Axiom Dynamics 4: Candidate Competition and Independence Audit—The Non-Additivity of Gain and the Legitimacy of Batches

The distortion of declared ledgers in candidate competition arises from two structural sources: overlaps among the gain sets of different candidates, and forwardcoupling of candidate firing or output through baseline rules. On a finite Horn deductive system, batch gain is defined as the cardinality of the difference betweenthe post-implementation closure and the baseline theorem set. If the candidate poolsatisfies no forward coupling relative to the baseline—candidate premises do not intersect the baseline-reachable outputs of other candidates, and baseline rules do notjointly consume the conclusions of candidates—then the gain function is a coveragefunction and hence monotone submodular; the declared ledger is not lower thanthe audited ledger, and the overestimate is nonnegative and equal to the overlapduplicate count. When this condition is violated, the coverage representation failsand submodularity can be destroyed: single-source synergy revives a candidate inbatches containing the premise provider and raises its marginal contribution, whilejoint synergy makes baseline rules jointly consume the outputs of multiple candidates and yields extra atoms. Attribution uses Shapley shares; it is the uniqueallocation scheme satisfying the four axioms of efficiency, symmetry, dummy, andadditivity, but it does not retroactively certify individual legitimacy. Selection undera budget constraint maximizes the audited ledger; in the submodular case greedyenjoys a multiplicative approximation guarantee, while in the non-submodular casethis guarantee fails. Readings for the toy pool give 15 submodularity violations andan attribution result of shares (2/3,2/3,2/3).