The claim specifies a meeting on 'April 14' brokered by 'Secretary of State Marco Rubio' revealing a 'language gap.' While multiple sources confirm direct talks occurred in Washington involving Rubio and that the talks focused on peace, none of the provided evidence specifically mentions the date 'April 14' or the conclusion that the difficulty was a 'language gap' rather than an objective gap. The evidence confirms the general event but not the specific details of the conclusion.

The evidence suggests a general alignment of interests regarding Hezbollah's military control, as seen in reports about ceasefire deals and the need for stability. However, no source explicitly states that 'Beirut and Jerusalem share an interest in preventing Hezbollah from exercising independent military control over southern Lebanon' as a confirmed, shared objective.

The evidence repeatedly frames the conflict around Hezbollah's armed status and its challenge to the Lebanese state. While the sentiment that peace requires addressing Hezbollah's military status is present across multiple reports, the claim's definitive statement that 'Achieving peace with Lebanon is impossible as long as Hezbollah exists...' is a strong, conclusive opinion/analysis not directly confirmed by multiple independent sources as an established fact.

The evidence provided for this claim is insufficient to confirm that the 'four pillars' framework was established or discussed across multiple independent sources. The web search results mention the concept of 'sovereign functions' but do not list or confirm the existence of a 'four pillars' structure.

The concept of Hezbollah's right to decide on war and peace is mentioned in the context of its resistance stance, but this specific pillar is not corroborated by multiple independent sources detailing the 'four pillars' framework.