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R0026-13863 · q-20261014-advance-20261028-r04-real-inner-product-space-0088
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[14, -1], [-1, 2]]. 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,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.
G001 — Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[14, -1], [-1, 2]]. 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,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.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=[[14, -1], [-1, 2]];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=[-1, -1]
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
Before
inner determinant diagonal product := mul(14, 2)
After
inner determinant diagonal product = 28
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
inner determinant off-diagonal product := mul(-1, -1)
After
inner determinant off-diagonal product = 1
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
inner signed determinant := sub(28, 1)
After
inner signed determinant = 27
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
inner completed-square slope := div(-1, 14)
After
inner completed-square slope = -1/14
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
inner completed-square remainder := div(27, 14)
After
inner completed-square remainder = 27/14
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
symmetric matrix=[[14, -1], [-1, 2]];positive leading=14;b/a=-1/14;det/a=27/14
After
q(s,t)=(14)*(s+(-1/14)*t)^2+(27/14)*t^2
Conditions: none
Complete the square only after a>0.If det>0,the second term is positive for t≠0;when t=0 and s≠0,the first is positive.Conversely strict positivity first gives a>0 at e1,then det/a>0 at s=−b/a,t=1.
Rule: R2_Sylvester_completed_square_identity · Uses: G001, S007
Before
symmetric=Yes;leading=14;determinant=27
After
strict symmetric R2 certificate=Yes
Conditions: none
Conjoin symmetry,positive leading minor and positive determinant.This is the stated symmetric R² inner-product protocol,with no additional completeness claim.
Rule: strict_Sylvester_conditions_conjunction · Uses: G001, S008
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Recompute inner completed-square remainder with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute inner completed-square slope with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute inner signed determinant with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute inner determinant off-diagonal product with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute inner determinant diagonal product with exact rational arithmetic, not rounded decimals.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.([[14, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)
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: 66f4bf47dfc44aaf6d99db33a4a065f37597dca1683d4a1ab8a3c6f20aa3128bcandidate record SHA-256: 5da5c26a52dff780d7606f0ee12992bda2020450ba15c7fb2a882766f63983ecsemantic review receipt SHA-256: f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798question-first witness SHA-256: cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77Lean: NOT_RUN · human review: NOT_PERFORMED