The Empty Occupant: A Mathematics Guide for Undergraduates

This mathematics guide accompanies Nicholas Cott’s monograph The Empty Occupant: Answerability, Testimony, and the Limits of Machine-Mediated Authority (manuscript dated 06 November 2026). It is designed to be read alongside The Empty Occupant: A Reading Guide for Undergraduates. It is written for undergraduates with no mathematical preparation beyond ordinary algebra: it assumes no calculus, no statistics beyond percentages and simple averages, and no training in formal logic. Contents. The guide explains every family of symbols, equations and quantitative comparisons in the monograph across ten lessons. Topics include: reading the book’s notation: conjunction, implication, equivalence, quantifiers, sets and forecasts; the four-relation Answerability conjunction and the three-capacity test for genuine reconsideration; maturation times, defended service intervals and the unit of Deontic Debt; stocks and flows of obligations, the weighted Deontic Debt index and its severity weights; why critical constraints cannot be traded off against a weighted total; the service-plan predicate, the no-plan formula for Institutional Insolvency and the maximum serviceable scale; the monograph’s four-week synthetic recovery calculation, including a derivation of its recovery inequality; the Robodebt staffing sensitivity calculation, and why it is not an insolvency verdict; what the equations can and cannot prove: lower bounds, forecasts and counterexamples. Approach. The guide reuses the invented Northvale rent-assistance council from the Reading Guide and keeps it separate from the monograph’s own 1,000-case synthetic scenario. Worked examples train students to keep stipulated, documentary and assumed numbers apart. They also teach students to check units and to tell a definition, a necessary condition, a lower bound and a sensitivity calculation apart. Steps added to aid learning, such as the stock–flow identity, are labelled as teaching illustrations. Structure. Each lesson includes learning objectives, a plain-English explanation, the relevant location in the monograph, worked calculations, key-term definitions, common mistakes, a summary and questions. The guide also provides: an eight-step checklist for interpreting any formula in the monograph; a practice workshop with three problems; answers to all lesson and workshop questions; a formula sheet and a complete symbol glossary; a suggested reading order for using it with the Reading Guide. Scope and currency. Section and equation references follow the 06 November 2026 manuscript. Historical figures follow the monograph and are not independently re-checked; the guide is not a new legal or documentary status check. Northvale and the practice problems are invented for teaching and describe no real organisation. This guide was drafted with AI assistance.