is_implies_ought
plain-language theorem explainer
Recognition Science identifies the ethical notions of moral goodness and betterness directly with the non-negativity and zero characterization of the defect functional. A foundationally oriented physicist or philosopher would cite the result as the formal dissolution of Hume's is-ought gap inside the J-cost framework. The proof is a compact term-mode refinement that applies the upstream defect lemmas to the structural clauses and unfolds the moral definitions to recover the zero-defect equivalence.
Claim. Let $d(x)$ be the defect of a positive real $x$, given by the J-cost function. Then $d(x) ≥ 0$ for every $x > 0$, and $d(x) = 0$ if and only if $x = 1$. An ethical action $a$ (a map from an initial positive real to a final positive real) is morally good precisely when its final state equals 1. Action $a$ is morally better than action $b$ precisely when $d(a_ {after}) ≤ d(b_ {after})$.
background
The module derives objective morality from J-cost minimization. Defect is the functional $d(x) = J(x)$ supplied by LawOfExistence, where J is the unique cost obeying the Recognition Composition Law. Upstream, CostFirstExistence.RSExists defines recognition existence as vanishing J-cost, while LawOfExistence supplies defect_nonneg (non-negativity for positive arguments), defect_zero_iff_one (zero only at unity), and defect_at_one (explicit zero at 1). The local setting is the claim that stable configurations are exactly those with zero defect, so that ethical ordering is the ordering by defect reduction.
proof idea
The term proof refines the four-way conjunction. The two IS clauses are discharged immediately by defect_nonneg and defect_zero_iff_one. For the MorallyGood biconditional the proof introduces the action, splits, unfolds MorallyGood and moral_cost, then applies defect_zero_iff_one in one direction and defect_at_one in the other. The MorallyBetter clause follows at once from the definition of moral_cost as defect of the after-state.
why it matters
The declaration is the central theorem of the ObjectiveMoralityStructure module and dissolves Hume's guillotine by equating good with zero defect. It realizes the identification good(x) ≡ RSExists(x) ≡ defect(x) = 0 forced by J-uniqueness (T5) and the existence law. The result sits downstream of the LawOfExistence defect lemmas and upstream of the uniqueness of the moral ideal; no further scaffolding is required.
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