# Question 1648: Inclusive Error Interval and a Candidate Boundary

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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.

R0002-01648 · q-20261013-advance-20261020-r03-numerical-approximation-error-0098



## Original question and givens

Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=5/3, tolerance=5/3≥0, candidate=45/3. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).


- G001 — Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=5/3, tolerance=5/3≥0, candidate=45/3. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).





## 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-numerical-approximation-error](https://ocme.evemisslab.com/en/mko/mko-numerical-approximation-error/)





## Step-by-step derivation

S001 · impliesBefore

reference=5/3

After

all values are exact rationals

Conditions: none

This is exact error arithmetic against a specified reference,without floating-point rounding that could alter an equality boundary.

Step identifiers and dependenciesRule: exact_error_model · Uses: G001





S002 · calculationBefore

reference−tolerance := sub(5/3, 5/3)

After

reference−tolerance = 0

Conditions: none

Compute the closed interval's lower endpoint.

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





S003 · calculationBefore

reference+tolerance := sum(5/3, 5/3)

After

reference+tolerance = 10/3

Conditions: none

Then compute its upper endpoint.

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





S004 · impliesBefore

|candidate−reference|≤tolerance

After

reference−tolerance≤candidate≤reference+tolerance

Conditions: none

Add the reference to the two-sided signed-error bound to obtain the closed interval;endpoints are included.

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





S005 · calculationBefore

candidate−reference := sub(15, 5/3)

After

candidate−reference = 40/3

Conditions: none

Calculate the actual deviation as a reverse check on interval membership.

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





S006 · impliesBefore

candidate=15; |deviation|=40/3

After

inside closed interval: No

Conditions: none

The absolute-error and closed-interval tests agree;zero tolerance allows only the reference itself.

Step identifiers and dependenciesRule: interval_membership · 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 candidate−reference with exact rational arithmetic, not rounded decimals.
- 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 reference+tolerance with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute reference−tolerance 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

(5/3, 5/3, 0, 10/3, 15, No)

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: 21c2b3c3bb8e67c5107ed3e1c8f1706aee5ae205c4c7fec75d18c1549501a2df
- candidate record SHA-256: 4e3c24ffd1c8de6e3c6f42a57a26d1b7886c102f8ea89a4ea1cbb570407f931d
- semantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861
- question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0098.json)