Chapter 10: Pandemic Triage as Manifold Allocation

Triage as Multi-Agent Pathfinding

A single clinical decision is pathfinding on the clinical decision complex \mathcal{C}: from a patient’s current state v_0 to a goal region G, minimizing total clinical friction \text{CF}(\gamma) along the path. The clinician computes f(n) = g(n) + h(n), and the minimum-cost path \gamma^* is the clinical geodesic.

Triage introduces two complications that transform the problem:

1. Multiple simultaneous patients: The clinician must compute geodesics for n patients at once, not one at a time.
2. Resource constraints: The total resources available (ventilators, ICU beds, surgical hours, medications) are insufficient for all patients to follow their individually optimal paths.

Definition 10.1 (Triage Problem). A triage problem is a tuple (N, \mathcal{C}, R, G) where:

• N = \{p_1, \ldots, p_n\} is a set of patients.
• \mathcal{C} = \{\mathcal{C}_1, \ldots, \mathcal{C}_n\} is the set of clinical decision complexes (one per patient, each carrying a nine-dimensional attribute vector).
• R is a resource constraint: a bound on the total resources available across all patients.
• G = \{G_1, \ldots, G_n\} is the set of goal regions (acceptable outcomes for each patient).

The triage decision is a feasible allocation that minimizes total clinical-moral friction:

(\gamma_1^*, \ldots, \gamma_n^*) = \arg\min_{\gamma_1, \ldots, \gamma_n} \sum_{i=1}^n \text{CF}(\gamma_i) \quad \text{subject to } \sum_{i=1}^n r(\gamma_i) \leq R

where \text{CF}(\gamma_i) is the total clinical friction for patient i’s path and r(\gamma_i) is the resource cost of that path.

What Changes Under Resource Constraints

Without resource constraints, each patient follows their individual geodesic \gamma_i^*. The total cost is \sum_i \text{CF}(\gamma_i^*) — the sum of individually optimal paths. Resource constraints force deviations from individual geodesics. Some patients receive suboptimal paths (delayed treatment, no ventilator, comfort care instead of ICU admission). The triage question is: which patients deviate, and how far?

[Figure 10.1: Schematic showing n = 30 patients on the clinical manifold, with individually optimal geodesics (dashed) and resource-constrained feasible paths (solid), highlighting the deviation between individual geodesics and constrained allocations]

Tractability

General multi-agent pathfinding is NP-hard. But clinical triage operates under constraints that make the problem tractable in practice:

• Bounded problem size: An ICU has 10–50 beds, not thousands. A pandemic surge involves hundreds to thousands of patients per hospital, but triage decisions are made locally within units.
• Few resource types: Ventilators, ICU beds, nursing hours, specific medications. Not a continuous resource space.
• Small per-patient action spaces: For each patient, the triage decision typically has a small number of options (admit to ICU / admit to floor / discharge / comfort care), not a combinatorial explosion.

The framework does not propose replacing existing triage heuristics (ESI, START, JumpSTART) with exact solvers. It proposes that the objective function should be total manifold cost \sum_i \text{CF}(\gamma_i), not total scalar outcome \sum_i d_1(\gamma_i). The existing algorithms work with the manifold objective — the change is in what is being optimized, not how the optimization is performed. The primary value is offline policy design: evaluating crisis standards of care against the manifold objective before the crisis arrives.

The Utilitarian Objective

The dominant approach to crisis standards of care is utilitarian: maximize the number of lives saved, or maximize total life-years saved. Emanuel et al. (2020), in their influential New England Journal of Medicine framework for COVID-19 resource allocation, proposed four values (maximizing benefits, treating people equally, promoting instrumental value, giving priority to the worst off) but operationalized the framework primarily through d_1 — clinical prognosis, expected survival, life-years gained.

The utilitarian triage objective is:

(\gamma_1^U, \ldots, \gamma_n^U) = \arg\max_{\gamma_1, \ldots, \gamma_n} \sum_{i=1}^n d_1(\gamma_i) \quad \text{subject to } \sum_{i=1}^n r(\gamma_i) \leq R

This is a well-defined optimization problem with a scalar objective. It is computationally tractable. It produces clear, defensible, protocol-driven decisions. And it is, in precisely the cases that matter most, manifold-suboptimal.

The Manifold Objective

The manifold-optimal triage objective minimizes total clinical friction on all nine dimensions:

(\gamma_1^*, \ldots, \gamma_n^*) = \arg\min_{\gamma_1, \ldots, \gamma_n} \sum_{i=1}^n \text{CF}(\gamma_i) \quad \text{subject to } \sum_{i=1}^n r(\gamma_i) \leq R

where:

\text{CF}(\gamma_i) = \sum_{(v_j, v_{j+1}) \in \gamma_i} \left[ \Delta \mathbf{a}^T \Sigma^{-1} \Delta \mathbf{a} + \sum_k \beta_k \cdot \mathbf{1}[\text{boundary } k \text{ crossed}] \right]

The manifold objective includes the utilitarian objective as a special case — when only d_1 is activated and no boundaries are crossed, the two objectives coincide. But the manifold objective also accounts for boundary-crossing costs, cross-dimensional interactions, and the moral costs imposed on both patients and clinicians by the triage process itself.

The Divergence Theorem

Theorem 10.1 (Divergence of Utilitarian and Manifold-Optimal Triage). Let \gamma_U^* be the utilitarian triage allocation (maximize total d_1) and let \gamma_{\mathcal{C}}^* be the manifold-optimal allocation (minimize total \text{CF}). Then:

1. \gamma_U^* = \gamma_{\mathcal{C}}^* when the triage decisions activate only d_1 and no moral boundaries are crossed.

2. \gamma_U^* \neq \gamma_{\mathcal{C}}^* when triage decisions cross moral boundaries or activate non-outcome dimensions. Specifically:

(a) Ventilator withdrawal from a current patient to reallocate to a new patient with better prognosis is utilitarian-optimal but manifold-suboptimal, because withdrawal crosses the abandonment boundary (\beta_{\text{abandon}}) and trust boundary (\beta_{\text{trust}}).

(b) Age-based deprioritization (younger patients first) is utilitarian-optimal (maximizes life-years) but manifold-suboptimal when it violates d_3 (justice: equal moral worth regardless of age).

(c) Lottery allocation (random assignment when prognoses are similar) is utilitarian-equivalent (no outcome difference) but manifold-superior: it preserves d_3 (fairness), d_7 (dignity), and d_8 (legitimacy) while avoiding boundary crossings.

Proof. (1) follows from the Mahalanobis distance reducing to the d_1 component when all other dimensions have zero activation and no boundary terms contribute. Under these conditions, minimizing total \text{CF} is identical to maximizing total d_1 (since \text{CF} is a monotonic function of deviation from optimal d_1).

(2a) Withdrawing a ventilator from a current patient crosses the abandonment boundary (\beta_{\text{abandon}}) and the trust boundary (\beta_{\text{trust}}). The utilitarian calculation considers only the d_1 gain of reallocation (the new patient has better prognosis). The manifold calculation adds \beta_{\text{abandon}} + \beta_{\text{trust}} to the cost of withdrawal. When these boundary penalties exceed the d_1 differential between the two patients, the manifold-optimal allocation keeps the ventilator with the current patient — even though the utilitarian allocation reassigns it.

(2b) Age-based deprioritization weights younger patients’ expected life-years more heavily. This is equivalent to applying a discount factor to d_1 based on age, which violates d_3 (equal moral worth). The manifold cost of the d_3 violation is:

\beta_{\text{justice}} \cdot \mathbf{1}[\text{age-based deprioritization}]

When \beta_{\text{justice}} exceeds the d_1 differential between the age groups, the manifold-optimal allocation is age-neutral.

(2c) When two patients have similar prognoses (|d_1(p_i) - d_1(p_j)| < \epsilon), the utilitarian objective is indifferent between them — any allocation is equally good on d_1. The manifold objective is not indifferent: a lottery allocation preserves d_3 (procedural fairness), d_7 (neither patient is devalued by the criterion), and d_8 (transparent, rule-based process), while a criterion-based allocation (by age, social value, or other discriminator) imposes costs on these dimensions. \square

Case A: Ventilator Withdrawal and Reallocation

The scenario that defined the moral crisis of the COVID-19 pandemic. A patient has been on a ventilator for seven days. Prognosis is poor but not hopeless — estimated 20% chance of survival. A new patient arrives in the emergency department with severe respiratory failure and an estimated 70% chance of survival if ventilated. There is one ventilator. The utilitarian calculation is clear: reallocate.

Utilitarian analysis:

d_1(\text{new patient, ventilated}) - d_1(\text{current patient, continued}) = 0.70 - 0.20 = +0.50

The utilitarian gain is 0.50 expected life-years per life-year at stake. Reallocate.

Manifold analysis:

The ventilator withdrawal crosses two boundaries:

• \beta_{\text{abandon}}: The clinician-patient relationship is a fiduciary bond. Withdrawing treatment from a current patient to benefit a stranger violates d_5 (trust) not just for this patient but for all patients who learn that ventilators can be reassigned. The boundary penalty reflects the systemic trust damage.

• \beta_{\text{rights}}: The current patient has a de facto claim on the ventilator (having already received it). Withdrawal may violate d_2 (rights and obligations) depending on the legal and ethical framework.

Additionally, the withdrawal imposes a cost on d_7 (dignity) — the current patient is effectively told that their life is worth less than the new arrival’s — and on d_8 (institutional legitimacy) — the hospital’s protocols did not warn the patient that ventilator support could be withdrawn.

\text{CF}(\text{reallocate}) = \Delta d_1 \text{ gain} + \beta_{\text{abandon}} + \beta_{\text{trust}} + \Delta d_7 + \Delta d_8

For \beta_{\text{abandon}} + \beta_{\text{trust}} > \Delta d_1, the manifold-optimal decision is to not reallocate — to continue ventilation for the current patient and find an alternative for the new arrival (transfer, improvised ventilation, prone positioning without intubation).

The key insight: The utilitarian analysis counts only the d_1 differential. The manifold analysis counts the full cost of the allocation process, including the boundaries crossed and the trust destroyed. The two analyses disagree precisely when the boundary costs are large — which is exactly the scenario that causes moral injury to clinicians.

Case B: Age-Based Deprioritization

A 32-year-old teacher and a 74-year-old retired engineer arrive simultaneously with comparable severity of illness. Both need ICU admission. One bed is available. The utilitarian protocol prioritizes the 32-year-old: more expected life-years saved.

Utilitarian analysis:

d_1(\text{teacher}) = 0.65 \times 45 \text{ life-years} = 29.25 \text{ QALYs} d_1(\text{engineer}) = 0.60 \times 12 \text{ life-years} = 7.20 \text{ QALYs}

The utilitarian gain of prioritizing the teacher is 29.25 - 7.20 = 22.05 QALYs. Prioritize the teacher.

Manifold analysis:

The age-based deprioritization activates d_3 (justice). The retired engineer has equal moral worth — his life is not less valuable because it is shorter. The d_3 cost depends on the community’s boundary penalty for age discrimination:

\beta_{\text{justice}} \cdot \mathbf{1}[\text{age used as triage criterion}]

The QALY Irrecoverability Theorem (Theorem 2) predicts exactly this failure: the QALY calculation discards the d_3 information. A QALY of 7.20 and a QALY of 29.25 differ only on d_1 — the eight dimensions discarded by the scalar include the justice dimension that makes the comparison problematic.

When prognoses are comparable (the survival probabilities differ by only 5 percentage points — 65% vs. 60%), the d_1 differential is small and the d_3 cost of age-based discrimination may exceed it:

\beta_{\text{justice}} > \frac{(0.65 - 0.60)^2}{\sigma_1^2}

In this case, the manifold-optimal allocation uses a lottery rather than an age-based criterion — preserving d_3 at negligible d_1 cost.

Case C: Lottery Allocation Among Similar Prognoses

Six patients with comparable severity and comparable prognoses need ICU admission. Three beds are available. The utilitarian algorithm can rank them by expected life-years, but the differences are within the uncertainty range — the ranking is dominated by measurement noise, not by genuine prognostic differences.

The manifold-superior allocation is a lottery.

A lottery preserves: - d_3 (fairness): every patient has an equal chance, regardless of age, race, social status, or insurance - d_7 (dignity): no patient is devalued by the allocation criterion (“you are less worth saving because…”) - d_8 (institutional legitimacy): the allocation process is transparent, defensible, and free from bias - d_5 (trust): the system treats everyone equally, reinforcing public trust in medical institutions

The d_1 cost of a lottery relative to the utilitarian optimum is, by hypothesis, negligible — the patients have comparable prognoses. The manifold benefit is substantial.

Proposition 10.1 (Manifold Superiority of Lottery Allocation). When the d_1 differential among patients is below threshold \epsilon (comparable prognoses), lottery allocation is manifold-superior to criterion-based allocation for any criterion that activates d_3, d_7, or d_8.

This result formalizes the intuition behind the widespread endorsement of lottery allocation in pandemic ethics guidelines (White et al., 2009; Emanuel et al., 2020). The utilitarian framework cannot justify lotteries — they are d_1-equivalent at best. The manifold framework can: lotteries are manifold-superior because they avoid boundary crossings on non-outcome dimensions.

COVID-19 and the Forced Contraction

The COVID-19 pandemic was the largest forced contraction in the history of clinical ethics. Across the world, clinicians were required to contract nine-dimensional clinical-moral tensors to binary decisions (ventilate / do not ventilate) under time pressure, resource scarcity, and extreme uncertainty.

The pandemic tested the framework’s predictions in real time:

Prediction 10.1: Clinician moral distress should be highest when triage decisions activate multiple manifold dimensions simultaneously.

The empirical evidence is consistent. The most morally distressing triage decisions were not those with clear d_1 differentials (patient A will clearly survive, patient B will clearly not) but those involving multiple dimensions: withdrawing ventilators from current patients (d_1 + d_5 + d_7), denying treatment to elderly patients (d_1 + d_3 + d_7), and making decisions under extreme prognostic uncertainty (d_1 + d_9). Single-dimension triage (patient A is more likely to survive than patient B, all else equal) caused less distress — exactly as the framework predicts.

Prediction 10.2: Moral injury should correlate with the number of boundary crossings, not just the severity of outcomes.

Greenberg et al. (2020) documented that moral injury in COVID-19 healthcare workers was not simply a function of patient mortality (a d_1 measure). Clinicians in high-mortality units with clear triage protocols reported less moral distress than clinicians in moderate-mortality units with ambiguous protocols that required repeated boundary crossings. The quantity of morally costly decisions — each one a boundary crossing on the manifold — mattered more than the clinical severity of any individual decision.

Prediction 10.3: Clinicians who disagreed with the triage protocol should show higher moral injury than clinicians who agreed with it, independent of clinical outcomes.

This prediction follows directly from the moral injury definition (Definition 5 in the GCE paper): moral injury occurs when the institutionally mandated geodesic requires crossing boundaries that the clinician’s personal manifold assigns high penalties. If the clinician’s \beta_k values are aligned with the protocol’s, no boundary is crossed. If they diverge, every triage decision is a forced boundary crossing.

The pandemic literature is consistent with this prediction. Rushton (2018) documented that moral distress was highest among clinicians whose personal ethical commitments conflicted with institutional protocols — not among clinicians who were simply exhausted.

What the Pandemic Revealed About Triage Design

The pandemic exposed a fundamental asymmetry in existing triage protocols: they were designed to optimize d_1 (clinical outcomes) under resource constraints, but the moral costs of implementation fell on dimensions (d_5: trust, d_7: dignity, d_3: justice) that the protocols did not model.

[Figure 10.2: Dimensional activation map of COVID-19 triage decisions, showing which of the nine clinical-moral dimensions were activated by different triage scenarios, with predicted and observed moral injury correlation]

The framework suggests three design principles for manifold-aware triage protocols:

1. Minimize unnecessary boundary crossings. Many pandemic triage protocols crossed boundaries that could have been avoided. For example, some protocols required explicit age-based deprioritization when a lottery among similar-prognosis patients would have achieved comparable d_1 outcomes without the d_3 violation.

2. Make boundary crossings explicit and anticipated. Clinicians reported less moral distress when boundary crossings were anticipated (part of a pre-established protocol) than when they were ad hoc (made up on the spot). Pre-established protocols reduce the clinician’s sense of personal responsibility for the boundary crossing — the institution crossed the boundary, not the individual.

3. Separate the decision from the implementation. Several institutions found that moral injury was reduced when triage decisions were made by a triage officer or committee who did not have a prior relationship with the patient, and the implementation was communicated by the treating clinician with appropriate emotional support. This separates the d_5 cost (trust damage from the triage decision) from the d_5 preservation (the treating clinician remains the patient’s advocate).

Scenario 1: The Surge — Thirty Patients, Twelve Ventilators

Sarah’s ICU has twelve ventilators and thirty patients who need them. The hospital has activated crisis standards of care. The triage protocol is utilitarian: maximize expected life-years saved.

The scalar computation: Rank all thirty patients by expected life-years gained from ventilation. Assign ventilators to the top twelve. Deny ventilators to the bottom eighteen.

The manifold computation: The scalar ranking is a starting point, not a final answer. The manifold computation additionally considers:

For each patient denied a ventilator: - \beta_{\text{abandon}}: Is the clinician-patient relationship severed by denial? - d_7 cost: Is the patient told their life is worth less than others’? - d_6 cost: What is the impact on the patient’s family? - d_9 cost: Was the prognostic estimate reliable enough to justify the ranking?

For the system as a whole: - d_3 cost: Does the ranking systematically disadvantage any demographic group? - d_8 cost: Was the protocol established through a legitimate process? - d_5 cost: Will public trust in the healthcare system survive this allocation?

The manifold-optimal triage may differ from the scalar ranking in several respects:

• Patients with very similar prognoses (rank 11, 12, 13, 14) should be allocated by lottery rather than by marginal d_1 differences that are within measurement error.
• Patients who have been receiving ventilation for several days should not be withdrawn unless the d_1 differential exceeds the withdrawal boundary costs.
• Patients from communities with historically low healthcare trust (d_5) should not be disproportionately denied — the bond index (Chapter 12) should be monitored.

Scenario 2: The Slow Surge — Accumulated Decisions Over Weeks

The acute surge is dramatic but conceptually simple — thirty patients, twelve ventilators, one decision. The slow surge is harder. Over three weeks, a steady stream of patients arrives, each requiring an allocation decision. The cumulative effect is more damaging than any single decision because:

1. Each decision is individually small but cumulatively large. Denying one patient a ventilator is a single boundary crossing. Denying thirty patients over three weeks is thirty boundary crossings — and the Moral Injury Accumulation Theorem (Chapter 13) says these accumulate monotonically.

2. The resource constraint shifts over time. As current patients recover or die, ventilators become available. A patient denied a ventilator on Day 3 might have received one on Day 5. The clinician who denied the ventilator on Day 3 carries the moral cost of a decision that subsequent events rendered unnecessary.

3. The heuristic degrades under sustained stress. After weeks of triage decisions, the clinician’s h(n) begins to change — either hypervigilance (holding ventilators in reserve “just in case,” which is manifold-suboptimal) or moral numbing (boundary penalties decrease, making future boundary crossings easier but degrading the quality of future decisions).

The manifold framework predicts: Slow-surge triage is more morally injurious than acute-surge triage, even if the total number of deaths is the same, because the slow surge produces more cumulative boundary crossings and more heuristic degradation.

Scenario 3: Pediatric Triage

The hardest triage scenario involves children. The d_1 calculation is different (children have more expected life-years), but the manifold cost of denying treatment to a child activates every non-outcome dimension:

• d_3 (justice): children cannot be responsible for their illness
• d_5 (trust): denying treatment to a child destroys parental trust in institutions
• d_6 (social impact): the death of a child devastates families and communities
• d_7 (dignity): children cannot understand or consent to the allocation
• \beta_{\text{sacred}}: many clinicians and many societies treat the death of a child as a near-sacred boundary

Proposition 10.3 (Pediatric Triage Asymmetry). The manifold cost of denying treatment to a child exceeds the manifold cost of denying treatment to an adult with identical d_1 prognosis, because the non-outcome dimensions (d_3, d_5, d_6, d_7) assign higher costs to pediatric cases. This is not age-based discrimination — it is the correct computation on a manifold that includes relational, justice, and dignity dimensions.

The asymmetry is precisely the opposite of the QALY-based asymmetry: QALYs favor children because they have more life-years to gain (d_1); the manifold favors children because non-outcome dimensions are more heavily activated. The two asymmetries point in the same direction, but for different reasons — and the distinction matters when the populations are adults of different ages (where the QALY asymmetry favors younger adults but the manifold calculation depends on the specific boundary costs, not on a generic age discount).

The Current Standard: Utilitarian Protocols

Most existing crisis standards of care follow the utilitarian framework:

1. Assess clinical prognosis (d_1) using validated scoring systems (SOFA, APACHE)
2. Rank patients by expected benefit from the scarce resource
3. Allocate to the highest-ranked patients
4. Re-evaluate periodically

These protocols are clear, defensible, and computationally simple. They are also systematically suboptimal on the manifold, because they ignore eight of nine dimensions.

A Manifold-Aware Protocol

The geometric framework suggests the following modifications to utilitarian triage protocols:

Step 1: Compute d_1 rankings as before. The utilitarian component of triage — clinical prognosis assessment — is valid and necessary. The manifold framework does not replace it; it supplements it.

Step 2: Identify boundary crossings. For each allocation in the utilitarian ranking, determine which clinical-moral boundaries are crossed: - Does the allocation require withdrawing treatment from a current patient? (\beta_{\text{abandon}}) - Does the allocation systematically disadvantage a demographic group? (\beta_{\text{justice}}) - Does the allocation violate specific patients’ advance directives or expressed preferences? (\beta_{\text{consent}})

Step 3: Compute manifold cost. For allocations that cross boundaries, add the boundary penalty to the d_1 cost. If the total manifold cost of the utilitarian allocation exceeds the total manifold cost of an alternative allocation, use the alternative.

Step 4: Apply lottery tiebreaking. When patients have comparable d_1 prognoses (within measurement uncertainty), use lottery allocation rather than marginal d_1 differences.

Step 5: Monitor the Bond Index. Compute the clinical Bond Index (Chapter 12) for the triage protocol stratified by demographics. If the Bond Index reveals systematic disparities, modify the protocol.

Step 6: Compute the Moral Injury Index. Before implementation, estimate the moral injury that the protocol will impose on clinicians (Chapter 13). If the Moral Injury Index exceeds a threshold, redesign the protocol to reduce unnecessary boundary crossings.

The Cost of Manifold-Aware Triage

The manifold-aware protocol is more complex than the utilitarian protocol. It requires more information (boundary penalties, demographic data, clinician surveys), more computation (manifold cost estimation, Bond Index monitoring), and more institutional infrastructure (ethics committees, triage oversight, lottery mechanisms).

Is the added complexity worth it?

The framework’s answer: the manifold cost is paid whether you compute it or not. A utilitarian protocol that ignores d_3, d_5, d_7, and d_8 does not eliminate the costs on those dimensions — it merely fails to account for them. The costs manifest as moral injury to clinicians, loss of public trust, demographic disparities in outcomes, and institutional legitimacy crises. These costs are real, even if the protocol does not measure them.

The manifold-aware protocol does not eliminate these costs. It makes them visible, computable, and — to the extent possible — minimizable.

From Larrey to the Present: A Dimensional History

Every triage system in the history of medicine is a specific contraction of the clinical tensor. Tracing the history of triage through the geometric lens reveals a gradual expansion of the dimensions that triage systems acknowledge — and a persistent failure to acknowledge all nine.

Baron Dominique Jean Larrey (1792): Contraction to d_1.

Napoleon’s chief surgeon introduced the first systematic battlefield triage: sort casualties by survivability, treat the most salvageable first, regardless of rank. Larrey’s system contracted the clinical tensor to a single dimension — clinical outcomes (d_1). It was revolutionary because it ignored social status (d_6), military rank (d_8), and political importance — dimensions that had previously dominated allocation. Larrey’s innovation was not just medical but geometric: he identified d_1 as the relevant dimension and discarded the rest.

Emergency Severity Index, ESI (2011): Contraction to d_1 + d_8.

The modern ESI system, used in most US emergency departments, assigns patients to five triage levels based on clinical acuity and anticipated resource needs. The system contracts the clinical tensor to two dimensions: d_1 (how sick is the patient?) and d_8 (how many resources will treatment require?). The system does not account for d_3 (justice: is the allocation equitable across demographics?), d_5 (trust: does the patient trust the emergency department?), or d_7 (dignity: is the triage process experienced as dehumanizing?).

START/JumpSTART (1983/2002): Mass Casualty Contraction.

The Simple Triage and Rapid Treatment (START) system, used for mass casualty events, contracts the clinical tensor to a binary — dead/alive — modified by respiratory rate, pulse, and mental status. JumpSTART adapts the system for pediatric patients. Both systems operate on d_1 alone, with no consideration of non-outcome dimensions. The systems are appropriate for their context (rapid field triage under extreme time pressure) but produce significant moral injury when applied to sustained allocation decisions where non-outcome dimensions are activated.

Pandemic Crisis Standards of Care (2020): Attempted expansion to d_1 + d_3.

COVID-19 triage guidelines (Emanuel et al., 2020; White et al., 2020) attempted to incorporate justice (d_3) alongside clinical outcomes (d_1): lotteries as tiebreakers, life-cycle considerations, essential worker prioritization. This represented an expansion from one-dimensional to two-dimensional triage. The geometric framework predicts that the expansion was insufficient — seven dimensions remained unaccounted for — and the pandemic moral injury literature confirms the prediction.

The Geometric Trend

The history of triage shows a slow expansion of the acknowledged dimensional space: from d_1 alone (Larrey) to d_1 + d_8 (ESI) to d_1 + d_3 (pandemic ethics). The geometric framework proposes the final step: triage on the full nine-dimensional manifold. This does not mean that every triage decision must explicitly compute nine-dimensional costs — in acute settings, rapid heuristic triage remains necessary. It means that the objective function against which triage systems are designed and evaluated should be the full manifold cost, not a scalar projection.

[Figure 10.3: Historical evolution of triage dimensionality, from Larrey (1D) through ESI (2D) and pandemic ethics (2D) to the manifold-optimal framework (9D)]

How the Decision Is Communicated Matters on the Manifold

Triage is not only a decision but a communication. The way a triage decision is communicated to the patient and family affects multiple dimensions of the clinical decision complex — and the manifold cost of communication is often larger than the manifold cost of the decision itself.

Consider two ways of communicating the same triage decision (denial of a ventilator):

Communication A (scalar): “Based on the protocol, your mother does not qualify for a ventilator. We will provide comfort care.”

Communication B (manifold-aware): “Your mother is very sick, and we have been working hard to help her. We have more patients who need ventilators than we have ventilators available. A triage team — not your mother’s doctors — has reviewed all the patients and made the allocation. Your mother will receive comfort care, which means we will focus on keeping her comfortable, managing her pain, and allowing you to be with her. We will not stop caring for her.”

Communication A is efficient but activates d_5 (trust damage: “the protocol doesn’t care about my mother”), d_7 (dignity damage: “my mother doesn’t qualify”), and d_8 (legitimacy damage: “what protocol? who decided?”). Communication B addresses each of these dimensions: it emphasizes the ongoing relationship (d_5), respects the patient’s worth (d_7), explains the process (d_8), and separates the triage decision from the treating team (\beta_{\text{abandon}} reduction).

Proposition 10.4 (Communication as Manifold Cost Modifier). The manifold cost of a triage decision is modulated by the quality of communication. A triage decision communicated with transparency (d_8), empathy (d_5), and dignity-preservation (d_7) has lower total manifold cost than the same decision communicated as a protocol output — even though the allocation is identical.

This proposition has immediate practical implications for triage protocol design: protocols should specify not only the allocation algorithm but the communication script. The communication is not a courtesy — it is a manifold cost intervention that reduces the total cost of the triage system.

Communication and Moral Injury

The communication burden falls on the clinician who delivers the message. A clinician who tells a patient “you don’t qualify for a ventilator” crosses the abandonment boundary in their own experience. A clinician who tells a patient “we will not stop caring for you — the form of care is changing, not the commitment” crosses the same boundary with lower subjective penalty.

Training clinicians in manifold-aware triage communication reduces both the patient’s manifold cost (better trust, dignity, and legitimacy preservation) and the clinician’s moral injury (lower subjective boundary-crossing penalty). The training is a dual intervention — benefiting both parties on the manifold.

Day 1: The First Triage

Twelve ventilators. Fourteen patients needing ventilation. The triage committee — two attendings and an ethicist — evaluates prognoses. The top twelve are ventilated. Two are assigned to comfort care.

The two patients assigned to comfort care: a 78-year-old retired teacher with diabetes and chronic kidney disease (estimated 15% survival with ventilation), and an 81-year-old retired postal worker with COPD (estimated 12% survival).

Sarah knows the 78-year-old. Mrs. Washington — she was Sarah’s patient on the general medicine ward two months ago. Sarah diagnosed her pneumonia, adjusted her diabetes medications, called her daughter every evening. Mrs. Washington trusts Sarah.

The triage committee assigns Mrs. Washington to comfort care. The protocol is utilitarian-optimal. On the scalar, the decision is correct.

Sarah walks to Mrs. Washington’s room. She sits beside her and takes her hand. “We don’t have enough ventilators,” she says. “I’m so sorry.”

Mrs. Washington looks at her. “Is there nothing you can do?”

On the manifold, this moment activates every dimension simultaneously. d_1 (clinical outcome: Mrs. Washington will likely die), d_2 (rights: she has a right to treatment that cannot be provided), d_3 (justice: she was deprioritized because of her age and comorbidities), d_4 (autonomy: she was not consulted in the allocation), d_5 (trust: Sarah promised to take care of her), d_6 (social impact: her daughter will grieve), d_7 (dignity: she is dying because a committee decided her life was worth less than others’), d_8 (institutional legitimacy: the hospital promised care it cannot deliver), d_9 (epistemic: the 15% estimate is uncertain — she might have survived).

Sarah crosses the abandonment boundary. She does not choose to cross it — the institution mandates it. The boundary penalty is added to her cumulative moral injury.

Week 2: The Withdrawal Decision

A patient has been on a ventilator for ten days. No improvement. The committee recommends withdrawal and reallocation to a newly admitted patient with better prognosis.

The utilitarian calculation is clear. The manifold calculation is not.

Sarah’s attending refuses to implement the withdrawal. “I started this patient on the ventilator,” he says. “I can’t be the one to take it away.” The triage committee assigns a different physician to implement the decision. The boundary crossing is the same; the assignment of responsibility changes.

Sarah notices that the attending is not being irrational. His h(n) correctly identifies that withdrawal crosses the abandonment boundary at high cost. The committee, operating on d_1 alone, does not see the cost. The attending, operating on the full manifold, does.

Week 4: The Heuristic Shifts

After four weeks of daily triage decisions, Sarah notices changes in herself. She no longer flinches when the triage pager sounds. She processes the numbers — SOFA score, prognosis estimate, ventilator availability — and makes the recommendation. The process has become routine.

She worries about this. Is she becoming more efficient or more numb?

The framework predicts two possible responses to sustained boundary crossing: hypervigilance (inflated \beta_k — the clinician becomes risk-averse) or moral numbing (deflated \beta_k — the boundary penalty decreases through repeated violation). Sarah is experiencing moral numbing. Her \beta_{\text{abandon}} has decreased. The boundary that once felt impassable now feels routine.

This is h(n) damage. It is not burnout — Sarah sleeps adequately, eats regularly, and has supportive colleagues. It is moral injury: the permanent alteration of her heuristic function by forced boundary crossings she did not choose.

The Toll

By the time the surge subsides, Sarah has participated in forty-seven triage decisions. She has denied ventilators to nineteen patients, twelve of whom died. She has recommended withdrawal of ventilation from three patients, all of whom died within hours. She has watched two colleagues leave medicine permanently.

She is not the clinician she was before the pandemic. Her h(n) has been altered on multiple dimensions. She is faster at triage — and she knows that the speed comes at a cost she cannot yet fully measure.

The manifold exists. Sarah has traversed its most costly regions. The mathematics of that traversal — the accumulation theorem, the heuristic damage modes, the moral injury index — is the subject of Part V.