# Question 6872: Finite Distinct Points and Least Containing Closed Interval

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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.

R0012-06872 · q-20261010-advance-20261014-r02-real-number-0047



## Original question and givens

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-21/2, 0/1, 22/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.


- G001 — Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-21/2, 0/1, 22/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.





## 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

[[-21, 2], [0, 1], [22, 3]]

After

[-21/2, 0/1, 22/3]

Conditions: none

Sort and deduplicate by rational value,not fraction strings;0/−2 and0/3 represent the same point0.

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





S003 · calculationBefore

smallest containing interval width := sub(22/3, -21/2)

After

smallest containing interval width = 107/6

Conditions: none

Rightmost minus leftmost point gives the nonnegative interval width.

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





S004 · impliesBefore

all points between -21/2 and 22/3

After

smallest containing closed interval=[-21/2,22/3]

Conditions: none

Every containing closed interval must include the smallest and largest points,hence this interval is minimal.This is not a claim that topological closure fills the gaps of a finite point set.

Step identifiers and dependenciesRule: containing_interval_not_topological_closure · Uses: G001, S003









## Reverse checks


- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Recompute smallest containing interval width with exact rational arithmetic, not rounded decimals.
- 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

([-21/2, 0/1, 22/3], -21/2, 22/3, 107/6)

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: 7a0611a57476734bb16e0d661a89c7f68d029404ab10bc7c03fa991b965858c9
- candidate record SHA-256: dd441e6039b945e0037138b6dd3a92a1f9d6e589be68dff02dbc0ad0e445083c
- 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-0047.json)