Question 8296: Stable Rational Sort with Original Index Witnesses

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=[8/3, 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=[8/3, 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

Step-by-step derivation

usual total order on R;all original denominators positive

closed endpoints included;negative scaling reverses;zero collapses

original pair0,1 left cross := mul(8, 2)

original pair0,1 left cross = 16

original pair0,1 right cross := mul(1, 3)

original pair0,1 right cross = 3

original pair0,1 cross residual := sub(16, 3)

original pair0,1 cross residual = 13

original pair0,1:fractions=[8/3, 1/2]

cross residual=13;comparison=>

original pair0,2 left cross := mul(8, 4)

original pair0,2 left cross = 32

original pair0,2 right cross := mul(-1, 3)

original pair0,2 right cross = -3

original pair0,2 cross residual := sub(32, -3)

original pair0,2 cross residual = 35

original pair0,2:fractions=[8/3, -1/4]

cross residual=35;comparison=>

original pair1,2 left cross := mul(1, 4)

original pair1,2 left cross = 4

original pair1,2 right cross := mul(-1, 2)

original pair1,2 right cross = -2

original pair1,2 cross residual := sub(4, -2)

original pair1,2 cross residual = 6

original pair1,2:fractions=[1/2, -1/4]

cross residual=6;comparison=>

sorted adjacent0 left cross := mul(-1, 2)

sorted adjacent0 left cross = -2

sorted adjacent0 right cross := mul(1, 4)

sorted adjacent0 right cross = 4

sorted adjacent0 cross residual := sub(-2, 4)

sorted adjacent0 cross residual = -6

sorted adjacent0:fractions=[-1/4, 1/2]

cross residual=-6;comparison=<

sorted adjacent1 left cross := mul(1, 3)

sorted adjacent1 left cross = 3

sorted adjacent1 right cross := mul(8, 2)

sorted adjacent1 right cross = 16

sorted adjacent1 cross residual := sub(3, 16)

sorted adjacent1 cross residual = -13

sorted adjacent1:fractions=[1/2, 8/3]

cross residual=-13;comparison=<

normalized original values=[8/3, 1/2, -1/4]

stable original indices=[2, 1, 0];sorted values=[-1/4, 1/2, 8/3]

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

([8/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 8/3], [<, <])

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.