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R0015-08203 · q-20261012-advance-20261017-r04-order-relation-0028
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. Intervals are [0/1, 0/1] and [0/1, 0/1]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).
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. Intervals are [0/1, 0/1] and [0/1, 0/1]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).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
lower bounds left cross := mul(0, 1)
After
lower bounds left cross = 0
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
lower bounds right cross := mul(0, 1)
After
lower bounds right cross = 0
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
lower bounds cross residual := sub(0, 0)
After
lower bounds cross residual = 0
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
lower bounds:fractions=[0/1, 0/1]
After
cross residual=0;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
upper bounds left cross := mul(0, 1)
After
upper bounds left cross = 0
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
upper bounds right cross := mul(0, 1)
After
upper bounds right cross = 0
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
upper bounds cross residual := sub(0, 0)
After
upper bounds cross residual = 0
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
upper bounds:fractions=[0/1, 0/1]
After
cross residual=0;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
intersection inner bounds left cross := mul(0, 1)
After
intersection inner bounds left cross = 0
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
intersection inner bounds right cross := mul(0, 1)
After
intersection inner bounds right cross = 0
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
intersection inner bounds cross residual := sub(0, 0)
After
intersection inner bounds cross residual = 0
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
intersection inner bounds:fractions=[0/1, 0/1]
After
cross residual=0;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
intervals=[[0/1, 0/1], [0/1, 0/1]]
After
max lower=0/1;min upper=0/1;nonempty=Yes
Conditions: none
Satisfying both lower bounds means using their maximum;both upper bounds means their minimum.Equal inner bounds still include that single point.
Rule: closed_interval_intersection_guard · Uses: G001, S013
S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 intersection inner bounds cross residual with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute intersection inner bounds right cross with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute intersection inner bounds 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 upper bounds cross residual with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute upper bounds right cross with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute upper bounds 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 lower bounds cross residual with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute lower bounds right cross with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute lower bounds 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.([0/1, 0/1], [0/1, 0/1], 0/1, 0/1, Yes, [0/1, 0/1])
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: 4569171c2f2cff25c75be40621f75445d30a0cc51da3bc328324afd0c9cd38bfcandidate record SHA-256: d8157505c06d55cf4e601a4beceb3495c6310767a44f44ede98e8d1ed54f5a83semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1bLean: NOT_RUN · human review: NOT_PERFORMED