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 struc
tural sources: overlaps among the gain sets of different candidates, and forward
coupling of candidate firing or output through baseline rules. On a finite Horn de
ductive system, batch gain is defined as the cardinality of the difference between
the post-implementation closure and the baseline theorem set. If the candidate pool
satisfies no forward coupling relative to the baseline—candidate premises do not in
tersect the baseline-reachable outputs of other candidates, and baseline rules do not
jointly consume the conclusions of candidates—then the gain function is a coverage
function and hence monotone submodular; the declared ledger is not lower than
the audited ledger, and the overestimate is nonnegative and equal to the overlap
duplicate count. When this condition is violated, the coverage representation fails
and submodularity can be destroyed: single-source synergy revives a candidate in
batches containing the premise provider and raises its marginal contribution, while
joint synergy makes baseline rules jointly consume the outputs of multiple candi
dates and yields extra atoms. Attribution uses Shapley shares; it is the unique
allocation scheme satisfying the four axioms of efficiency, symmetry, dummy, and
additivity, but it does not retroactively certify individual legitimacy. Selection under
a budget constraint maximizes the audited ledger; in the submodular case greedy
enjoys a multiplicative approximation guarantee, while in the non-submodular case
this guarantee fails. Readings for the toy pool give 15 submodularity violations and
an attribution result of shares (2/3,2/3,2/3).