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R0026-13899 · q-20261014-advance-20261028-r04-real-inner-product-space-0049
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[25, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).
G001 — Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[25, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).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
candidate real R2 form;matrix=[[25, 2], [2, 5]];B(x,y)=x^T*A*y
After
matrix distributivity supplies bilinearity;other inner-product obligations remain separate
Conditions: none
The matrix formula is bilinear without presuming symmetry or positive definiteness.This protocol is real R²,not a complex-conjugate or arbitrary-dimensional claim.
Rule: inner_candidate_bilinear_not_presumed_inner_product · Uses: G001
Before
opposite off-diagonal entries=[2, 2]
After
symmetric=Yes
Conditions: none
Standard basis vectors read off matrix entries,so symmetry is equivalent to equality of the two off-diagonal entries.One pairwise equality cannot establish universal symmetry.
Rule: inner_symmetry_additional_to_bilinearity · Uses: G001, S001
S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.([[25, 2], [2, 5]], [2, 2], Yes)
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.
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