# Question 6815: Ordered Rational Midpoint and Endpoint Gaps

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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.

R0012-06815 · q-20261010-advance-20261014-r02-real-number-0015



## Original question and givens

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-14/2≤b=15/3. Give (midpoint, midpoint−a, b−midpoint). 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. a=-14/2≤b=15/3. Give (midpoint, midpoint−a, b−midpoint). 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 · calculationBefore

endpoint sum := sum(-7, 5)

After

endpoint sum = -2

Conditions: none

Add the two ordered endpoints completely.

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





S003 · calculationBefore

ordered midpoint := div(-2, 2)

After

ordered midpoint = -1

Conditions: none

Divide by2 for the exact midpoint;equal endpoints are allowed.

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





S004 · calculationBefore

midpoint minus left endpoint := sub(-1, -7)

After

midpoint minus left endpoint = 6

Conditions: none

The left distance is midpoint minus the left endpoint.

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





S005 · calculationBefore

right endpoint minus midpoint := sub(5, -1)

After

right endpoint minus midpoint = 6

Conditions: none

The right distance is the right endpoint minus midpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

a=-7≤midpoint=-1≤b=5

After

two nonnegative distances equal 6

Conditions: none

Endpoint order makes both differences nonnegative distances;when a=b both are0,so no degenerate-interval exclusion is needed.

Step identifiers and dependenciesRule: ordered_midpoint_equidistance · Uses: G001, S005









## Reverse checks


- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Recompute right endpoint minus midpoint with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute midpoint minus left endpoint with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute ordered midpoint with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute endpoint sum 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.





## Result

(-1/1, 6/1, 6/1)

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: 53366e0000c7dc0eb58796e29174ebcf097f055c2845c780f37b4cded59cb4cf
- candidate record SHA-256: d24ff32d7b65486a24cffb0c5e3fe12a26eae70fd3b999b7e5564c364aad802a
- semantic review receipt SHA-256: 90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00
- question-first witness SHA-256: b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9
- Lean: NOT_RUN · human review: NOT_PERFORMED

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