arXiv:2611.08740v1 Announce Type: new Abstract: We study unital chain-map multiplications on minimal free resolutions of quotient rings. We prove that Katth\"an's example admits no associative multiplication over any field, without assuming graded commutativity or compatibility with an internal grading. The proof gives a relative obstruction in which associativity on a specified finite collection of triples forces failure at another triple. The argument extends to a family that includes nonmonomial ideals. We also give criteria for uniform non-associativity, where a fixed triple fails for every multiplication, by comparing either associators or their primitives under changes of multiplication. These criteria persist from monomial elimination ideals and give explicit obstructions for Avramov's example and the known eight-vertex Bier sphere over every field. As an application, we show that the edge ideal of the eight-cycle admits an injective Taylor comparison on five selected generators, while every selection of that size fails to admit a multiplicative comparison. Four generators admit a multiplicative embedding.
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