A Kernel-Clean Lean Mechanization of Classical Lottery in Action and the Wakker--Debreu--Koopmans Representation Layer

Jingyuan Li, Ilia Tsetlin and Fan Wang

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Abstract: We present a Lean 4/Mathlib formalization of the additive representation theory behind Classical Lottery in Action and the Wakker-Debreu-Koopmans (WDK) layer it relies on. Our central result is a machine-checked proof that the cross-pair Thomsen / double-cancellation (hexagon) condition is irreducible from the ordinal axioms of additive conjoint measurement (weak order, restricted solvability, Archimedean condition, and tradeoff consistency). We exhibit an explicit verified counter-model (additiveRealBoolPref) satisfying all ordinal axioms yet failing the cross-pair condition, with every strict standard sequence being an arithmetic progression and hence non-dense. Around this boundary we mechanize the full derivable construction: continuous Debreu/Eilenberg utility from separability, standard-sequence grids, bisection methods from connectedness, and global additive gluing. All public theorems are sorry-free conditional wrappers over this single irreducible structural input. The development is kernel-clean, depending only on standard Lean foundations (propext, Classical.choice, Quot.sound). The companion file ClassicalLotteryInAction.lean formalizes local classical-lottery constructions, average-utility results, matching-frequency lemmas, and ambiguity-attitude statements used by the Management Science paper. This draws a precise, machine-certified line between what additive conjoint measurement can prove and what it must assume.

Date: 2026-06
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