1.8. Findings🔗

• The law of marginal utility rests on one praxeological premise, and that premise is subjunctive preference: what the agent would serve at each supply. No axiom about actual action is cited. Rothbard's derivation therefore uses what his demonstrated-preference doctrine forbids. This is Nozick's cost argument (1977, §III) transposed to marginal utility and machine-checked.

• Rothbard's own derivation is one step from an asserted premise (p. 24). The one-step proof is faithful to him; the audit point is that "derived from the fundamental axiom of human action" (p. 27) rests on an assertion.

• Nothing structural about preference is forced: no transitivity, no totality. Rothbard presupposes a linear value scale (Figure 3, pp. 25–26); the encoding needs none, because the disposition carries the order.

• The law is strict, as Rothbard's is. Non-increasing marginal utility is the neoclassical form, not his.

• Determinacy of the drop is not assumed. Rothbard's "the marginal unit" presupposes exactly one end goes; the law holds over all of them.

• The marginality of the end at the smaller supply is unused by the proof. The law holds for every end served there against every end the next unit adds; Rothbard's statement is weaker than his premise delivers. Stronger still: a served end beats any serviceable unserved end, not only the one the next unit reaches.

• Interchangeability of units is not needed for the ordering. With the plan indexed by which units, the urgency principle and the law along a chain of named units hold without it. It is needed once, to state the law by supply size: "the plan at n units" is a function of n only if two sub-stocks of the same size serve the same ends. Nozick's remark that without a unit the law cannot be stated (1977, p. 371) lands here, on the statement and not the derivation. Interchangeability is a hypothesis, not an axiom: Rothbard makes it definitional of a supply (p. 23), and where it fails the units are not one good.

• Rothbard defines a supply with the words Nozick attacks, "valued equally", "regards ... indifferently" (p. 23), and withdraws them on the next page: interchangeability "does not mean that the concrete units are actually valued equally" (p. 24). The encoding follows p. 24. Indifference between units is asserted nowhere, and the law survives; what it needs instead, that the plan ignores which units, is subjunctive, the same kind of commitment as swap dominance. The units trap and the demonstrated-preference collision are one collision.

• Independence of uses is a situational hypothesis, statable only once preference ranges over bundles. Where uses are complementary the law is silent, and the statement says so.

• Decidable identity of ends is a suppressed premise of "the bundle minus this end", surfaced by refusing classical logic.

• The action axiom decomposes into a definition, a bridge, and an existence claim, and only the bridge could do deductive work. Nothing has needed it.

• A universal claim about "the agent's X" must not quantify over the type of X-shaped structures: given one, rivals are definable and the axiom refutes itself. Two such crashes are in the archaeology.