Axiom Dynamics 2: Transition Timing Criterion—Net Gain in Description Length and a Typology of Theory Change

When should an axiom base undergo a transition and admit a new axiom? This
paper gives a decidable answer using net gain as the sole currency. The criterion
adopts a three-stage total ledger: syntactic cost, derivation cost, and the running ac
count. Syntactic cost charges each registered rule by “1 plus the number of premises”;
derivation cost takes the shortest derivation depth of a phenomenon, and underiv
able phenomena pay a penalty equal to the size of the atom set; the running account
charges by the selector’s actual firing steps and the cumulative number of registra
tions in the candidate pool, and divergence is entered as a preregistered truncated
reading rather than as infinity. The addition of a candidate is legitimate if and
only if the total ledger after the candidate is strictly smaller than the baseline total
ledger; an empty transition is not legitimate. Typology determines only the ledger
structure: purely expansionary candidates do not pay contraction cost, revisionary
candidates must pay contraction cost for withdrawing old rules, and the running
account is nonzero only when the selector actually executes open-target simulation.
The conditional structure of the criterion unfolds accordingly. The termination
gain of meta-rule candidates depends on the openness of the system: in a closed
system the selector terminates on finitely many candidates, and the termination ac
count is the negative meta-rule cost; in an open system the default roaming selector
misses the target within the truncation window, whereas a well-ordered selector hits
it in finitely many steps, and the termination account is a finite positive reading
under preregistered constants. Compression common benefit depends on system
scale: on chain systems the net gain of the expansion reading is n2 2 −7n+2, which
is nonpositive for n ≤ 6 and strictly positive at n = 7, and the empty-transition
equation has no integer solution; the revision variant is n2 2 −9n+4, and the thresh
old moves to n = 9. The comprehensive judgment of the three candidates shows:
the meta-rule candidate is conditionally legitimate, legitimate when openness detec
tion is positive or when scale crosses the threshold; the higher-order rule candidate
is legitimate with a thin margin; the bottom-level reconstruction candidate is not
legitimate, and its net gain is robustly negative against a positive rewriting fee.
The self-reference trap is dismantled by a three-layer structure: the criterion is a
preregistered counting procedure, candidates are counted objects, and the encoding
scheme is locked before candidates; the running account is the third stage within
the snapshot, not an off-ledger simulation. The criterion does not promise uncondi
tional robustness across families of encoding schemes, nor does it cover deletion-type
transitions; these two directions are registered as subsequent interfaces.