The Structure of Exploitation on Heterogeneous-Value Loops: Gauge Symmetry, Holonomy, and the Loop Deficit

A charge of exploitation is made in some description of an economy, and the party charged answers in another. Where the two descriptions differ only in bookkeeping, a charge whose truth changes between them can be defeated without any transfer being altered. This paper takes that observation as a principle and follows it. Admissible charges are the ones constant under the symmetry group of re-descriptions, so the group fixes which charges can be made at all. For value of a single kind the group is abelian and leaves one invariant on each closed circuit, the loop deficit, which registers whether circulation returns. For value of several kinds the group is larger, and the invariant becomes the conjugacy class of a matrix holonomy. Two mechanisms of extraction appear there with no counterpart in the single-kind theory. The first is spectral: a circuit whose holonomy has determinant one closes on every aggregate measure while one kind of value compounds without limit and another is extinguished, so aggregate closure criteria are unable to register kind-wise divergence. The second is sequential: two orders of the same exchanges, with identical rates and identical aggregates, leave a contributor in different positions, so an arrangement can redistribute value by sequencing alone. Conditions of exploitation are then restated on the enlarged group, with kind capture and spectral drift added to the absorption, schema power and blocked return of the single-kind case. The contribution is threefold. The first is a criterion of admissibility for exploitation claims, namely invariance under re-description. The second identifies the spectrum, rather than the determinant, as the object that carries kind-wise information around a circuit. The third is the treatment of the order of exchange as an independent locus of extraction, detectable by comparing two sequences of the same exchanges without observation of a whole circuit.