j_submult
plain-language theorem explainer
The submultiplicativity bound on J-cost asserts that for positive reals x and y the cost of their product satisfies J(xy) ≤ 2J(x) + 2J(y) + 2J(x)J(y). Number theorists modeling multiplicative ledgers in Recognition Science cite this when bounding composite transaction costs from prime factors. The argument is a one-line wrapper that invokes the underlying Jcost_submult lemma derived from the d'Alembert identity.
Claim. For positive real numbers $x$ and $y$, the Recognition Science cost function satisfies $J(xy) ≤ 2J(x) + 2J(y) + 2J(x)J(y)$.
background
The J-cost function is defined by J(z) = (z + z^{-1})/2 - 1 for z > 0 and obeys the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). Nonnegativity of J on the quotient then yields the stated submultiplicative inequality. This module frames primes as irreducible ledger transactions whose unique factorization supplies the balance sheet of the multiplicative group, while drawing a structural parallel between J-cost symmetry and the functional equation of the completed zeta function.
proof idea
The proof is a one-line wrapper that applies the Jcost_submult lemma from the Cost module. That lemma extracts the inequality directly from the d'Alembert identity together with the nonnegativity of Jcost on the quotient x/y.
why it matters
The bound supplies the multiplicative control required to link ledger costs to the zeta functional equation inside the prime-ledger model. It supports the module hypothesis that d'Alembert zero-line constraints on the completed zeta function would confine all zeros to the critical line, aligning with the Recognition Science prediction of the Riemann Hypothesis. The result sits downstream of J-uniqueness (T5) and the Recognition Composition Law.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.