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? Thispaper gives a decidable answer using net gain as the sole currency. The criterionadopts a three-stage total ledger: syntactic cost, derivation cost, and the running account. Syntactic cost charges each registered rule by “1 plus the number of premises”;derivation cost takes the shortest derivation depth of a phenomenon, and underivable phenomena pay a penalty equal to the size of the atom set; the running accountcharges by the selector’s actual firing steps and the cumulative number of registrations in the candidate pool, and divergence is entered as a preregistered truncatedreading rather than as infinity. The addition of a candidate is legitimate if andonly if the total ledger after the candidate is strictly smaller than the baseline totalledger; an empty transition is not legitimate. Typology determines only the ledgerstructure: purely expansionary candidates do not pay contraction cost, revisionarycandidates must pay contraction cost for withdrawing old rules, and the runningaccount is nonzero only when the selector actually executes open-target simulation.The conditional structure of the criterion unfolds accordingly. The terminationgain of meta-rule candidates depends on the openness of the system: in a closedsystem the selector terminates on finitely many candidates, and the termination account is the negative meta-rule cost; in an open system the default roaming selectormisses the target within the truncation window, whereas a well-ordered selector hitsit in finitely many steps, and the termination account is a finite positive readingunder preregistered constants. Compression common benefit depends on systemscale: on chain systems the net gain of the expansion reading is n2 2 −7n+2, whichis nonpositive for n ≤ 6 and strictly positive at n = 7, and the empty-transitionequation has no integer solution; the revision variant is n2 2 −9n+4, and the threshold moves to n = 9. The comprehensive judgment of the three candidates shows:the meta-rule candidate is conditionally legitimate, legitimate when openness detection is positive or when scale crosses the threshold; the higher-order rule candidateis legitimate with a thin margin; the bottom-level reconstruction candidate is notlegitimate, 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 apreregistered counting procedure, candidates are counted objects, and the encodingscheme is locked before candidates; the running account is the third stage withinthe snapshot, not an off-ledger simulation. The criterion does not promise unconditional robustness across families of encoding schemes, nor does it cover deletion-typetransitions; these two directions are registered as subsequent interfaces.