arXiv:2605.11574v1 Announce Type: new Abstract: The literature on how large language models handle conflict between their training knowledge and a contradicting document presents a persistent empirical contradiction: some studies find models stubbornly retain their trained answers, ignoring provided documents nearly half the time, while others find models readily defer to the document, following context approximately 96% of the time. We argue these contradictions dissolve once one recognises that prior experiments have studied three qualitatively distinct processing situations without distinguishing them. We propose a three-regime framework: Regime 1 (single-source updating, dominant predictor: evidence coherence), Regime 2 (competitive integration, dominant predictor: parametric certainty), and Regime 3 (task-appropriate selection, dominant predictor: task knowledge requirement). We formalise a distinction between parametric strength (exposure frequency) and parametric uniqueness (encoding consistency), showing empirically that these are orthogonal dimensions (r = -0.002, p = .97) with strength as the operative predictor in stable factual domains. We validate the framework across Claude Sonnet 4.6, GPT-5.5, Gemini 2.5 Flash, Llama 4 Maverick, and DeepSeek V3 using 9,970 API calls in three experimental phases. GEE logistic regression confirms the predicted Regime 2 certainty gradient for all five models (beta = -0.38 to -0.50, all p <= .013, BH-FDR corrected). A Regime 3 ablation shows task framing alone flips context-following from near-100% (contextual knowledge condition) to 6-71% (parametric knowledge condition), with all five models significant (p < .001). The certainty gradient is robust to multinomial outcome modeling, sensitivity analyses for hedging responses, and FDR correction.