# Question 4702: Original Rational-Expression Domain Guard

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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.

R0008-04702 · q-20261008-advance-20261011-r03-variable-expression-0077



## Original question and givens

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(5)/(x−(-2)), excluding x=-2. Ordered candidates=[-2/1, -2/1, -2/1]. Give (three domain-eligibility values, three E values or undefined).


- G001 — All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(5)/(x−(-2)), excluding x=-2. Ordered candidates=[-2/1, -2/1, -2/1]. Give (three domain-eligibility values, three E values or undefined).





## 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-variable-expression](https://ocme.evemisslab.com/en/mko/mko-variable-expression/)





## Step-by-step derivation

S001 · impliesBefore

typed rational expression and explicitly given environment

After

evaluation,not solving an equation

Conditions: none

Evaluating an expression is not solving an equation;substitute only the supplied environment without inventing roots or changing bindings.

Step identifiers and dependenciesRule: expression_environment_scope · Uses: G001





S002 · calculationBefore

candidate0 original denominator := sub(-2, -2)

After

candidate0 original denominator = 0

Conditions: none

Evaluate the original denominator x−b first;do not cancel a hole just because the numerator is0.

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





S003 · impliesBefore

x=-2;x−b=0

After

legal=No;E(x)=undefined

Conditions: none

Eligibility comes from the original expression's domain;even a=0 does not legalize the excluded point,and its value remains undefined.

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





S004 · calculationBefore

candidate1 original denominator := sub(-2, -2)

After

candidate1 original denominator = 0

Conditions: none

Evaluate the original denominator x−b first;do not cancel a hole just because the numerator is0.

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





S005 · impliesBefore

x=-2;x−b=0

After

legal=No;E(x)=undefined

Conditions: none

Eligibility comes from the original expression's domain;even a=0 does not legalize the excluded point,and its value remains undefined.

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





S006 · calculationBefore

candidate2 original denominator := sub(-2, -2)

After

candidate2 original denominator = 0

Conditions: none

Evaluate the original denominator x−b first;do not cancel a hole just because the numerator is0.

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





S007 · impliesBefore

x=-2;x−b=0

After

legal=No;E(x)=undefined

Conditions: none

Eligibility comes from the original expression's domain;even a=0 does not legalize the excluded point,and its value remains undefined.

Step identifiers and dependenciesRule: original_expression_domain_guard · Uses: G001, S006









## Reverse checks


- S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S006 · premise_and_result_audit — Recompute candidate2 original denominator with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Recompute candidate1 original denominator with exact rational arithmetic, not rounded decimals.
- 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 — Recompute candidate0 original denominator 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

([No, No, No], [undefined, undefined, undefined])

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: 23d7afab8cc0a4f7b8935ce5ea1a5a1989c475ada08c8ad3f25fe6430a4690fe
- candidate record SHA-256: a68d81e0bdef746d826e07dc36e53be99ec07c625a523ef74c503055fb4ffafe
- semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289
- question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261011-r03-variable-expression-0077.json)