One partner says "Fine" meaning "resolution"; the other hears "surrender." The word is shared; the affective uptake is not. We formalize this as affective meaning divergence (AMD), the total-variation distance between interlocutors' anchor-conditioned affect distributions. Building on speech-act theory, common-ground accumulation, and entropy-regularized game theory, we derive a logit best-response map whose dynamics undergo a saddle-node bifurcation: when $βα> 4$, a monotone increase in AMD-driven load produces an abrupt, hysteretic collapse of repair coordination. On Conversations Gone Awry (CGA-Wiki; $N = 652$), derailing conversations exhibit critical-slowing-down (CSD) signatures across multiple levels: lexical divergence variance ($p < 0.001$, $d = 0.36$), AMD variance ($p = 0.001$, $d = 0.26$), and dialog-act repair variance ($p = 0.016$, $d = 0.20$), all significant after correction and stronger than toxicity and sentiment baselines. AMD provides a distinct temporal signature, with retrospectively measured variance peaking at the bifurcation point while toxicity variance peaks earlier, and is the only indicator grounded in the theoretical framework. Boundary-condition analysis on CGA-CMV ($N = 1,169$) yields mixed but directionally consistent evidence.