The claim
Read a currency on a set X as a function u : X → ℝ that represents a relation ⪰ on X: x ⪰ y exactly when u(x) ≥ u(y). Read an ordering as a complete preorder (complete and transitive).
1. On a finite set, every ordering is a currency. Let ⪰ be a complete preorder on a finite X and put u(x) = |{z ∈ X : x ⪰ z}|. If x ⪰ y, then y ⪰ z gives x ⪰ z by transitivity, so {z : y ⪰ z} ⊆ {z : x ⪰ z} and u(x) ≥ u(y). If not x ⪰ y, completeness gives y ⪰ x, so by the same argument {z : x ⪰ z} ⊆ {z : y ⪰ z}. The inclusion is strict: y is in {z : y ⪰ z}, since a complete relation is reflexive, and y is not in {z : x ⪰ z}, since not x ⪰ y. So u(y) > u(x), and u represents ⪰.
2. On the unit square, the lexicographic order is not a currency. Let (a, b) ⪰ (a′, b′) when a > a′, or a = a′ and b ≥ b′, on [0, 1]². Suppose u represented it. For each a ∈ [0, 1], (a, 1) ≻ (a, 0), so the open interval I_a = (u(a, 0), u(a, 1)) is non-empty. If a < a′, then (a′, 0) ≻ (a, 1), so u(a′, 0) > u(a, 1): the intervals are pairwise disjoint. Each contains a rational, and distinct intervals contain distinct rationals, so there would be uncountably many rationals. No such u exists.
3. A continuous ordering is a currency. A complete preorder on a connected, separable topological space whose upper and lower contour sets are closed has a continuous real-valued representation. The theorem is essentially Eilenberg’s (1941); Debreu (1954, LIT-1454) removed the connectedness assumption for second-countable spaces, and states both in Theory of Value (1959, LIT-1435, read in NOTE-1112, pp. 56, 72–73), with the lexicographic order of R² as his example of a complete preordering that “cannot be represented by a real-valued function” (p. 72). The lexicographic order of part 2 fails the hypothesis: its contour sets are not closed.
3a. Continuity is sufficient, not necessary. A discontinuous ordering can still have a representation. Take cost–resource pairs (c, r) with c, r ∈ [0, 1], a continuous trade-off f(c, r) with values in [0, 1] for costs at or below a threshold c*, and above it a ranking by cost alone, resource ignored. Then u(c, r) = f(c, r) for c ≤ c* and u(c, r) = −c for c > c* represents it: every option above the threshold is worse than every option below. The ordering is discontinuous at c* unless f takes its least value at every point of the line c = c*; with f(c, r) = (1 + r − c)/2 and c* < 1, for instance, the lower contour set of (0, 0) contains every option above the threshold but not (c*, 1), which they approach. What has no representation is a graded override, in which the overriding dimension decides at every level across a continuum and another dimension still breaks its ties. Above the threshold, a ranking by cost with resource breaking ties at equal cost is the order of part 2 again.
3b. A threshold override with a discrimination threshold is not an ordering. Part 3a’s threshold is exact. A real override decides by its own dimension only when the difference on it is large enough to register, and lets another dimension decide otherwise. With cost c, resource r and a discrimination threshold δ > 0, let x be chosen over y when c(y) − c(x) > δ, or when |c(x) − c(y)| ≤ δ and r(x) > r(y). This is Tversky’s lexicographic semiorder, built on Luce’s semiorders, in which indifference within a threshold is not transitive. Its choices can cycle. Take x₀ = (0, 0), x₁ = (δ/2, 1/3), x₂ = (δ, 2/3), x₃ = (3δ/2, 1). Each neighbour is within δ on cost and has more resource, so x₁ is chosen over x₀, x₂ over x₁ and x₃ over x₂; but x₃ costs 3δ/2 more than x₀, past the threshold, so x₀ is chosen over x₃. No complete preorder on the four options agrees with all four choices, so the rule admits no ordering of them, and so no currency. Every pairwise choice is still decided: each pair, faced alone, is an ordering of two options. The cycle shows only across choices, which is part 4’s case.
4. Successive currencies need not share exchange rates. If R_t is an ordering at each t with representation u_t, nothing relates u_t to u_s: any strictly increasing transformation of u_t represents R_t equally well, and R_t and R_s may disagree. A sequence of orderings can even cycle across time, x above y at t, y above z at s, z above x at r, with each instant’s ordering a currency. A currency that holds its exchange rates over time is a further fact, not a consequence of having one at each instant.
What it means for the reframing
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Where options are few and discrete, the reframing is a relabelling at any single instant, by part 1. Its content there is part 4: the record’s word “ordering” does not promise stable rates.
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Where magnitudes are continuous, as in the trade-off experiments Veit’s criterion uses (a noxious cost against a resource, in proportion to both), “admits an ordering” is strictly weaker than “admits a currency”. A continuous trade-off, Veit’s form (THEORY-101), has a continuous currency. An override, ALARM’s “100% weight” (THEORY-090), has no currency if it is graded, the alarm deciding at every level of its own intensity; an exact threshold override has a discontinuous one; and a threshold override with a discrimination threshold, the realistic case, admits no ordering at all over a chain of nearby options, and shows itself as cycles in choice (part 3b). CLAIM-435 draws the consequence for their rivalry.
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“Currency” has four senses, and this claim separates two. Ordinal extendibility, ordinal comparability, scalar currency and cardinal trade-off structure are successively stronger (TERM-077). “Ordering” here is the second, a complete preorder; a potential needs only the first, acyclic wins that some complete ordering extends. “Currency” here is the third. Part 4’s exchange rates belong to the fourth: a representation fixed up to an affine transform, which nothing in parts 1 to 3 supplies. TERM-076 once defined a common currency as a complete, transitive relation, running the second and third together; it was revised before merge to use only the first.
What it does not say
- That every ordering without a currency is lexicographic. Others exist (orders on non-separable spaces, among them); the claim says only that the graded lexicographic one lies in the gap.
- That every override lacks a currency. An exact threshold override has one (part 3a); with a discrimination threshold it has no ordering (part 3b).
- That a negotiation always produces an ordering. It need not (THEORY-027, CLAIM-353); that is a separate fact, used in CLAIM-440.
- That anything here is new mathematics. Parts 1 to 3b are textbook results of utility theory (Eilenberg; Debreu; Luce; Tversky), written out so the record can check them.