Distance asks how far apart two positions are. Persuasion asks how much has to come loose at once. Pick a framework and walk the route.
A step moves to an adjacent complete labeling, and its price is the number of arguments whose label changes. A route costs what its worst step costs, not what its steps add up to. That is the whole model: a long negotiation of small concessions can succeed where one large demand is refused.
Releasing a commitment releases everything whose only standing justification it was. Those arguments are the ripple of the step. Watch the framework at the top: arguments both parties agree on are drawn with a dashed outline, and the demo rings one at the moment it goes undecided.
Watch what happens when you add an outside applicant that both parties reject: the disagreement does not change, and the incline goes up anyway.
Dinner: the ripple is not our example. It is Example 1 of Alfano, Greco, Parisi and Trubitsyna (KR 2024), built to study counterfactual explanations. The effect was already in it.
Take An outside applicant: the framework has exactly three coherent positions, Ada's, Ben's, and the one where everything is undecided. There is nowhere else to stand. Ada's and Ben's positions each commit to something the other denies, and a move in this model only ever suspends commitments or makes them; reversing a judgment ("a is in" to "a is out") is not a move one can ask of anyone. Reversal has to be realized as suspend, then re-commit, and the only position below both is total suspension. Once there, completeness drags c along: c stays out only while some attacker of it stands, and at the bottom neither a nor b does.
The dashed arc is the direct demand: "adopt my position, now." Here it is even cheaper by count (it changes two labels; the gentlest route must move three at once), but it is not a persuasion move. It asks Ada to flip from believing a to believing b without ever passing through doubt, and no coherent position realizes that state.
Agreed claims that hang on the contested one are ballast. Raise the slider and the incline rises with it without bound, though the parties still disagree about exactly two arguments.
On The ridge, the gentlest route refuses to meet in the middle. It goes through a cross-party position that neither side holds, chosen so the shared rejection never comes loose.
Suppose you and a friend disagree about exactly one thing. Giving in should cost exactly that one thing: your side of the disagreement. That would be a local concession, with the price paid right where you disagree.
But opinions lean on each other. Some things you both agree on are held up only by the thing you are arguing about. Ada and Ben both reject the fish, each for the same reason: my own main course is better. Ask Ada to reopen her main and her reason goes with it; the fish is back on the table, though nobody wanted it there.
So a concession can cost more than the disagreement, and the extra is paid outside it, on things everyone thought were settled. That is why concessions are not local. The two counters above make it precise: distance counts the disagreement; the incline counts what must actually come loose at once, on the gentlest possible route. Whenever the incline exceeds the distance, someone is paying outside the disagreement.