# Question 29054: A Complex Pair versus a Real-Embedding Candidate

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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.

R0057-29054 · q-20261013-advance-20261022-r03-complex-number-0079



## Original question and givens

Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[-9, 0], real candidate=-8. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).


- G001 — Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[-9, 0], real candidate=-8. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).





## 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-complex-number](https://ocme.evemisslab.com/en/mko/mko-complex-number/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-str22-complex-real-embedding-audit, complex-real-embedding-audit]

After

derived = [[value, real_candidate, is_real_embedding, matches]]

Conditions: none

Use this bank exact ordered tuple and hyphenated operation.Rational outputs reduce with positive denominators and OMIT/1 for integers,unlike the preceding legacy bank.Do not change original bilingual snapshots or meaning.

Step identifiers and dependenciesRule: STR22_exact_interface_and_source_ORDERED_tuple_roles · Uses: G001





S002 · impliesBefore

source roles = [[-9, 0], -8]

After

derived = [Yes, No]

Conditions: none

a+bi is in the real embedding IFF b=0.Matching THIS candidate also requires a=candidate.The first flag may be True while matching is False;the candidate bound is±1e6,not the pair component bound1000.

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









## Reverse checks


- S002 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
- S001 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.





## Result

([-9, 0], -8, Yes, 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: 9eea89edfd008d804917032f32283d3cdc3e622a03cc489924cd81633b355ca3
- candidate record SHA-256: 1fcc043c955396e12dce5151c55ac72c7246023266cf2eb5e26ecb437c8fe8be
- semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6
- question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261022-r03-complex-number-0079.json)