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R0012-06826 · q-20261010-advance-20261014-r02-real-number-0051
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[0/1, 1/1], J=[1/1, 2/1] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
G001 — Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[0/1, 1/1], J=[1/1, 2/1] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.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
explicit rational points embedded in the real line
After
exact finite order and distance data,not real completeness or irrational classification
Conditions: none
Use exact rational values for this finite slice of the real line,without rounded decimals or claims about real completeness or classification of irrationals.
Rule: rational_real_slice_scope · Uses: G001
Before
I=[0,1];J=[1,2]
After
intersection bounds=[max left=1,min right=1]
Conditions: none
Closed intervals include endpoints;the intersection uses the larger left and smaller right bound.Even lo=hi gives a common point.
Rule: closed_interval_inner_bounds · Uses: G001, S001
Before
lo=1;hi=1;lo≤hi=Yes
After
intersection=[1/1, 1/1];separation=0/1
Conditions: none
Endpoint contact is not empty and has gap0;only lo>hi is empty.Do not replace the closed-interval condition with strict inequality.
Rule: closed_contact_not_empty · Uses: G001, S002
S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(Yes, [1/1, 1/1], 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: 752191794f7d75c23ad52a3e71bdb0e33abb0ea4403078d27136fffc66c9dc80candidate record SHA-256: 5f4561280b4cead4180a033408910409ae124cb545ea41e3e2394672ec0c87b6semantic review receipt SHA-256: 90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00question-first witness SHA-256: b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9Lean: NOT_RUN · human review: NOT_PERFORMED