Use the unary DFA for lengths divisible by modulus=6, with accepted word length=12. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).
G001 — Use the unary DFA for lengths divisible by modulus=6, with accepted word length=12. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).
Definition and scope
Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings.
The DFA state is length mod m;every unary transition adds1 modulo m.The first m+1 states repeat0 at positions0,m.The ORIGINAL word length is divisible by m,so x empty,y length m and the remaining z form a legal accepted word.
derived = [EVERY integer i>=0: pumped length=i*modulus+z, z is divisible by modulus, state remains0 after each loop;accepted, Yes]
Conditions: none
For arbitrary i>=0,traverse the same complete loop i times and then z.Since z is also divisible by m,the final state remains0.Full transitions and this quantified invariant support EVERY i,not sampled pump counts.
S004 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S003 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S002 · premise_and_result_audit — Recompute Pumping ORIGINAL unary z length with exact rational arithmetic, not rounded decimals.
S001 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
This question review and source binding
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.