Picture two halves that have come apart. Reconcile is the move that brings them back so they fit again: two people friendly once more, two sets of figures that finally agree. One way to hold irreconcilable is that this move is simply not available. The edges are shaped so that no amount of pushing makes them meet, so the gap stays open. That one picture covers the whole word — two sides whose aims will not meet, two ideas that will not sit together, and a person who will not come back.
From the word’s history
Formed from ir- (not) + reconcilable, from reconcile; borrowed via Middle French irreconciliable, from Late Latin irreconciliabilis.
由 ir-（不）＋ reconcilable 組成，底層是 reconcile（讓兩邊重新合得上）。經中古法文 irréconciliable，來自晚期拉丁文 irreconciliābilis。字面上就是「拼不回去的」，跟今天的核心畫面剛好對上。
Wiktionary, "irreconcilable"
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