Axiom Dynamics 6: Between Sequential and Batch—Execution Equivalence Criteria for Axiom Transitions and the Normal Form of Replay Verdicts

This paper closes two still-open problems in the criterion system of axiom transi
tions on finite Horn deductive systems. On the execution problem: can a candidate
batch that passes the holistic transition-legitimacy biconditional be implemented as
a sequential execution that is legal at every step? We prove a deadlock criterion:
a pure expansion batch has a step-ignition-legal sequential implementation if and
only if the batch contains no deadlock subset, that is, no proper subset that, af
ter activation, has already generated the entire closure of the batch. The criterion
yields a counting necessary condition, namely that batch gain is no less than the
number of candidates, and proves a strict separation of three tiers: holistic legality
does not imply step-ignition reachability; step-ignition reachability does not imply
that all orderings are step-ignition legal; and the counting condition is necessary
but not sufficient. On the verdict problem: what is the canonical record of a replay
verdict? We prove a verdict normal form: under the replay protocol in which the
protected core is fixed throughout registration and the archive is append-complete,
every history uniquely decomposes, up to adjacent-compression congruence, into an
alternating sequence of maximal clean blocks and non-clean segments; the normal
form is a complete invariant of the verdict, the comparison object for verdict equal
ity is the normal-form structure rather than the number of raw segment records,
and it is finitely decidable. An insertion-sensitivity counterexample proves that this
congruence is an exact boundary and cannot be coarsened; bleach impossibility is
a definitional corollary of archive appendability. All readings are recomputed by a
single self-contained script in exact integer arithmetic; the subset-criterion engine
and the ordering depth-first engine agree in two routes with zero conflicts; all in
puts are hardcoded, with no external data source and no random source. The above
conclusions hold within the fragment of finite Horn systems, no negation, no quan
tifiers, and no self-referential encoding; minimal interleaved implementation with
contraction batches, the complexity classification of the deadlock criterion, and the
refinement direction of the congruence are registered as open problems.