# Question 13843: A Symmetric Sylvester Certificate, Not Positive Sample Values

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0026-13843 · q-20261014-advance-20261028-r04-real-inner-product-space-0068



## Original question and givens

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[19, 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=[[19, 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.





## Definition and scope

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.


- [mko-real-inner-product-space](https://ocme.evemisslab.com/en/mko/mko-real-inner-product-space/)





## Step-by-step derivation

S001 · impliesBefore

candidate real R2 form;matrix=[[19, 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.

Step identifiers and dependenciesRule: inner_candidate_bilinear_not_presumed_inner_product · Uses: G001





S002 · impliesBefore

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.

Step identifiers and dependenciesRule: inner_symmetry_additional_to_bilinearity · Uses: G001, S001





S003 · calculationBefore

inner determinant diagonal product := mul(19, 1)

After

inner determinant diagonal product = 19

Conditions: none

Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

inner signed determinant := sub(19, 0)

After

inner signed determinant = 19

Conditions: none

Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

inner completed-square slope := div(0, 19)

After

inner completed-square slope = 0

Conditions: none

Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

inner completed-square remainder := div(19, 19)

After

inner completed-square remainder = 1

Conditions: none

Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · impliesBefore

symmetric matrix=[[19, 0], [0, 1]];positive leading=19;b/a=0;det/a=1

After

q(s,t)=(19)*(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.

Step identifiers and dependenciesRule: R2_Sylvester_completed_square_identity · Uses: G001, S007





S009 · impliesBefore

symmetric=Yes;leading=19;determinant=19

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.

Step identifiers and dependenciesRule: strict_Sylvester_conditions_conjunction · Uses: G001, S008









## Reverse checks


- 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.





## Result

([[19, 0], [0, 1]], Yes, 19, 19, Yes, Sylvester_R2_universal_certificate)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 7746b404a4f75b264bec2fa314f81799649b57fa08b8cf361762b1815f12e9aa
- candidate record SHA-256: a5ce5b40a3fba307b07c78a2ca9b9cbcdc29353dfce3a682d01756ab73672d1b
- semantic review receipt SHA-256: f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798
- question-first witness SHA-256: cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77
- Lean: NOT_RUN · human review: NOT_PERFORMED

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