defect

In the Law of Existence module the defect functional formalizes the statement that x exists if and only if defect(x) = 0. The J-cost is the canonical cost functional J(x) = ½(x + x⁻¹) - 1, which satisfies the Recognition Composition Law, is nonnegative on the positive reals, and attains its minimum of zero at unity. This definition sits inside the Recognition Science framework that derives all physics from a single functional equation, with J appearing as the T5 uniqueness map (J(x) = cosh(log x) - 1). Upstream the toy Defect from CPM.LNALBridge returns zero when structural checks pass, while the main J is imported from the Cost module.

This is a one-line definition that directly aliases the J functional. No lemmas are applied; the body simply unfolds to the expression (x + x⁻¹)/2 - 1.

This definition anchors the Law of Existence theorems such as defect_zero_iff_one and law_of_existence, which state that x exists precisely when defect(x) = 0 and therefore that 1 is the unique existent. It feeds directly into the cost algebra certificate that verifies J satisfies RCL, normalization at 1, reciprocal symmetry, and non-negativity. In the framework it connects to the T5 J-uniqueness step of the forcing chain and to the eight-tick octave. It touches the open question of extending the defect to non-positive reals in empirical programs.