Quotient Critical Geometry: Collapse-Time Laws and Bifurcated Failure Modes in Affective Governance

This paper formalizes collapse risk in adaptive governance systems as a quotient geometry in loading–resilience space. Rather than treating collapse as a function of load, misalignment, or endurance alone, the paper defines the collapse-control parameter κ (t) =A (t) /H (t) =S (t) δ (t) /H (t) (t) =A (t) /H (t) =S (t) (t) /H (t) κ (t) =A (t) /H (t) =S (t) δ (t) /H (t), where instability, misalignment, and affective endurance jointly determine proximity to collapse. The paper derives collapse-time laws under quasi-static loading, showing how time-to-collapse depends on the initial quotient state, the critical threshold, and the depletion rate of affective endurance. It also corrects the critical interaction-density relation, showing that stronger misalignment coupling lowers the safe coordination density, while higher capacity and endurance raise it. A central contribution is the classification of Freeze and Runaway as the two primitive collapse-mode classes of affective governance systems. Freeze corresponds to suppressed excitation gain and silent criticality, while Runaway corresponds to amplification cascades under preserved or elevated sensitivity. Mixed-mode collapses are treated as composite regimes composed of local Freeze and Runaway dynamics. Controlled mechanism tests support the associated predictions, including critical slowing down, finite-size scaling, and early-warning behavior. The paper positions quotient critical geometry as the formal collapse-law layer of the broader AGP/AGM series.