The claim
1. A potential exists exactly when winning moves never cycle. Let X be a finite set of outcomes and write x → y when the negotiation can move from x to y by a move that wins a conflict. A potential is a function P : X → ℝ with P(y) > P(x) whenever x → y.
- If P exists, no chain x₀ → x₁ → … → x_k = x₀ exists, since it would give P(x₀) > P(x₀).
- If no chain cycles, the graph of winning moves on the finite set X is acyclic, so every chain is finite, of length at most |X| − 1. Let P(x) be the length of the longest chain ending at x. If x → y, a longest chain ending at x extends to one ending at y, so P(y) ≥ P(x) + 1.
So “some ordering prevails”, read as “the negotiation has a potential”, is exactly “no chain of winning moves cycles”, and integer values suffice. This is the finite case of what Monderer and Shapley call the finite improvement property and a generalized ordinal potential (THEORY-896), proved here because their paper is unread (LIT-1462).
2. A transitive binary record has a cyclic share of (n − 2)/3n, in a complete tournament. Take n options, every pair contested once, and the record Y on the complete graph with Y = +1 on each pair in the direction of the winner, the winners forming a strict ranking 1 ≻ 2 ≻ … ≻ n. The least-squares potential of the Hodge decomposition (THEORY-894) on the complete graph is s = (1/n) div Y up to a constant, and option i’s net wins are (n − i) − (i − 1) = n + 1 − 2i, so s_i − s_j = 2(j − i)/n. On the pair i ≻ j with d = j − i the residual is 1 − 2d/n, and there are n − d such pairs. With A = Σ(n − d), B = Σ(n − d)d and C = Σ(n − d)d², summed over d = 1, …, n − 1,
A = n(n − 1)/2, B = n(n − 1)(n + 1)/6, C = n²(n − 1)(n + 1)/12,
and the squared residual is A − (4/n)B + (4/n²)C = (n − 1)(n − 2)/6. Divided by ‖Y‖² = n(n − 1)/2, the cyclic share is (n − 2)/3n: 1/9 for three options, rising towards 1/3. It was also computed numerically for n = 3, 4, 5, 8 and 20.
So a binary record’s cyclic share is not ambivalence. A perfect ranking already leaves one, because binary outcomes are not differences of any scores. The baseline is exact only for the complete, equally weighted tournament; part 6 shows it does not carry to sparse records. Ambivalence is better counted without the least-squares step, as circular triads or as the least fraction of wins whose reversal leaves them acyclic (TERM-077).
3. Coherence is not alignment. A record can be wholly gradient while no single perspective wins: a stable compromise that splits every conflict the same way is as coherent as one perspective deciding all of them. Whether a particular perspective’s ordering ≿_v is the potential is a further fact, measured ordinally by the share of contested wins that agree with ≿_v, not by the gradient share. With perspectives v: A ≻ B ≻ C and w: C ≻ B ≻ A and a compromise settling every conflict as B ≻ A ≻ C, the record is wholly acyclic, and its agreement is 2/3 with v and 1/3 with w; where v decides everything, it is 1 and 0 (valence.py, case 6).
4. The Hodge gradient can reverse a comparison the wins settle. Take margins A over B by 1, B over C by 10 and A over C by 1. The wins are transitive, A ≻ B ≻ C, and the longest-chain potential of part 1 keeps that order. On the complete graph the least-squares potential is s = (1/3) div Y, and div Y = (1 + 1, 10 − 1, −10 − 1) = (2, 9, −11), so s = (2/3, 3, −11/3): B above A, though A won their contest. The cyclic share is 0.33, though nothing cycles. So the gradient is not an ordinal potential, and its cyclic share mixes uneven margins with cycles. The counterexample is the review’s; it was checked by hand and in valence.py, case 1.
5. Netting a record erases alternation. Let one pair be contested six times, won by A and B in turn. The netted flow on the pair is zero, the record of a pair never in conflict. Kept as a sequence, it has six contests and five reversals in five transitions, the most variable record the pair can have (case 4). Direction, the number of contests and their variability are three quantities; a measure built on the net flow sees only the first.
6. The baseline does not carry to sparse records. Take four options and a transitive record A ≻ B ≻ C ≻ D on every pair except (A, C), never contested. Its cyclic share is 0.200, against 1/6 for the complete tournament; with the same graph and orientations drawn from other orderings it ranges from 0.1 to 0.2 (case 3). The baseline for a sparse record is a null model that keeps its graph of contested pairs, not (n − 2)/3n.
7. A record fixes a class of potentials, not one. Every strictly increasing transform of a potential is one. And where pairs are not contested, distinct orderings can agree with every win: on wins A over B, C over D and A over D, both A ≻ B ≻ C ≻ D and C ≻ A ≻ D ≻ B agree with all three (case 7). Only a contest on a pair the two rank differently, such as B against C, separates them.
8. A vacuous record has every potential. With no contested pairs the relation of wins is empty and acyclic, so by part 1 every ordering is a potential (case 5). Agreement, as in part 3, is undefined there, having no wins to count.
9. A best-fitting ordering of a cycle is no Lyapunov function. Let the moves recur A → B → C → A. The best-fitting ordering misfits one move in three, a reversal fraction of 1/3. But a function non-decreasing along all three moves has P(A) ≤ P(B) ≤ P(C) ≤ P(A), so it is constant: no ordering is a Lyapunov function of these dynamics, however well it fits (valence.py, case 10). Fit within a tolerance and monotonicity along the dynamics are different properties; the second is what resumption needs.
10. The sign a class gives can be set-valued. Take the two orderings of part 7, both potentials of the same record, a state B and a reference state C. A ≻ B ≻ C ≻ D puts the state ahead of its reference and C ≻ A ≻ D ≻ B puts it behind (case 12). So {sgn_P(x, r) : P in the class} is {+, −}, and the record does not fix the sign.
11. An ordering fitted to a record overstates its agreement there. Draw every pair’s winner among five options at random, so that no ordering prevails, and fit the best ordering to one such record. Over 200 trials it agrees with 0.87 of the contests it was fitted to, and with 0.50 of a fresh record drawn the same way, which is chance (case 13). Agreement of a selected ordering is scored on held-out contests.
What it does not say
- That a negotiation is a game in Monderer and Shapley’s sense. A potential game has players changing their own coordinates of a joint profile; a negotiation over one indivisible choice has no such coordinates. Part 1 needs only the relation of winning moves, so it holds for either; the game-theoretic reading fits where the organization’s choice is divided into jurisdictions, as Sen’s is (THEORY-027).
- That the potential’s existence fixes the dynamics. One-at-a-time winning moves climb it and stop at a local maximum; simultaneous moves can cycle even in an exact potential game.
- That the Hodge decomposition is useless here. It measures cardinal inconsistency, and where margins are intensities that is worth measuring. Parts 4 to 6 say only that it is not the test of an ordinal potential.
- That the Hodge shares are invariant. They depend on the weights given to pairs, the aggregation of repeated contests and the norm; on continuous magnitudes a lexicographic override has no real-valued potential at all (CLAIM-436), so the term is read ordinally there. CLAIM-369 makes the same point for population dynamics: which part of a flow counts as gradient depends on a chosen metric.
Note of 2026-10-27: the edge says what the claim uses
The status review of this date (ADR-057, ADR-058) moved THEORY-896 from grounds to corroborated_by. Part 1 is proved here in full, “proved here because their paper is unread”. Monderer and Shapley’s general result corroborates it, and the proof does not use it.