Three people have nine chores to share. They may disagree completely about which jobs are unpleasant. Can the chores always be divided with a strong fairness guarantee?

This Evidence Press candidate says yes, provided each person’s costs add across chores. The guarantee is called envy-freeness up to any chore. Remove any one chore from your own bundle, even a chore you consider costless. What remains must be no more costly, by your own standards, than either person’s entire bundle. It does not mean everyone receives the same number of chores, or that their total burdens are equal.

The proof first narrows any hypothetical counterexample to twenty-one real variables. There are nineteen thousand six hundred and eighty-three complete assignments. The formula handles assignments where everyone has at least one chore. A written argument covers the other assignments. Separate checking software checks each step of the contradiction, using exact fractions and elementary logical deductions. It does not simply accept a solver’s verdict.

The package also includes an allocation finder. Given rational costs, it returns a division and the eighteen inequalities that let a reader check it directly. The result extends an earlier eight-chore existence theorem. It does not settle arbitrary numbers of chores, strategic reporting or real-world measurement of inconvenience.

This remains an unrefereed candidate. Internal replay is not independent external reproduction, and the complete proof has not been formalised in a proof assistant. The paper, source code and proof objects accompany Nine chores for three agents, released on the eighth of October twenty twenty-six. This briefing uses an AI-generated voice and is not additional mathematical evidence.