# Question 23736: Domain Extensions and Premature Methods Are Different Legendre Boundaries [Return to catalogue page 119](https://ocme.evemisslab.com/en/questions/page/119/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261112-r06-legendre-symbol-foundations/#q-20261014-advance-20261112-r06-legendre-symbol-0086) 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. R0046-23736 · q-20261014-advance-20261112-r06-legendre-symbol-0086 ## Original question and givens Use mode=composite_modulus, modulus=27, original a=11. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary). - G001 — Use mode=composite_modulus, modulus=27, original a=11. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, 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-legendre-symbol](https://ocme.evemisslab.com/en/mko/mko-legendre-symbol/) ## Step-by-step derivation S001 · impliesBefore source operation=legendre-domain-method-boundary;original snapshot and mathematics retained After explicit ordered source output fields=[mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary] Conditions: none Keep the source snapshot while listing ordered fields without JavaScript.Radius uses its original mode to select6 or11 fields without padding;distinguish nonclaims from actual negations. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · calculationBefore Legendre ORIGINAL boundary class quotient times modulus := mul(0, 27) After Legendre ORIGINAL boundary class quotient times modulus = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Legendre ORIGINAL boundary class ORIGINAL signed value minus multiple := sub(11, 0) After Legendre ORIGINAL boundary class ORIGINAL signed value minus multiple = 11 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore Legendre ORIGINAL boundary class:ORIGINAL a=11,positive modulus=27;actual quotient=0 After normalized class=11;a=quotient*modulus+class;ORIGINAL a retained in output where declared Conditions: none Retain the same class for negative a and normalize into0..p−1,rather than using a negative language-specific remainder. Step identifiers and dependenciesRule: legendre_ORIGINAL_signed_class_Euclidean_normalization · Uses: G001, S003 S005 · impliesBefore Legendre composite boundary modulus:ORIGINAL modulus=27;ALL possible trial divisors through square root=[2, 3, 4, 5] After ALL [divisor,remainder] rows=[[2, 1], [3, 0], [4, 3], [5, 2]];prime=No Conditions: none Qualify primality over the full trial range,not by guessing from Legendre/Euler classification.This is a modulus premise,not the square status of a. Step identifiers and dependenciesRule: legendre_complete_prime_trial_not_Euler_classification · Uses: G001, S004 S006 · impliesBefore ORIGINAL mode=composite_modulus;modulus=27,a=11,normalized class=11 After unsupported_domain_or_method_certificate fixed No;modulus outside declared odd-prime Legendre domain Conditions: none For premature_euler,the No neither refutes Euler criterion nor puts a valid input out of domain.Modulus2/composite cases have a domain restriction.No Jacobi or other extension is silently adopted. Step identifiers and dependenciesRule: legendre_DOMAIN_or_METHOD_No_responsibilities_distinct · 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 Legendre ORIGINAL boundary class ORIGINAL signed value minus multiple with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Legendre ORIGINAL boundary class 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, 27, 11, No, Legendre_symbol_domain_requires_an_odd_prime_modulus) 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: cbb0036a28f0c642a44104ba7ab7eead57c6b5ff5348f9e6c7a77510cc6a9de8 - candidate record SHA-256: 77e1e48236c15e6844e736fea42698d24bf83bb8b8b68066f5fcbbf0a27bda33 - semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1 - question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261112-r06-legendre-symbol-0086.json)