# Question 30293: Normalized Rational Absolute-Value Branch [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-0018) 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. R0059-30293 · q-20261010-advance-20261014-r06-absolute-value-0018 ## Original question and givens Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=-5/-3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch. - G001 — Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=-5/-3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch. ## 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-rational-two-branches, density14_absolute_rational_two_branches, finite_rational_absolute_protocol] After derived = [[normalized_value, sign_branch, absolute_value]] 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 input x, [-5, -3]] After derived = [5/3, 5, 3] 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 x, 5/3] After derived = [Yes, 5/3] 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, S002 S004 · impliesBefore source roles = [5/3] After derived = [5/3, nonnegative, 5/3] Conditions: none Return normalized x,the branch LABEL chosen by its VALUE,and abs(x).Zero belongs to the nonnegative branch,and a negative denominator does not mean a negative value.This is the declared rational slice,not a new certificate for all real absolute-value properties. Step identifiers and dependenciesRule: Absolute_normalized_rational_value_CORRECT_branch_label · Uses: G001, S003 ## Reverse checks - 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 (5/3, nonnegative, 5/3) 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: 71f671d1f83ebfce54d4e8753a43f3c6971c8dcdaca97a4279e51e603c9adb01 - candidate record SHA-256: 13fd48246447529e8aaa4129d9a3b15c75957c517d37242cd91efd7bf7a073f1 - 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-0018.json)