Axiom Dynamics 3: Conservativeness and Invariance Rigidity—Dual Conditions for the Legitimacy of Axiom Transitions

Changes of the rule set of an axiom system—expansion, contraction, and revision—
change the theorem set in the closure, but “what a change must preserve and what
it must gain” has previously lacked a machine-checkable compositional criterion.
This paper establishes, on a finite Horn deductive system, an object-level three-way
decomposition ledger for transitions: any rule-set change induces a theorem set T to
T′, and the difference is decomposed into the object-level gain set Gainobj = T′\T,
the loss set Loss = T \ T′, and the invariant set Inv = T ∩ T′. The three-way
decomposition is complete and pairwise disjoint. Legitimacy is decided by a dual
condition: the protection core Core ⊆ Inv (Condition 1, losslessness), and Gainobj
or meta-level gain Gainmeta is nonempty (Condition 2, gainfulness). Failure of
Condition 1 is an immediate veto and is not compensated by Condition 2 or by
syntactic-level gains; after Condition 1 holds, Condition 2 can fail independently.
The criterion for meta-level gain is based on the carrier function Car(P,S): a
property P constitutes meta-level gain if and only if P holds in the post-change
system, its carrier is nonempty, and it belongs to Type A genuine acquisition (P
had a carrier before the change but did not hold) or Type B level addition (P had
no applicable carrier before the change and acquires a carrier after the change).
Type B must be marked “level addition.” The termination of closure computation
has carriers and holds at both ends of the change and is not counted as meta-level
gain; savings in derivation depth and description length belong to the syntactic-level
ledger and are also not counted.
On the finite Horn fragment, universal conservativeness (closure equality for ev
ery fact set) is equivalent to redundancy (the conclusion of a rule is already derivable
from its premises via existing rules). This equivalence is proved by syntactic induc
tion and recomputed by full enumeration with seven atoms, a six-rule baseline, 448
candidate rules, and 128 fact sets, with zero counterexamples. Universal conserva
tiveness is the syntactic counterpart of deductive conservativeness at the level of rule
redundancy, not an equivalent of the model-expansion property; model expansion
implies deductive conservativeness, but the converse generally fails. Fixed-fact-set
conservativeness for rule incorporation holds automatically by monotonicity and
provides no detection power; the two readings are distinguished by qualifiers and
are not conflated.
Membership rigidity resides in the layer of theorem-set membership, not in the
syntactic layer. Chain–star rewiring, with the empty fact input as baseline, keeps
the theorem set pointwise invariant; the syntactic ledger is the same, 15, the sums
of shortest derivation depths are 28 and 7 respectively, and the two-stage total
ledgers are 43 and 22 respectively. This case displays the separation between the
membership layer and the syntactic layer, and does not by itself constitute a positive
verdict of legitimacy: its object-level gain is empty, and the depth saving does not
satisfy Condition 2. The condition for membership rigidity is closure preservation;
if a rule transition causes some atom to fall into Loss, rigidity does not apply, and
the dual-condition verdict must be entered. If a falsifiable prediction is registered
as a member of the theorem set, a closure-preserving rule reorganization does not
change its membership; changes of the fact set are not within the jurisdiction of
the three-way decomposition, and under forward-chaining Horn semantics fact-set
expansion is monotonically nondecreasing, so empirical-level elimination must be
borne by fact-set contraction, fact replacement, or rule change.
The four types of verdicts show the logical independence of the two branches:
an uncompensated contraction dies by Condition 1 because Loss touches the pro
tection core; a compensated revision keeps the theorem set unchanged but has
no object-level gain and no meta-level gain satisfying the carrier condition, and
dies by Condition 2; an object-level genuine expansion bears Condition 2 through
new theorems; the incorporation of a meta-rule bears Condition 2 through Type B
level-addition meta-level gain, and its legitimacy depends on the meta-rule indeed
introducing a candidate-selection operator in the rule-application semantics. What
is determinately established is the completeness of the three-way decomposition, the
conjunctive structure of the dual condition, the logical independence of Condition 1
and Condition 2, the equivalence of universal conservativeness and redundancy on
the finite Horn fragment, and the above stratified readings; what is open or to be
verified is the membership of the list of meta-level properties, the boundary of the
carrier function in real theories, the theory of choice of Core, and the interaction
between the dual condition and derivation-cycle theorems. The legitimacy crite
rion and the timing criterion apply independently: legitimate does not equal worth
making, and worth making does not equal legitimate.