Question 4708: Original Rational-Expression Domain Guard

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0008-04708 · q-20261008-advance-20261011-r03-variable-expression-0083

Original question and givens

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(6)/(x−(0)), excluding x=0. Ordered candidates=[0/1, 8/2, -9/1]. 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)=(6)/(x−(0)), excluding x=0. Ordered candidates=[0/1, 8/2, -9/1]. 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

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: expression_environment_scope · Uses: G001

S002 · calculation

Before

candidate0 original denominator := sub(0, 0)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

x=0;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 dependencies

Rule: original_expression_domain_guard · Uses: G001, S002

S004 · calculation

Before

candidate1 original denominator := sub(4, 0)

After

candidate1 original denominator = 4

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

candidate1 E(x) := div(6, 4)

After

candidate1 E(x) = 3/2

Conditions: none

Evaluate the original expression only with a nonzero denominator.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

x=4;x−b=4

After

legal=Yes;E(x)=3/2

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 dependencies

Rule: original_expression_domain_guard · Uses: G001, S005

S007 · calculation

Before

candidate2 original denominator := sub(-9, 0)

After

candidate2 original denominator = -9

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

candidate2 E(x) := div(6, -9)

After

candidate2 E(x) = -2/3

Conditions: none

Evaluate the original expression only with a nonzero denominator.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · implies

Before

x=-9;x−b=-9

After

legal=Yes;E(x)=-2/3

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 dependencies

Rule: original_expression_domain_guard · Uses: G001, S008

Reverse checks

S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S008 · premise_and_result_audit — Recompute candidate2 E(x) with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute candidate2 original denominator with exact rational arithmetic, not rounded decimals.
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 candidate1 E(x) with exact rational arithmetic, not rounded decimals.
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, Yes, Yes], [undefined, 3/2, -2/3])

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: c07bf517038b026c611b507e6ac27b0df05df57d9e5905750bd79164dd4634fe
candidate record SHA-256: 371090c48811064bdea83a445a7f88e47a90aaf6f1059043d941bc790fb4a38e
semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289
question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67
Lean: NOT_RUN · human review: NOT_PERFORMED

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