# Question 6829: Closed Interval Intersection and Separation Gap

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0012-06829 · q-20261010-advance-20261014-r02-real-number-0054



## Original question and givens

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-3/2, 3/2], J=[0/1, 5/2] 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=[-3/2, 3/2], J=[0/1, 5/2] 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.





## 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-real-number](https://ocme.evemisslab.com/en/mko/mko-real-number/)





## Step-by-step derivation

S001 · impliesBefore

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.

Step identifiers and dependenciesRule: rational_real_slice_scope · Uses: G001





S002 · impliesBefore

I=[-3/2,3/2];J=[0,5/2]

After

intersection bounds=[max left=0,min right=3/2]

Conditions: none

Closed intervals include endpoints;the intersection uses the larger left and smaller right bound.Even lo=hi gives a common point.

Step identifiers and dependenciesRule: closed_interval_inner_bounds · Uses: G001, S001





S003 · impliesBefore

lo=0;hi=3/2;lo≤hi=Yes

After

intersection=[0/1, 3/2];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.

Step identifiers and dependenciesRule: closed_contact_not_empty · Uses: G001, S002









## Reverse checks


- 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.





## Result

(Yes, [0/1, 3/2], 0/1)

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 32fbf1e97cce46d40eaf39c68f181cb932cad04196f0baa0ccd7d21aac9952a5
- candidate record SHA-256: e5586b47a0714a4cb6fdb62ade8dca7e8a6f05418dfe96304c2711c90425fdef
- semantic review receipt SHA-256: 90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00
- question-first witness SHA-256: b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261010-advance-20261014-r02-real-number-0054.json)