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R0026-13827 · q-20261014-advance-20261028-r04-real-inner-product-space-0052
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[3, 0], [0, 1]]. 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,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.
G001 — Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[3, 0], [0, 1]]. 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,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.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=[[3, 0], [0, 1]];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=[0, 0]
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(3, 1)
After
inner determinant diagonal product = 3
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(0, 0)
After
inner determinant off-diagonal product = 0
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(3, 0)
After
inner signed determinant = 3
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(0, 3)
After
inner completed-square slope = 0
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(3, 3)
After
inner completed-square remainder = 1
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=[[3, 0], [0, 1]];positive leading=3;b/a=0;det/a=1
After
q(s,t)=(3)*(s+(0)*t)^2+(1)*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=3;determinant=3
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.([[3, 0], [0, 1]], Yes, 3, 3, Yes, Sylvester_R2_universal_certificate)
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: 0f2325ff42948b5fcde5f6b493557bab9bce9b1597386706e94a0eaada89b7efcandidate record SHA-256: fa51de0a2924627db70d9ad10c0d21ca0f8f25465580c7406e965b738c453099semantic review receipt SHA-256: f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798question-first witness SHA-256: cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77Lean: NOT_RUN · human review: NOT_PERFORMED