# Question 8244: Scaled Rational Values and Order Direction [Return to catalogue page 42](https://ocme.evemisslab.com/en/questions/page/42/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261017-r04-order-relation-foundations/#q-20261012-advance-20261017-r04-order-relation-0094) 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. R0015-08244 · q-20261012-advance-20261017-r04-order-relation-0094 ## Original question and givens Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=6/3, b=7/2, t=0/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t). - G001 — Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=6/3, b=7/2, t=0/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t). ## 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-order-relation](https://ocme.evemisslab.com/en/mko/mko-order-relation/) ## Step-by-step derivation S001 · impliesBefore usual total order on R;all original denominators positive After closed endpoints included;negative scaling reverses;zero collapses Conditions: none These exact rationals use the usual real total order,not complex numbers or an arbitrary partial order.Original denominators must be positive;normalizing a negative input denominator does not waive that contract. Step identifiers and dependenciesRule: real_rational_order_scope · Uses: G001 S002 · calculationBefore scaled first value := mul(0, 2) After scaled first value = 0 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore scaled second value := mul(0, 7/2) After scaled second value = 0 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore original order left cross := mul(2, 2) After original order left cross = 4 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore original order right cross := mul(7, 1) After original order right cross = 7 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore original order cross residual := sub(4, 7) After original order cross residual = -3 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore original order:fractions=[2/1, 7/2] After cross residual=-3;comparison=< Conditions: none Reduced denominators are positive,so the cross-difference sign determines order.Retain equality and use original indices separately to break equal-value sorting ties. Step identifiers and dependenciesRule: rational_positive_denominator_comparison · Uses: G001, S006 S008 · calculationBefore scaled order left cross := mul(0, 1) After scaled order left cross = 0 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore scaled order right cross := mul(0, 1) After scaled order right cross = 0 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore scaled order cross residual := sub(0, 0) After scaled order cross residual = 0 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore scaled order:fractions=[0/1, 0/1] After cross residual=0;comparison== Conditions: none Reduced denominators are positive,so the cross-difference sign determines order.Retain equality and use original indices separately to break equal-value sorting ties. Step identifiers and dependenciesRule: rational_positive_denominator_comparison · Uses: G001, S010 S012 · impliesBefore scale=0/1;old comparison=< After new comparison==;direction=merged Conditions: none The direction label depends on the scale's sign.Positive preserves,negative reverses,and zero collapses both values to0 even if originally equal.Equality remains equality under negative scaling. Step identifiers and dependenciesRule: order_scale_sign_three_cases · Uses: G001, S011 ## Reverse checks - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute scaled order cross residual with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute scaled order right cross with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute scaled order left cross with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute original order cross residual with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute original order right cross with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute original order left cross with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute scaled second value with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute scaled first value 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 (2/1, 7/2, 0/1, 0/1, 0/1, <, =, collapsed) 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: 918171ed520a9a45c891ba190be4d875fd7645d4af19db320d3c3f27b4c61fe2 - candidate record SHA-256: 6d36a0d41cce40260e82aeaaf60c1baf126dd0b57db0176e68eda220f8b9964d - semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18 - question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261017-r04-order-relation-0094.json)