Reasoning Geometry of Language Models: How Prompting, Training, and Scale Shape Judgment Trajectories

Language-model reasoning exhibits behaviorally anchored functional geometry, but prompting and post-training reshape it in fundamentally different ways. Controlled contrasts identify reasoning-associated ($u$) and judgment-associated ($v$) factors: reasoning is $u$-weighted ($S_R=0.96$), judgment is $v$-weighted ($S_J=-0.36$), yielding $\Gamma =1.33$. A $2\times 2$ factorial reveals asymmetric interaction: reasoning remains $u$-weighted across policy states, while judgment geometry depends on reasoning context. Prompting and post-training act as distinct geometric operators: prompting reweights engagement of this functional organization, whereas post-training reorients the representation frame in which it is expressed. Held-out Procrustes alignment removes 95% of the final-layer $v$–$u$ rotation gap, showing that most angular change reflects shared frame reorientation rather than loss of functional structure. Despite coordinate variation, relative functional organization persists across probes, training states, and tasks. Interventions along either factor are no more consequential than norm-matched random directions, separating behavioral readability from selective steerability. Endpoint shift predicts reflection success (AUROC $0.85$). Together, these results identify relational trajectory geometry as a substrate for model observability, distinguishing functional change from representational reorientation across prompting and post-training.