1.3. The other findings🔗

What else the audit turned up, beyond the one claim.

Rothbard's 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.
The law is strict, as Rothbard's is. Non-increasing marginal utility is the neoclassical form, not his.
Nothing assumes that exactly one want is given up. Rothbard's phrase "the marginal unit" takes that for granted; the law here holds for every want that goes.
One unit to one end is not needed at all. Rothbard assumes it and says he is assuming it — "each unit of means is capable of serving one of the ends", introduced with "We assume for simplicity" (p. 26) — and it reads like a premise of the derivation. It is not a premise of anything here. It was built into what a plan is, where no statement showed it; taken out, no theorem asked for it back. Where an extra unit adds no new end the law is simply silent, and the six horses supply a case where it is not silent, so nothing is lost. A simplification its own author flagged, and the ordering does not want it.
The proof never uses the fact that the end at the smaller supply is the marginal one. The law holds for every end served there against every end the next unit would add — and stronger still, a served end beats any unserved end that could be served, not only the one the next unit reaches. Rothbard's statement claims less than his premise delivers.

What the law does not need, which is most of what it usually gets.

No property of preference is forced: not transitivity, not completeness. Rothbard assumes a single ranked value scale (Figure 3, pp. 25–26). None of it is needed, because the plan already carries the ordering.
Interchangeability of units is not needed for the ordering. With the plan indexed by which units, both the urgency principle and the law along a chain of named units go through without it. It is needed exactly once, to state the law by size of supply: the phrase "the plan at n units" only picks out one thing if two piles of the same size serve the same ends. So it bears on the wording, not on the derivation — which is why it is a hypothesis and not a praxeological claim. Rothbard makes it part of what a supply is (p. 23), and where it fails the units are not one good. But his definition does not deliver it. It speaks of units "equally capable of rendering the same service" — a fact about the units, where the wording needs a fact about the man's plan over them. One stock carries plans of both kinds (unitsAlike_not_entail_homogeneous), so no condition on the units can decide it.
Indifference between units is needed nowhere. Rothbard defines a supply with the words "valued equally" and "regards ... indifferently" (p. 23), then takes them back on the next page: interchangeability "does not mean that the concrete units are actually valued equally" (p. 24). The formalization follows p. 24.
Independence of uses is a condition on the situation, and it can only be stated at all once preference ranges over bundles of ends rather than single ends. Where uses are complementary the law says nothing, and the statement admits as much.

Findings about the formalizing rather than the doctrine.

Being able to tell two ends apart is a suppressed premise of the phrase "this bundle, minus this end". It only came to light because the proofs refuse classical logic.
The claim that people act splits into three: a definition, a bridge, and an existence claim. Only the bridge could do deductive work, and nothing has needed it.
Never assert a claim about every structure of a given shape — say, every plan the man might have. Given one plan, a rival can always be built that breaks such a claim, so the claim is refutable by construction. Assert it of the single plan a theorem is handed instead, and that same construction becomes harmless. Better: it turns into a result. One of the theorems below proves that two plans satisfying the claim cannot differ by a single swap — so "the" value scale is something proved here, not something assumed.