# Question 30265: Absolute-Value Target Roots and Candidate Audit [Return to catalogue page 152](https://ocme.evemisslab.com/en/questions/page/152/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261010-advance-20261014-r06-absolute-value-foundations/#q-20261010-advance-20261014-r06-absolute-value-0090) 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. R0059-30265 · q-20261010-advance-20261014-r06-absolute-value-0090 ## Original question and givens Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=13/2, ordered queries=[-13/2, 13/2, 15/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values). - G001 — Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=13/2, ordered queries=[-13/2, 13/2, 15/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values). ## 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-absolute-value](https://ocme.evemisslab.com/en/mko/mko-absolute-value/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-density14-absolute-preimage-rational-audit, density14_absolute_preimage_rational_audit, finite_rational_absolute_protocol] After derived = [[target_nonnegative, ALL_increasing_roots, ALL_absolute_query_values, ALL_equal_target_flags]] Conditions: none Retain the ORIGINAL bank full operation prefix,domain and ordered answer roles.Use orthogonal_equal_unit only where a frame is declared,and degrees where an angle unit is declared.Integer fractions retain/1.Do not invent parameters or alter original bilingual sources. Step identifiers and dependenciesRule: R59_EXACT_family_domain_FULLprefix_operation_and_answer_roles · Uses: G001 S002 · impliesBefore source roles = [Absolute target r, [13, 2]] After derived = [13/2, 13, 2] Conditions: none Require a RAW SAFE pair with nonzero denominator.This bank ALLOWS negative denominators;normalize sign/gcd BEFORE deciding the value sign.Do not import a primitive positive-denominator rule or a STR22 component cap. Step identifiers and dependenciesRule: Absolute_RAW_SAFE_pair_NONZERO_den_normalize_SIGN_and_gcd · Uses: G001, S001 S003 · impliesBefore source roles = [Absolute ORIGINAL queries slot0, [-13, 2]] After derived = [-13/2, -13, 2] Conditions: none Require a RAW SAFE pair with nonzero denominator.This bank ALLOWS negative denominators;normalize sign/gcd BEFORE deciding the value sign.Do not import a primitive positive-denominator rule or a STR22 component cap. Step identifiers and dependenciesRule: Absolute_RAW_SAFE_pair_NONZERO_den_normalize_SIGN_and_gcd · Uses: G001, S002 S004 · impliesBefore source roles = [Absolute ORIGINAL queries slot1, [13, 2]] After derived = [13/2, 13, 2] Conditions: none Require a RAW SAFE pair with nonzero denominator.This bank ALLOWS negative denominators;normalize sign/gcd BEFORE deciding the value sign.Do not import a primitive positive-denominator rule or a STR22 component cap. Step identifiers and dependenciesRule: Absolute_RAW_SAFE_pair_NONZERO_den_normalize_SIGN_and_gcd · Uses: G001, S003 S005 · impliesBefore source roles = [Absolute ORIGINAL queries slot2, [15, 2]] After derived = [15/2, 15, 2] Conditions: none Require a RAW SAFE pair with nonzero denominator.This bank ALLOWS negative denominators;normalize sign/gcd BEFORE deciding the value sign.Do not import a primitive positive-denominator rule or a STR22 component cap. Step identifiers and dependenciesRule: Absolute_RAW_SAFE_pair_NONZERO_den_normalize_SIGN_and_gcd · Uses: G001, S004 S006 · calculationBefore Absolute ORIGINAL queries slot0 ORIGINAL negative branch negation := sub(0, -13/2) After Absolute ORIGINAL queries slot0 ORIGINAL negative branch negation = 13/2 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [Absolute ORIGINAL queries slot0, -13/2] After derived = [No, 13/2] Conditions: none Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator. Step identifiers and dependenciesRule: Absolute_value_normalized_sign_branch_ZERO_is_nonnegative · Uses: G001, S006 S008 · impliesBefore source roles = [Absolute ORIGINAL queries slot1, 13/2] After derived = [Yes, 13/2] Conditions: none Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator. Step identifiers and dependenciesRule: Absolute_value_normalized_sign_branch_ZERO_is_nonnegative · Uses: G001, S007 S009 · impliesBefore source roles = [Absolute ORIGINAL queries slot2, 15/2] After derived = [Yes, 15/2] Conditions: none Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator. Step identifiers and dependenciesRule: Absolute_value_normalized_sign_branch_ZERO_is_nonnegative · Uses: G001, S008 S010 · calculationBefore Absolute ORIGINAL magnitude minus target slot0 := sub(13/2, 13/2) After Absolute ORIGINAL magnitude minus target slot0 = 0 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Absolute ORIGINAL magnitude minus target slot1 := sub(13/2, 13/2) After Absolute ORIGINAL magnitude minus target slot1 = 0 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Absolute ORIGINAL magnitude minus target slot2 := sub(15/2, 13/2) After Absolute ORIGINAL magnitude minus target slot2 = 1 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore source roles = [13/2, [-13/2, 13/2, 15/2]] After derived = [Yes, [-13/2, 13/2], [13/2, 13/2, 15/2], [Yes, Yes, No]] Conditions: none Since |x|>=0,a negative target is a LEGAL No with an empty root list.At0 the ONLY root is0/1;a positive target gives increasing-r,r.STILL compute all three ORIGINAL absolute query values and masks;do not reject the negative target or deduplicate query positions. Step identifiers and dependenciesRule: Absolute_preimage_NEGATIVE_target_LEGAL_EMPTY_roots_ZERO_singleton · Uses: G001, S012 ## Reverse checks - S013 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. - S012 · premise_and_result_audit — Recompute Absolute ORIGINAL magnitude minus target slot2 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Absolute ORIGINAL magnitude minus target slot1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Absolute ORIGINAL magnitude minus target slot0 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. - S008 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. - S007 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. - S006 · premise_and_result_audit — Recompute Absolute ORIGINAL queries slot0 ORIGINAL negative branch negation with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. - S004 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. - S003 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. - S002 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. - S001 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. ## Result (Yes, [-13/2, 13/2], [13/2, 13/2, 15/2], [Yes, Yes, No]) 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: 5fc4da465b037d2c85ac3855ad27d595e26184857f023ac3a2f5efa97b6a8551 - candidate record SHA-256: 0771d76bab94ed6e9060c69cb781a608afa737f3c295a6ba07a5167eab80e218 - semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5 - question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261010-advance-20261014-r06-absolute-value-0090.json)