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R0015-08287 · q-20261012-advance-20261017-r04-order-relation-0012
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=[-1/3, 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=[-1/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.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
original pair0,1 left cross := mul(-1, 2)
After
original pair0,1 left cross = -2
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
original pair0,1 right cross := mul(1, 3)
After
original pair0,1 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, S002
Before
original pair0,1 cross residual := sub(-2, 3)
After
original pair0,1 cross residual = -5
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 pair0,1:fractions=[-1/3, 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.
Rule: rational_positive_denominator_comparison · Uses: G001, S004
Before
original pair0,2 left cross := mul(-1, 4)
After
original pair0,2 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, S005
Before
original pair0,2 right cross := mul(-1, 3)
After
original pair0,2 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, S006
Before
original pair0,2 cross residual := sub(-4, -3)
After
original pair0,2 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, S007
Before
original pair0,2:fractions=[-1/3, -1/4]
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, S008
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
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.
Rule: rational_positive_denominator_comparison · Uses: G001, S012
Before
sorted adjacent0 left cross := mul(-1, 4)
After
sorted adjacent0 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, S013
Before
sorted adjacent0 right cross := mul(-1, 3)
After
sorted adjacent0 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, S014
Before
sorted adjacent0 cross residual := sub(-4, -3)
After
sorted adjacent0 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, S015
Before
sorted adjacent0:fractions=[-1/3, -1/4]
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, S016
Before
sorted adjacent1 left cross := mul(-1, 2)
After
sorted adjacent1 left cross = -2
Conditions: none
Substitute the source data exactly in the stated operand order;division requires a nonzero denominator.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
sorted adjacent1 right cross := mul(1, 4)
After
sorted adjacent1 right 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, S018
Before
sorted adjacent1 cross residual := sub(-2, 4)
After
sorted adjacent1 cross residual = -6
Conditions: none
Substitute the source data exactly in the stated operand order;division requires a nonzero denominator.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
sorted adjacent1: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.
Rule: rational_positive_denominator_comparison · Uses: G001, S020
Before
normalized original values=[-1/3, 1/2, -1/4]
After
stable original indices=[0, 2, 1];sorted values=[-1/3, -1/4, 1/2]
Conditions: none
Preserve original index order when values tie.Repeated values remain separate entries rather than being deduplicated as a set.
Rule: stable_rational_original_index_sort · Uses: G001, S021
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.([-1/3, 1/2, -1/4], [0, 2, 1], [-1/3, -1/4, 1/2], [<, <])
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: 48ccacc15535f73deec263d9d34c0e5cb76947980df634b6186de06cc3013ca8candidate record SHA-256: 8e6c6072a77c627287319734734db8974cc0b8679d25eb25b974bf2af5e04943semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1bLean: NOT_RUN · human review: NOT_PERFORMED