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R0019-10301 · q-20261013-advance-20261023-r01-complex-modulus-conjugate-0001
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-12, -3]. Give (value,conjugate).
G001 — Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-12, -3]. Give (value,conjugate).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.
Before
declared complex ordered pair=[-12, -3]
After
exact integer coefficients in the algebra with i^2=-1
Conditions: none
Use the declared complex algebra,not componentwise real-vector multiplication.The zero complex number is allowed.
Rule: exact_complex_pair_scope · Uses: G001
Before
given value negate imaginary coefficient := sub(0, -3)
After
given value negate imaginary coefficient = 3
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
given value:original ordered pair=[-12, -3]
After
conjugate pair=[-12, 3]
Conditions: none
The pair(a,b) denotes a+bi.Conjugation fixes the real part and negates only b;it does not exchange coordinates.
Rule: complex_conjugate_real_fixed_imaginary_negated · Uses: G001, S002
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 given value negate imaginary coefficient 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.([-12, -3], [-12, 3])
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 21c41f444f135aabe4578d1fd3e5a29fd399fe6b6af0dcfa08ab49c638e7b6a0candidate record SHA-256: 445e381a1d925db105d53330d27256ce4c4c42145041388c4dea6ede71bd21fbsemantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419Lean: NOT_RUN · human review: NOT_PERFORMED