrites rcl with Jcost_G_eq_cosh_sub_one to express everything in cosh terms. A ring tactic shows that 2*((cosh a - 1)(cosh b - 1)) + 2((cosh a - 1) + (cosh b - 1)) equals 2*cosh a * cosh b - 2. Linarith then yields the target identity.
why it matters
The result is invoked directly by the downstream theorem cosh_sum_via_dAlembert to obtain the product-to-sum decomposition. The module doc-comment states that the RCL forcing the cost function also proves the Love optimality theorem. It supplies the hyperbolic identity needed for thermodynamic instability arguments in the ethics module and confirms consistency with the J-uniqueness step (T5) and RCL in the Recognition Science forcing chain.
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