j_dalembert
plain-language theorem explainer
The d'Alembert identity holds for the J-cost function on positive reals: J(xy) + J(x/y) equals 2J(x) + 2J(y) + 2J(x)J(y). Number theorists modeling primes as irreducible ledger transactions in Recognition Science would cite this when connecting cost symmetries to functional equations. The proof is a direct one-line application of the dalembert_identity theorem from the Cost module.
Claim. For all positive real numbers $x$ and $y$, the J-cost satisfies $J(xy) + J(x/y) = 2J(x) + 2J(y) + 2J(x)J(y)$.
background
In Recognition Science the J-cost is the function J(x) = (x + x^{-1})/2 - 1 on positive reals, giving the cost of a recognition event. The module treats natural numbers as multiplicative ledger transactions, with primes as the irreducible entries whose factorization corresponds to ledger balance. The upstream dalembert_identity theorem states: 'The d'Alembert identity: J(xy) + J(x/y) = 2J(x) + 2J(y) + 2J(x)J(y)' and supplies the algebraic verification under the positivity hypotheses.
proof idea
The proof is a one-line wrapper that applies the dalembert_identity theorem from the Cost module directly to the hypotheses hx and hy.
why it matters
This theorem places the d'Alembert identity inside the prime-ledger model, supporting the structural parallel between J-cost symmetry and the zeta functional equation. It supplies the proved d'Alembert zero-structure component listed in the module. It instantiates the Recognition Composition Law and sits on the forcing chain from T5 J-uniqueness onward. The module leaves open whether the Euler product imposes further d'Alembert-type constraints on the completed zeta function.
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