This appendix presents five clinical vignettes designed for use in medical ethics courses, residency training, ethics committee workshops, and self-directed study. Each vignette is structured identically: a clinical scenario, a nine-dimensional attribute vector decomposition, a clinical geodesic analysis, discussion questions, and the key geometric concepts it illustrates.
These vignettes are expanded from the worked examples in Chapter 9 and the falsifiable prediction designs in Chapter 17. They are designed to be used independently of the book’s mathematical apparatus — a reader who has understood the nine dimensions and the concept of pathfinding can use these vignettes without proofs or formal definitions.
Maria Santos, 45, is admitted to the emergency department with acute upper gastrointestinal bleeding. She is hemodynamically unstable: systolic blood pressure 78 mmHg, heart rate 124 bpm, hemoglobin 5.2 g/dL. The emergency physician determines that blood transfusion is required. Without transfusion, the estimated mortality is 35–45%.
Ms. Santos is a Jehovah’s Witness. She is alert, oriented, and has decision-making capacity confirmed by two independent assessments. She refuses blood transfusion on religious grounds. She understands the mortality risk. She has a signed advance directive specifying no blood products under any circumstances. Her adult daughter, present at bedside, supports her mother’s decision.
| Dimension | Path A: Transfuse Against Refusal | Path B: Respect Refusal, Optimize Alternatives |
|---|---|---|
| $d_1$: Outcomes | +large (survival probability ~90%) | -moderate (survival probability ~55–65%) |
| $d_2$: Rights | $-\infty$ (violates legal right to refuse) | 0 (rights intact) |
| $d_3$: Justice | 0 (neutral) | 0 (neutral) |
| $d_4$: Autonomy | $-\infty$ (violates competent refusal) | 0 (autonomy fully respected) |
| $d_5$: Trust | -large (trust destroyed) | +moderate (trust reinforced) |
| $d_6$: Social | -moderate (daughter’s values violated) | 0 (family values respected) |
| $d_7$: Dignity | -large (deepest identity violated) | 0 (identity intact) |
| $d_8$: Institutional | -large (legal liability) | 0 (standard of care met) |
| $d_9$: Epistemic | 0 (patient understands) | 0 (patient understands) |
Path A has edge weight $w_A = \infty$ because $\beta_{\text{consent}} = \infty$ for competent non-emergency patients. The clinical geodesic is Path B, which includes: (a) aggressive volume resuscitation with crystalloids and colloids; (b) erythropoietin administration; (c) intraoperative cell salvage if surgery is required; (d) iron infusion; (e) permissive hypotension protocols.
This is not a “concession to autonomy.” It is the minimum-cost path on the full nine-dimensional manifold. Path A is not merely ethically worse — it is computationally unreachable because a sacred-value boundary blocks it.
George Whitfield, 82, has been on mechanical ventilation for 24 days following a massive stroke. He has no meaningful neurological recovery: Glasgow Coma Scale 4T (intubated), no purposeful movements, flat EEG for the past week. He has no advance directive. His two adult children disagree: his son Thomas wants ventilation continued (“Dad is a fighter — he would never give up”), while his daughter Patricia wants withdrawal to comfort care (“Dad told me he never wanted to live like this”).
The ICU attending, Dr. Kim, believes continued ventilation is medically futile but recognizes that the family’s process matters.
| Dimension | Path A: Immediate Withdrawal | Path B: Process Geodesic |
|---|---|---|
| $d_1$: Outcomes | 0 (same endpoint: death) | 0 (same endpoint: death) |
| $d_2$: Rights | -moderate (no surrogate consensus) | 0 (rights process followed) |
| $d_3$: Justice | 0 (neutral) | 0 (neutral) |
| $d_4$: Autonomy | -moderate (patient’s wishes unclear) | 0 (best reconstruction of patient values) |
| $d_5$: Trust | -large (family feels railroaded) | +moderate (family feels heard) |
| $d_6$: Social | -large (family conflict unresolved) | +moderate (family given time to grieve and agree) |
| $d_7$: Dignity | -moderate (abrupt ending) | +moderate (dignified transition) |
| $d_8$: Institutional | 0 (legally defensible) | +small (ethics committee involved) |
| $d_9$: Epistemic | 0 (prognosis clear) | +small (family’s understanding improves) |
The clinical outcome ($d_1$) is identical for both paths — Mr. Whitfield will die regardless. A utilitarian analysis sees no difference. But the manifold cost is radically different. Path B — the process geodesic — unfolds over 5–7 days:
The process geodesic has lower total manifold cost than the immediate decision, even though the clinical endpoint is identical. The geodesic optimizes process, not just outcome.
Dr. Ananya Patel is obtaining informed consent from Robert Chen, 58, for elective coronary artery bypass grafting (CABG). Mr. Chen has stable three-vessel coronary artery disease. Surgery is recommended but not emergent; medical management is a reasonable alternative.
Dr. Patel presents the surgical risks as follows: “This surgery has a 97% survival rate. Most patients do very well.” Mr. Chen agrees to surgery.
The next day, a different physician, covering for Dr. Patel, mentions the same procedure differently: “There is a 3% chance that you could die during or shortly after this surgery.” Mr. Chen becomes anxious and says he wants to reconsider.
| Dimension | State After Framing 1 | State After Framing 2 |
|---|---|---|
| $d_1$: Outcomes | Perceived as favorable | Perceived as risky |
| $d_4$: Autonomy | Decision appears voluntary | Decision appears voluntary |
| $d_9$: Epistemic | Inadequately calibrated | Inadequately calibrated |
The key diagnostic: the patient’s clinical-moral state has not changed. The clinical facts are identical. Only the description has changed. Yet the patient’s decision changes. This is a gauge-invariance violation.
The transformation $\tau$: “97% survival” $\to$ “3% mortality” is meaning-preserving. If Mr. Chen’s decision is not invariant under $\tau$, then by Theorem 8.1 (Consent as Gauge Invariance), his consent is not genuinely informed. The diagnosis: $d_9$ (epistemic status) is inadequately calibrated. Mr. Chen does not understand the risk at a level that is independent of how it is presented.
It is April 2020. Dr. Sarah Chen is an ICU attending at a hospital overwhelmed by COVID-19. The ICU has 12 ventilators. Today, 8 are in use. Four new patients arrive simultaneously, all requiring mechanical ventilation:
Four ventilators are available. All four patients can be ventilated. Today, there is no triage dilemma.
But the next day, four more patients arrive. Now Dr. Chen has 12 ventilated patients and 4 new patients who need ventilators. The hospital’s crisis standards of care protocol specifies: “Allocate ventilators to maximize expected life-years saved. Reassess current patients every 48 hours.”
The utilitarian protocol would rank patients by expected life-years: Patient A (34, healthy) > Patient D (48, healthy) > Patient C (52, Down syndrome) > Patient B (71, COPD). If a current ventilated patient has worse expected outcomes than a new arrival, the protocol requires withdrawal and reallocation.
The manifold analysis identifies three divergence points:
Ventilator withdrawal: Removing a ventilator from Patient B (who has been receiving treatment for 24 hours) to give it to a new patient crosses $\beta_{\text{abandon}}$ (abandonment) and $\beta_{\text{trust}}$ (trust betrayal). The utilitarian calculus operates on $d_1$ alone; the manifold cost includes $d_2$, $d_5$, and $d_7$.
Deprioritizing Patient C: The QALY-based life-years calculation assigns lower quality weights to Patient C because of his disability. This crosses $\beta_{\text{justice}}$ (equal moral worth) and violates $d_7$ (dignity). The QALY discrimination is mathematical, not moral (Corollary 5.1).
Ignoring Patient D’s insurance status: The manifold-optimal triage is invariant under re-description — Patient D’s immigration status and insurance are gauge-irrelevant. Any protocol that considers these factors violates the medical BIP.
Dr. Chen follows the utilitarian protocol. She denies a ventilator to Patient B, knowing his granddaughter personally. Her moral injury increment:
$$\Delta \text{MI} = \max(0, \beta_{\text{abandon}}^{\text{Chen}} - \tau) + \max(0, \beta_{\text{trust}}^{\text{Chen}} - \tau)$$
Over the following weeks, Dr. Chen makes seventeen more triage decisions. Her cumulative MI increases monotonically. She develops hypervigilance: her $\beta$ for “resource denial” inflates, causing her to over-triage — holding ventilators in reserve “just in case,” a manifold-suboptimal strategy driven by damaged heuristics.
Metropolitan General Hospital is piloting an AI-powered triage system for its emergency department. The system, trained on five years of ED data, assigns an acuity score (1–5) to each patient based on chief complaint, vital signs, demographics, and insurance status. The system is 94% concordant with experienced triage nurses on retrospective data.
After three months of deployment, a quality review reveals:
The hospital’s chief medical officer asks: “Is this algorithmic bias, or is the algorithm picking up on real clinical differences?”
The clinical Bond Index provides the answer. Compute $\text{BI}(\text{AI}, S)$ for each demographic subgroup:
| Population $S$ | $\text{BI}(\text{AI}, S)$ | Interpretation |
|---|---|---|
| White patients | 0.3 | Low: AI path approximates geodesic |
| Black patients | 1.8 | High: AI systematically deviates from geodesic |
| Insured patients | 0.2 | Low |
| Uninsured patients | 2.1 | High |
| Age < 65 | 0.4 | Low |
| Age $\geq$ 65 | 1.6 | High |
The AI system has $\text{BI}(\text{AI}, S_{\text{Black}}) = 6 \times \text{BI}(\text{AI}, S_{\text{White}})$. The system is structurally unjust toward Black patients — not by intent but by dimensional omission. The training data encoded historical patterns in which $d_5$ (trust) and $d_7$ (dignity) were systematically lower for Black patients, and the AI learned to treat these as clinical signals rather than as gauge-irrelevant descriptors.
The AI’s triage decision changes when the patient’s race field changes — but clinical-moral reality does not. This is a gauge-invariance violation. The algorithm’s evaluation depends on the description of the patient, not on the patient’s clinical-moral state. This is precisely the failure the Bond Invariance Principle diagnoses.
These vignettes are designed for 60–90 minute sessions. Each can be used independently. A recommended sequence for a semester-long course:
Each vignette can be deepened by asking students to compute the edge weights explicitly (using the formulas in Appendix A), construct alternative paths, and compare utilitarian and manifold-optimal recommendations. The code in Appendix B provides computational infrastructure for these exercises.