# Question 23304: Zero, Modulus Two and Composite Moduli Need Separate Conventions [Return to catalogue page 117](https://ocme.evemisslab.com/en/questions/page/117/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261111-r06-quadratic-residue-foundations/#q-20261014-advance-20261111-r06-quadratic-residue-0079) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0045-23304 · q-20261014-advance-20261111-r06-quadratic-residue-0079 ## Original question and givens Use mode=zero_class, modulus=11, original a=0. Give (mode, modulus, normalized_class, declared_model_qualified, boundary). - G001 — Use mode=zero_class, modulus=11, original a=0. 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. - [mko-quadratic-residue](https://ocme.evemisslab.com/en/mko/mko-quadratic-residue/) ## Step-by-step derivation S001 · impliesBefore source operation=quadratic-domain-boundary;original snapshot and mathematics retained After explicit ordered source output fields=[mode, modulus, normalized_class, declared_model_qualified, boundary] Conditions: none 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. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · calculationBefore Quadratic ORIGINAL boundary class Euclidean quotient times modulus := mul(0, 11) After Quadratic ORIGINAL boundary class Euclidean quotient times modulus = 0 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Quadratic ORIGINAL boundary class ORIGINAL signed value minus multiple := sub(0, 0) After Quadratic ORIGINAL boundary class ORIGINAL signed value minus multiple = 0 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore Quadratic ORIGINAL boundary class:ORIGINAL signed value=0;positive modulus=11;actual quotient=0 After canonical residue=0;value=quotient*modulus+residue and0<=residue1 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. Step identifiers and dependenciesRule: finite_prime_ALL_trial_divisors_through_square_root · Uses: G001, S004 S006 · impliesBefore ORIGINAL mode=zero_class,modulus=11,a=0;normalized class=0 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. Step identifiers and dependenciesRule: quadratic_boundary_No_only_declared_nonzero_odd_prime_model · Uses: G001, S005 ## Reverse checks - 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 (zero_class, 11, 0, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: c9ff70dfc4ffe1021f5f3039cf5306843bd4c1025f388ddcd0c2909917184968 - candidate record SHA-256: cf97a11ce0e50df51e17ba9cec45b618068a4a6a0eff49522e80bda21c1d596a - semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438 - question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261111-r06-quadratic-residue-0079.json)