# Question 8297: Stable Rational Sort with Original Index Witnesses [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-0022) 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. R0015-08297 · q-20261012-advance-20261017-r04-order-relation-0022 ## 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. Ordered inputs=[9/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order. - 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. Ordered inputs=[9/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order. ## 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 original pair0,1 left cross := mul(3, 2) After original pair0,1 left cross = 6 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 original pair0,1 right cross := mul(1, 1) After original pair0,1 right cross = 1 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 pair0,1 cross residual := sub(6, 1) After original pair0,1 cross residual = 5 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 · impliesBefore original pair0,1:fractions=[3/1, 1/2] After cross residual=5;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, S004 S006 · calculationBefore original pair0,2 left cross := mul(3, 4) After original pair0,2 left cross = 12 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 · calculationBefore original pair0,2 right cross := mul(-1, 1) After original pair0,2 right cross = -1 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, S006 S008 · calculationBefore original pair0,2 cross residual := sub(12, -1) After original pair0,2 cross residual = 13 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 · impliesBefore original pair0,2:fractions=[3/1, -1/4] After cross residual=13;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, S008 S010 · calculationBefore original pair1,2 left cross := mul(1, 4) After original pair1,2 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, S009 S011 · calculationBefore original pair1,2 right cross := mul(-1, 2) After original pair1,2 right cross = -2 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, S010 S012 · calculationBefore original pair1,2 cross residual := sub(4, -2) After original pair1,2 cross residual = 6 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, S011 S013 · impliesBefore original pair1,2:fractions=[1/2, -1/4] After cross residual=6;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, S012 S014 · calculationBefore sorted adjacent0 left cross := mul(-1, 2) After sorted adjacent0 left cross = -2 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, S013 S015 · calculationBefore sorted adjacent0 right cross := mul(1, 4) After sorted adjacent0 right 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, S014 S016 · calculationBefore sorted adjacent0 cross residual := sub(-2, 4) After sorted adjacent0 cross residual = -6 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, S015 S017 · impliesBefore sorted adjacent0:fractions=[-1/4, 1/2] After cross residual=-6;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, S016 S018 · calculationBefore sorted adjacent1 left cross := mul(1, 1) After sorted adjacent1 left cross = 1 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, S017 S019 · calculationBefore sorted adjacent1 right cross := mul(3, 2) After sorted adjacent1 right cross = 6 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, S018 S020 · calculationBefore sorted adjacent1 cross residual := sub(1, 6) After sorted adjacent1 cross residual = -5 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, S019 S021 · impliesBefore sorted adjacent1:fractions=[1/2, 3/1] After cross residual=-5;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, S020 S022 · impliesBefore normalized original values=[3/1, 1/2, -1/4] After stable original indices=[2, 1, 0];sorted values=[-1/4, 1/2, 3/1] Conditions: none Preserve original index order when values tie.Repeated values remain separate entries rather than being deduplicated as a set. Step identifiers and dependenciesRule: stable_rational_original_index_sort · Uses: G001, S021 ## Reverse checks - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Recompute sorted adjacent1 cross residual with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute sorted adjacent1 right cross with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute sorted adjacent1 left cross with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Recompute sorted adjacent0 cross residual with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute sorted adjacent0 right cross with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute sorted adjacent0 left cross with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Recompute original pair1,2 cross residual with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute original pair1,2 right cross with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute original pair1,2 left cross with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute original pair0,2 cross residual with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute original pair0,2 right cross with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute original pair0,2 left cross with exact rational arithmetic, not rounded decimals. - 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 — Recompute original pair0,1 cross residual with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute original pair0,1 right cross with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute original pair0,1 left cross 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 ([3/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 3/1], [<, <]) 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: ea37988edc9c0d1784d91a1e0414cb25d95ecb1d2d0926369f93060b3a60659d - candidate record SHA-256: 4ff031a07855629cc39943507680248aba6da08dc8223255befc90aca9bcbaae - 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-0022.json)