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R0026-13833 · q-20261014-advance-20261028-r04-real-inner-product-space-0058
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[8, 0], [0, 0]]. 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=[[8, 0], [0, 0]]. 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=[[8, 0], [0, 0]];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(8, 0)
After
inner determinant 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, 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(0, 0)
After
inner signed determinant = 0
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, 8)
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(0, 8)
After
inner completed-square remainder = 0
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=[[8, 0], [0, 0]];positive leading=8;b/a=0;det/a=0
After
q(s,t)=(8)*(s+(0)*t)^2+(0)*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
nonpositive determinant witness matrix second slot:matrix=[[8, 0], [0, 0]];input=[0, 8]
After
input dimension=2;output dimension=2;rows dot input
Conditions: none
Retain column-input/row-output roles for rectangular matrices.Computed values can exceed raw input bounds and are not clipped.
Rule: matrix_row_column_roles · Uses: G001, S008
Before
nonpositive determinant witness matrix second slot row0 column0 product := mul(8, 0)
After
nonpositive determinant witness matrix second slot row0 column0 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, S009
Before
nonpositive determinant witness matrix second slot row0 column1 product := mul(0, 8)
After
nonpositive determinant witness matrix second slot row0 column1 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, S010
Before
nonpositive determinant witness matrix second slot row0 sum := sum(0, 0)
After
nonpositive determinant witness matrix second slot row0 sum = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
nonpositive determinant witness matrix second slot row1 column0 product := mul(0, 0)
After
nonpositive determinant witness matrix second slot row1 column0 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, S012
Before
nonpositive determinant witness matrix second slot row1 column1 product := mul(0, 8)
After
nonpositive determinant witness matrix second slot row1 column1 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, S013
Before
nonpositive determinant witness matrix second slot row1 sum := sum(0, 0)
After
nonpositive determinant witness matrix second slot row1 sum = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
nonpositive determinant witness first coordinate0 product := mul(0, 0)
After
nonpositive determinant witness first coordinate0 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, S015
Before
nonpositive determinant witness first coordinate1 product := mul(8, 0)
After
nonpositive determinant witness first coordinate1 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, S016
Before
nonpositive determinant witness bilinear sum := sum(0, 0)
After
nonpositive determinant witness bilinear sum = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
nonzero witness=[0, 8];q(witness)=0=leading*determinant
After
strict positivity fails;zero determinant is not strict positive definiteness
Conditions: none
Positive a makes this vector nonzero;det≤0 gives a nonpositive value,including the zero-value failure of semidefiniteness.Positive samples do not repair it.
Rule: nonpositive_determinant_actual_countervector · Uses: G001, S018
Before
symmetric=Yes;leading=8;determinant=0
After
strict symmetric R2 certificate=No
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, S019
S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S018 · premise_and_result_audit — Recompute nonpositive determinant witness bilinear sum with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute nonpositive determinant witness first coordinate1 product with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute nonpositive determinant witness first coordinate0 product with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute nonpositive determinant witness matrix second slot row1 sum with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute nonpositive determinant witness matrix second slot row1 column1 product with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute nonpositive determinant witness matrix second slot row1 column0 product with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute nonpositive determinant witness matrix second slot row0 sum with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute nonpositive determinant witness matrix second slot row0 column1 product with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute nonpositive determinant witness matrix second slot row0 column0 product with exact rational arithmetic, not rounded decimals.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.([[8, 0], [0, 0]], Yes, 8, 0, No, 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: 03a1015169e991c21a125c43fd5316d1bdebffa00efbb19eda525d10af69ad20candidate record SHA-256: 9427b5ff31c7d269e1701caeff9ae4812735670ea28b2955f231020eb701b01bsemantic review receipt SHA-256: f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798question-first witness SHA-256: cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77Lean: NOT_RUN · human review: NOT_PERFORMED