Quotient Critical Geometry: Boundary-crossing times, recovery basins and actionable intervention margins in adaptive governance

Abstract

A load-to-reserve quotient can mark a boundary without determining failure time or an executable rescue. Quotient Critical Geometry (QCG) specifies this reduced interface for the Affective Gain Module (AGM). Conditional reciprocal-drain results give crossing times, approximation error, omitted-drift bounds and a moving-load solution; recovery basins can prevent crossing. Paid, delayed relief yields an initiation deadline depending on absolute load, price and latency. In a finite reference retaining reserve, load, deadline and task obligations, a planner robustly completes 68 of 96 eligible constructed states. A declared quotient trigger completes 43 and the best trigger with at most one switch point completes 54. No stationary rule observing only the exact quotient and admissible operations exceeds 66; adding task count or remaining slots reaches 68. A full-state two-slot controller with an absolute-margin tie-break also reaches 68. These limits concern the declared policy classes, not every quotient-indexed policy or affect-specific efficacy. Two legacy scripts reproduce without establishing universal critical scaling or a threefold warning advantage. Converter, energy-harvesting and yeast studies provide mechanism and measurement comparisons, not fitted AGM parameters. Prospective tests distinguish conditional timing, response slowing, predictive scaling and route restoration. The results separate a monitor index, its trajectory law and a feasible intervention. Note on Version 2.0. This version replaces Version 1 (May 2026; then subtitled "Collapse-Time Laws and Bifurcated Failure Modes in Affective Governance") and is a full rewrite (about 3,200 to 11,400 words). The load-to-reserve quotient is kept as a monitor index but is separated from its trajectory law and from a feasible intervention, and a finite reference retaining reserve, load, deadline and task obligations is added. Two legacy scripts reproduce without establishing universal critical scaling or a threefold warning advantage. Files: the manuscript as PDF and a supplement archive (48 files) with the frozen finite model, analytical identities, the finite enumerations and optimizations, and the legacy generators with their outputs. The Version 1 files remain available in the previous version of this record. Series. Paper B of the Affective Gain Module (AGM) programme. The integrative flagship, Affective Governance in Adaptive Systems, and the other companion papers (A and C to F) are archived separately. AI use disclosure. Generative AI (GPT-6.0, OpenAI; Claude Opus 5.5, Anthropic) was used substantively in preparing this work, including source comparison, drafting and editing, and, where applicable, mathematical and counterexample checks and the writing and running of supplementary code. The research questions, framework and final claims were directed and reviewed by the author, who takes full responsibility for the content, including the accuracy of all references and reported numbers. Repository metadata were prepared with assistance from Claude (Anthropic).