When the Manifold Collapses: A Geometric Theory of Constraint Conflict in Large Language Models

Mabrok (2026) proved that the expressibility gap of LLM hidden-statemanifolds obeys a linear volume scaling law under regularity conditions in-cluding a positive margin gradient lower bound. We prove that this regular-ity fails under contextual constraint conflict. Using the Riemannian coareaformula, we show that when the empirical coarea density becomes singularnear the Voronoi boundary — AC(s)∼s−α with α>0 — the linear scalingη(ε)∼ε transitions to a sub-linear power law η(ε)∼ε1−α (Theorem 1).We prove that where constraint conflict diffuses probability mass across Kcompetingalternatives,theFishermetricdegeneratesalongmargin-relevantdirections, with Fisher sensitivity bounded by O(1/K) (Theorem 2); thisK-diffusion condition is treated as a model-dependent empirical premiserather than a universal consequence of conflict. Experiments on six trans-former architectures (124M–1.5B parameters) across five constraint conflictlevels plus two randomized-prefix controls confirm the theory: two of thethirty conflict cells survive Bonferroni correction for β <1 (OPT-1.3B ex-treme, β = 0.664 [0.595,0.748]; GPT-2 extreme, β = 0.809), while zeroManifold Collapse in LLMs 2of twelve randomized controls show significant collapse, confirming con-flict specificity. The theory identifies two channels — metric degenera-tion and density concentration — that are empirically dissociated: the cellexhibiting the strongest Fisher contraction (fivefold dF collapse, condition-numberreductionfrom17.5to3.9,entropy–margindecoupling∆ρ= +0.29;OPT-125M strong conflict) shows no scaling collapse, while the Bonferroni-surviving collapse cells show unchanged Fisher distances. We propose athree-phase classification of manifold response: Navigation (connected, en-riching), Collapse (degenerate), and Fragmentation (disconnected, halluci-natory).