The Bankruptcy of Trolley Calculus: Why the Utilitarian Arithmetic of Persons Is Formally Incoherent | Utilitarismus 404, Part I/IV

Position preprint. Companion to "Who Watches the Watchmen?" & TIK meta-evaluation framework | [wAI ~ (prhps) wErrors] Trolley calculus — any decision procedure, philosophical or algorithmic, that assigns each person a nonnegative real number, ranks outcomes by the sum of those numbers, and mandates the maximising option — presupposes four things: that the scalar exists, that the sum is coherent, that optimising the sum is stable, and that the calculus can vouch for its own axioms. We show that each presupposition fails on formal grounds. Existence: under Kant's price/dignity dichotomy taken literally, the induced preference order is lexicographic and admits no real-valued representation (a Debreu-type non-representability theorem); independently, empirically observed framing cycles make marginal moral value a non-exact 1-form, so no scalar potential exists. Closure: Arrow's and Arrhenius' impossibility theorems, unbounded-utility paradoxes, and transfinite absorption (ℵ + ℵ = ℵ) void additive aggregation across persons. Stability: Goodhart dynamics guarantee that any deployable proxy for the "value of a person" self-destructs under arg max. Foundation: Gödel II, Tarski, and Moore's open question bar the calculus from certifying the principle of utility from within — a limit Sidgwick already confessed. A Fungibility Lemma formalises the dehumanisation objection: additive calculi assign persons a price by construction, and price is precisely what Kant defines dignity against. The conjunction is not a set of edge-case anomalies but a structural bankruptcy. The constructive residue is developed in the companion TIK framework: maximality over partial orders, a "do not do wrong" floor, and explicit external anchoring in an invariant kernel Φ. Design principle: one figure and one table carry the argument. Every formal result cited is established mathematics; every ethical bridge premise is stated as a numbered, criticisable axiom. S.V.E. Meta-License v4.0 (share-alike)[CC BY-NC-SA 4.0 + Symmetric Addendum]