Use mode=composite_modulus, modulus=9, original a=3. Give (mode, modulus, normalized_class, declared_model_qualified, boundary).
G001 — Use mode=composite_modulus, modulus=9, original a=3. Give (mode, modulus, normalized_class, declared_model_qualified, boundary).
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.
Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof.
Retain the original signed value and congruence class via Euclidean division;do not mistake a language-specific negative remainder for the canonical class.
Quadratic composite boundary modulus:ORIGINAL integer=9;complete possible trial divisors=[2, 3]
After
ALL [divisor,remainder] rows=[[2, 1], [3, 0]];prime=No;a composite>1 has a factor at most its square root
Conditions: none
Check the entire square-root divisor range.The empty range qualifies prime2,but not0 or1.Samples,square tables or an Euler criterion do not replace primality qualification.
ORIGINAL mode=composite_modulus,modulus=9,a=3;normalized class=3
After
declared_model_qualified=No;zero,p2 and composite conventions handled separately;NOT a claim this class has no square root under another convention
Conditions: none
Zero has root0 but is excluded by this nonzero convention.Modulus2 or composite classes may also have roots.The No marks model scope,not absence of roots or an out-of-domain two-root theorem.
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S003 · premise_and_result_audit — Recompute Quadratic ORIGINAL boundary class ORIGINAL signed value minus multiple with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute Quadratic ORIGINAL boundary class Euclidean quotient times modulus with exact rational arithmetic, not rounded decimals.
S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
Result
(composite_modulus, 9, 3, No, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)
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.