Negotiating a cease-fire is a diplomat’s nightmare. The side gaining ground doesn’t want to give up its advantage, while the defenders don’t want to make concessions at gunpoint. Therein lies the challenge for State Department officials as they try to broker a stable truce in the nasty war between Armenia and Azerbaijan.

The Trump administration has tried admirably this week to mediate a settlement to this ancient battle over Nagorno-Karabakh, an enclave in the faraway Caucasus that’s within Azerbaijan’s borders but is populated by self-governing Armenians.

Karabakh for three decades has been one of those “frozen conflicts,” locked in a status quo that has been favorable to Armenia and its strong ground forces. But the conflict was suddenly unfrozen Sept. 27, when Azerbaijan — using armed drones supplied by Turkey and Israel — was able to neutralize Armenia’s air defenses, artillery and tanks. To Armenia’s distress, the status quo vanished.

A stable long-term outcome for this craggy enclave would be an autonomous status, independent from either neighbor, what Armenians like to call the “Republic of Artsakh.” But any such final-status issues are a very long way off while the guns are still firing.

Though this conflict is remote for most Americans, it offers a case study in how regional problems left unresolved can eventually explode into much wider crises: Turkey is boasting that it’s ready to join Azerbaijan on the battlefield, Russian forces in Armenia under a defense pact could be drawn in, and Iranian forces are inching toward the border. This faraway war could quickly get very hot.

Thankfully, for a Trump administration whose foreign policy sometimes resembles go-it-alone diktats, this mediation has been different — a careful, multilateral effort working in tandem with Russia and France. The three countries, operating as the Minsk Group Co-Chairs, have been trying to settle the Karabakh impasse since 1992.

The truce that Washington announced Sunday hasn’t succeeded so far, any better than two earlier cease-fires negotiated this month by Moscow and Paris. But U.S. officials are pushing the right buttons, aided (yes, that’s right) by a tweet from President Trump. They propose a meeting Thursday in Geneva to organize international monitoring of a real cease-fire (which Armenia wants) and negotiations about a “timeline” for a “comprehensive settlement” (which Azerbaijan seeks).

The Trump administration saw warning signs of the brewing conflagration back on Sept. 25 when Deputy Secretary of State Stephen Biegun cautioned the ambassadors of Armenia and Azerbaijan against a military buildup the United States had detected. Both swore they had no intention of going to war, but 48 hours later, Azerbaijan launched an attack and Armenia immediately countered.

Russia negotiated a cease-fire Oct. 10, but it broke down before the ink was dry; France reaffirmed the truce on Oct. 17, again to no effect. Secretary of State Mike Pompeo then stepped in, summoning foreign ministers of the two combatants to Washington last Friday for what became joint talks. By Saturday night, Biegun had hammered out a cease-fire deal, monitored by two international groups, and talks for a lasting settlement starting Thursday in Geneva, organized by the United States, Russia and France.

The United States stressed that both sides must recognize that the status quo had changed. To make sure Armenians understood the need for compromise, this message was passed to Armenian American representatives by top Republicans and Democrats alike. “Of course, we are ready for reasonable compromises,” Varuzhan Nersesyan, the Armenian ambassador to the United States, told me Tuesday.

But the Azerbaijanis, sensing they have the upper hand, have resisted. At 3 a.m. Sunday, State got word that Azerbaijan President Ilham Aliyev had nixed the deal because it didn’t specify that Armenia must withdraw. New language was added referring to a “timetable” and other buzzwords for the “comprehensive solution” Aliyev wants. The new version was blessed in Baku and Yerevan, and in Moscow and Paris. Trump tweeted “congratulations” and said “many lives will be saved” when the truce took effect Sunday night.

But Sunday’s cease-fire quickly went up in smoke. Aliyev still seemed to smell an imminent Azerbaijani victory on the battlefield, and the Armenians were determined to prevent the Azeris from seizing what’s known as the “Lachin Corridor” connecting Armenia with Karabakh.

Combatants don’t stop fighting unless the costs of continuing are too great. The United States should be thinking — urgently — about how to raise the cost of prolonged fighting. An Israeli arms cutoff to Baku? Russian muscle-flexing to support Armenia? A U.S. statement blasting Azerbaijan and its ally Turkey for ignoring the peace deal?

Realpolitik, Karabakh version: This cease-fire won’t work unless the alternative is more painful. On the way to peace, diplomats need to turn the screws.

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