Question 8226: Scaled Rational Values and Order Direction

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=-1/2, b=1/3, t=0/1. 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=-1/2, b=1/3, t=0/1. 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

Step-by-step derivation

usual total order on R;all original denominators positive

closed endpoints included;negative scaling reverses;zero collapses

scaled first value := mul(0, -1/2)

scaled first value = 0

scaled second value := mul(0, 1/3)

scaled second value = 0

original order left cross := mul(-1, 3)

original order left cross = -3

original order right cross := mul(1, 2)

original order right cross = 2

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

original order cross residual = -5

original order:fractions=[-1/2, 1/3]

cross residual=-5;comparison=<

scaled order left cross := mul(0, 1)

scaled order left cross = 0

scaled order right cross := mul(0, 1)

scaled order right cross = 0

scaled order cross residual := sub(0, 0)

scaled order cross residual = 0

scaled order:fractions=[0/1, 0/1]

cross residual=0;comparison==

scale=0/1;old comparison=<

new comparison==;direction=merged

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

(-1/2, 1/3, 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.