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R0015-08236 · q-20261012-advance-20261017-r04-order-relation-0086
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=-2/3, b=-1/2, t=3/3. 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=-2/3, b=-1/2, t=3/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).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.
Before
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.
Rule: real_rational_order_scope · Uses: G001
Before
scaled first value := mul(1, -2/3)
After
scaled first value = -2/3
Conditions: none
Substitute the source data exactly in the stated operand order;division requires a nonzero denominator.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
scaled second value := mul(1, -1/2)
After
scaled second value = -1/2
Conditions: none
Substitute the source data exactly in the stated operand order;division requires a nonzero denominator.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
original order right cross := mul(-1, 3)
After
original order right cross = -3
Conditions: none
Substitute the source data exactly in the stated operand order;division requires a nonzero denominator.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
original order cross residual := sub(-4, -3)
After
original order cross residual = -1
Conditions: none
Substitute the source data exactly in the stated operand order;division requires a nonzero denominator.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
original order:fractions=[-2/3, -1/2]
After
cross residual=-1;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.
Rule: rational_positive_denominator_comparison · Uses: G001, S006
Before
scaled order left cross := mul(-2, 2)
After
scaled order left cross = -4
Conditions: none
Substitute the source data exactly in the stated operand order;division requires a nonzero denominator.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
scaled order right cross := mul(-1, 3)
After
scaled order right cross = -3
Conditions: none
Substitute the source data exactly in the stated operand order;division requires a nonzero denominator.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
scaled order cross residual := sub(-4, -3)
After
scaled order cross residual = -1
Conditions: none
Substitute the source data exactly in the stated operand order;division requires a nonzero denominator.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
scaled order:fractions=[-2/3, -1/2]
After
cross residual=-1;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.
Rule: rational_positive_denominator_comparison · Uses: G001, S010
Before
scale=1/1;old comparison=<
After
new comparison=<;direction=order preserved
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.
Rule: order_scale_sign_three_cases · Uses: G001, S011
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.(-2/3, -1/2, 1/1, -2/3, -1/2, <, <, order-preserved)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 877fd5467b3a5663e778380fc5e64478c31855464e5010e886674b3908f0df2bcandidate record SHA-256: bb2a29bb2b92ef0f8e6f7c55f796edb32d307fef65886104c4b3b356b19e7a9bsemantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1bLean: NOT_RUN · human review: NOT_PERFORMED