Bilinear Evaluation, Symmetry and Strict Sylvester Inner-Product Qualification
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261014-advance-20261028-r04-real-inner-product-space-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[2, -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. x=[-2, -3], y=[-5, 12]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[2, -1], [-1, 2]], [-2, -3], [-5, 12], -43)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[2, -1], [-1, 2]], [-2, -3], [-5, 12], -43).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
2. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
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. x=[-1, -2], y=[-4, 11]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[3, 0], [0, 1]], [-1, -2], [-4, 11], -10)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[3, 0], [0, 1]], [-1, -2], [-4, 11], -10).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
3. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[4, 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. x=[0, -1], y=[-3, 10]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[4, 1], [1, 2]], [0, -1], [-3, 10], -17)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[4, 1], [1, 2]], [0, -1], [-3, 10], -17).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
4. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[5, 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. x=[1, 0], y=[-2, 9]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[5, 2], [2, 5]], [1, 0], [-2, 9], 8)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[5, 2], [2, 5]], [1, 0], [-2, 9], 8).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
5. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[6, -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. x=[2, 1], y=[-1, 8]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[6, -1], [-1, 2]], [2, 1], [-1, 8], -11)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[6, -1], [-1, 2]], [2, 1], [-1, 8], -11).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
6. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[7, 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. x=[-2, 2], y=[0, 7]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[7, 0], [0, 1]], [-2, 2], [0, 7], 14)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[7, 0], [0, 1]], [-2, 2], [0, 7], 14).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
7. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[8, 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. x=[-1, 3], y=[1, 6]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[8, 1], [1, 2]], [-1, 3], [1, 6], 25)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 1], [1, 2]], [-1, 3], [1, 6], 25).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
8. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[9, 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. x=[0, -3], y=[2, 5]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[9, 2], [2, 5]], [0, -3], [2, 5], -87)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[9, 2], [2, 5]], [0, -3], [2, 5], -87).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
9. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[10, -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. x=[1, -2], y=[3, 4]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[10, -1], [-1, 2]], [1, -2], [3, 4], 16)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[10, -1], [-1, 2]], [1, -2], [3, 4], 16).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
10. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[11, 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. x=[2, -1], y=[4, 3]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[11, 0], [0, 1]], [2, -1], [4, 3], 85)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[11, 0], [0, 1]], [2, -1], [4, 3], 85).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
11. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[12, 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. x=[-2, 0], y=[5, 2]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[12, 1], [1, 2]], [-2, 0], [5, 2], -124)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[12, 1], [1, 2]], [-2, 0], [5, 2], -124).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
12. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[13, 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. x=[-1, 1], y=[-5, 1]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[13, 2], [2, 5]], [-1, 1], [-5, 1], 58)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[13, 2], [2, 5]], [-1, 1], [-5, 1], 58).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
13. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
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. x=[0, 2], y=[-4, 0]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[14, -1], [-1, 2]], [0, 2], [-4, 0], 8)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[14, -1], [-1, 2]], [0, 2], [-4, 0], 8).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
14. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 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. x=[1, 3], y=[-3, -1]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[15, 0], [0, 1]], [1, 3], [-3, -1], -48)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 1]], [1, 3], [-3, -1], -48).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
15. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[16, 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. x=[2, -3], y=[-2, -2]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[16, 1], [1, 2]], [2, -3], [-2, -2], -50)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[16, 1], [1, 2]], [2, -3], [-2, -2], -50).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
16. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[17, 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. x=[-2, -2], y=[-1, -3]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[17, 2], [2, 5]], [-2, -2], [-1, -3], 80)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[17, 2], [2, 5]], [-2, -2], [-1, -3], 80).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
17. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[18, -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. x=[-1, -1], y=[0, -4]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[18, -1], [-1, 2]], [-1, -1], [0, -4], 4)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[18, -1], [-1, 2]], [-1, -1], [0, -4], 4).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
18. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
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. x=[0, 0], y=[1, -5]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[19, 0], [0, 1]], [0, 0], [1, -5], 0)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[19, 0], [0, 1]], [0, 0], [1, -5], 0).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
19. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[20, 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. x=[1, 1], y=[2, -6]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[20, 1], [1, 2]], [1, 1], [2, -6], 24)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[20, 1], [1, 2]], [1, 1], [2, -6], 24).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
20. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[21, 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. x=[2, 2], y=[3, -7]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[21, 2], [2, 5]], [2, 2], [3, -7], 40)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[21, 2], [2, 5]], [2, 2], [3, -7], 40).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
21. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[22, -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. x=[-2, 3], y=[4, -8]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[22, -1], [-1, 2]], [-2, 3], [4, -8], -252)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[22, -1], [-1, 2]], [-2, 3], [4, -8], -252).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
22. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[23, 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. x=[-1, -3], y=[5, -9]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[23, 0], [0, 1]], [-1, -3], [5, -9], -88)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[23, 0], [0, 1]], [-1, -3], [5, -9], -88).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
23. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[24, 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. x=[0, -2], y=[-5, -10]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[24, 1], [1, 2]], [0, -2], [-5, -10], 50)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[24, 1], [1, 2]], [0, -2], [-5, -10], 50).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
24. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
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. x=[1, -1], y=[-4, -11]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[25, 2], [2, 5]], [1, -1], [-4, -11], -59)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[25, 2], [2, 5]], [1, -1], [-4, -11], -59).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
25. Evaluate a Candidate Bilinear Form without Presuming an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[26, -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. x=[2, 0], y=[-3, -12]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.
View answer and explanation
([[26, -1], [-1, 2]], [2, 0], [-3, -12], -132)
Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[26, -1], [-1, 2]], [2, 0], [-3, -12], -132).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-evaluate · candidate
26. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[2, -1], [0, 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,[A_01,A_10],symmetric?).
View answer and explanation
([[2, -1], [0, 2]], [-1, 0], No)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[2, -1], [0, 2]], [-1, 0], No).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
27. Symmetry Alone Does Not Establish an Inner Product
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,[A_01,A_10],symmetric?).
View answer and explanation
([[3, 0], [0, 1]], [0, 0], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[3, 0], [0, 1]], [0, 0], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
28. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[4, 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,[A_01,A_10],symmetric?).
View answer and explanation
([[4, 1], [1, 2]], [1, 1], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[4, 1], [1, 2]], [1, 1], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
29. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[5, 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?).
View answer and explanation
([[5, 2], [2, 5]], [2, 2], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[5, 2], [2, 5]], [2, 2], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
30. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[6, -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,[A_01,A_10],symmetric?).
View answer and explanation
([[6, -1], [-1, 2]], [-1, -1], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[6, -1], [-1, 2]], [-1, -1], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
31. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[7, 0], [1, 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,[A_01,A_10],symmetric?).
View answer and explanation
([[7, 0], [1, 1]], [0, 1], No)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[7, 0], [1, 1]], [0, 1], No).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
32. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[8, 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,[A_01,A_10],symmetric?).
View answer and explanation
([[8, 1], [1, 2]], [1, 1], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 1], [1, 2]], [1, 1], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
33. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[9, 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?).
View answer and explanation
([[9, 2], [2, 5]], [2, 2], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[9, 2], [2, 5]], [2, 2], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
34. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[10, -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,[A_01,A_10],symmetric?).
View answer and explanation
([[10, -1], [-1, 2]], [-1, -1], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[10, -1], [-1, 2]], [-1, -1], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
35. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[11, 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,[A_01,A_10],symmetric?).
View answer and explanation
([[11, 0], [0, 1]], [0, 0], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[11, 0], [0, 1]], [0, 0], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
36. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[12, 1], [2, 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,[A_01,A_10],symmetric?).
View answer and explanation
([[12, 1], [2, 2]], [1, 2], No)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[12, 1], [2, 2]], [1, 2], No).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
37. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[13, 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?).
View answer and explanation
([[13, 2], [2, 5]], [2, 2], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[13, 2], [2, 5]], [2, 2], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
38. Symmetry Alone Does Not Establish an Inner Product
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,[A_01,A_10],symmetric?).
View answer and explanation
([[14, -1], [-1, 2]], [-1, -1], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[14, -1], [-1, 2]], [-1, -1], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
39. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 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,[A_01,A_10],symmetric?).
View answer and explanation
([[15, 0], [0, 1]], [0, 0], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 1]], [0, 0], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
40. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[16, 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,[A_01,A_10],symmetric?).
View answer and explanation
([[16, 1], [1, 2]], [1, 1], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[16, 1], [1, 2]], [1, 1], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
41. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[17, 2], [3, 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?).
View answer and explanation
([[17, 2], [3, 5]], [2, 3], No)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[17, 2], [3, 5]], [2, 3], No).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
42. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[18, -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,[A_01,A_10],symmetric?).
View answer and explanation
([[18, -1], [-1, 2]], [-1, -1], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[18, -1], [-1, 2]], [-1, -1], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
43. Symmetry Alone Does Not Establish an Inner Product
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,[A_01,A_10],symmetric?).
View answer and explanation
([[19, 0], [0, 1]], [0, 0], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[19, 0], [0, 1]], [0, 0], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
44. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[20, 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,[A_01,A_10],symmetric?).
View answer and explanation
([[20, 1], [1, 2]], [1, 1], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[20, 1], [1, 2]], [1, 1], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
45. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[21, 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?).
View answer and explanation
([[21, 2], [2, 5]], [2, 2], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[21, 2], [2, 5]], [2, 2], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
46. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[22, -1], [0, 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,[A_01,A_10],symmetric?).
View answer and explanation
([[22, -1], [0, 2]], [-1, 0], No)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[22, -1], [0, 2]], [-1, 0], No).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
47. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[23, 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,[A_01,A_10],symmetric?).
View answer and explanation
([[23, 0], [0, 1]], [0, 0], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[23, 0], [0, 1]], [0, 0], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
48. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[24, 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,[A_01,A_10],symmetric?).
View answer and explanation
([[24, 1], [1, 2]], [1, 1], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[24, 1], [1, 2]], [1, 1], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
49. Symmetry Alone Does Not Establish an Inner Product
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?).
View answer and explanation
([[25, 2], [2, 5]], [2, 2], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[25, 2], [2, 5]], [2, 2], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
50. Symmetry Alone Does Not Establish an Inner Product
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[26, -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,[A_01,A_10],symmetric?).
View answer and explanation
([[26, -1], [-1, 2]], [-1, -1], Yes)
Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[26, -1], [-1, 2]], [-1, -1], Yes).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-symmetry · candidate
51. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[1, 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.
View answer and explanation
([[1, 0], [0, 0]], Yes, 1, 0, No, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[1, 0], [0, 0]], Yes, 1, 0, No, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
52. A Symmetric Sylvester Certificate, Not Positive Sample Values
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.
View answer and explanation
([[3, 0], [0, 1]], Yes, 3, 3, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[3, 0], [0, 1]], Yes, 3, 3, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
53. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[4, 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,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.
View answer and explanation
([[4, 1], [1, 2]], Yes, 4, 7, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[4, 1], [1, 2]], Yes, 4, 7, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
54. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[5, 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,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.
View answer and explanation
([[5, 2], [2, 5]], Yes, 5, 21, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[5, 2], [2, 5]], Yes, 5, 21, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
55. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[6, -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,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.
View answer and explanation
([[6, -1], [-1, 2]], Yes, 6, 11, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[6, -1], [-1, 2]], Yes, 6, 11, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
56. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[7, 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.
View answer and explanation
([[7, 0], [0, 1]], Yes, 7, 7, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[7, 0], [0, 1]], Yes, 7, 7, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
57. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[8, 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,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.
View answer and explanation
([[8, 1], [1, 2]], Yes, 8, 15, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 1], [1, 2]], Yes, 8, 15, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
58. A Symmetric Sylvester Certificate, Not Positive Sample Values
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.
View answer and explanation
([[8, 0], [0, 0]], Yes, 8, 0, No, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 0], [0, 0]], Yes, 8, 0, No, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
59. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[10, -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,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.
View answer and explanation
([[10, -1], [-1, 2]], Yes, 10, 19, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[10, -1], [-1, 2]], Yes, 10, 19, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
60. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[11, 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.
View answer and explanation
([[11, 0], [0, 1]], Yes, 11, 11, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[11, 0], [0, 1]], Yes, 11, 11, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
61. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[12, 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,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.
View answer and explanation
([[12, 1], [1, 2]], Yes, 12, 23, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[12, 1], [1, 2]], Yes, 12, 23, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
62. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[13, 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,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.
View answer and explanation
([[13, 2], [2, 5]], Yes, 13, 61, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[13, 2], [2, 5]], Yes, 13, 61, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
63. A Symmetric Sylvester Certificate, Not Positive Sample Values
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,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.
View answer and explanation
([[14, -1], [-1, 2]], Yes, 14, 27, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[14, -1], [-1, 2]], Yes, 14, 27, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
64. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 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.
View answer and explanation
([[15, 0], [0, 1]], Yes, 15, 15, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 1]], Yes, 15, 15, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
65. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 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.
View answer and explanation
([[15, 0], [0, 0]], Yes, 15, 0, No, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 0]], Yes, 15, 0, No, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
66. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[17, 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,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.
View answer and explanation
([[17, 2], [2, 5]], Yes, 17, 81, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[17, 2], [2, 5]], Yes, 17, 81, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
67. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[18, -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,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.
View answer and explanation
([[18, -1], [-1, 2]], Yes, 18, 35, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[18, -1], [-1, 2]], Yes, 18, 35, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
68. A Symmetric Sylvester Certificate, Not Positive Sample Values
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.
View answer and explanation
([[19, 0], [0, 1]], Yes, 19, 19, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[19, 0], [0, 1]], Yes, 19, 19, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
69. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[20, 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,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.
View answer and explanation
([[20, 1], [1, 2]], Yes, 20, 39, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[20, 1], [1, 2]], Yes, 20, 39, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
70. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[21, 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,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.
View answer and explanation
([[21, 2], [2, 5]], Yes, 21, 101, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[21, 2], [2, 5]], Yes, 21, 101, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
71. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[22, -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,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.
View answer and explanation
([[22, -1], [-1, 2]], Yes, 22, 43, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[22, -1], [-1, 2]], Yes, 22, 43, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
72. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[22, 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.
View answer and explanation
([[22, 0], [0, 0]], Yes, 22, 0, No, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[22, 0], [0, 0]], Yes, 22, 0, No, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
73. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[24, 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,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.
View answer and explanation
([[24, 1], [1, 2]], Yes, 24, 47, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[24, 1], [1, 2]], Yes, 24, 47, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
74. A Symmetric Sylvester Certificate, Not Positive Sample Values
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,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.
View answer and explanation
([[25, 2], [2, 5]], Yes, 25, 121, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[25, 2], [2, 5]], Yes, 25, 121, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
75. A Symmetric Sylvester Certificate, Not Positive Sample Values
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[26, -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,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.
View answer and explanation
([[26, -1], [-1, 2]], Yes, 26, 51, Yes, Sylvester_R2_universal_certificate)
Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[26, -1], [-1, 2]], Yes, 26, 51, Yes, Sylvester_R2_universal_certificate).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-positive-definite · candidate
76. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[1, 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,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.
View answer and explanation
([[1, 0], [0, 0]], Yes, Yes, No, No, not_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[1, 0], [0, 0]], Yes, Yes, No, No, not_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
77. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
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,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.
View answer and explanation
([[3, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[3, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
78. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[4, 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.
View answer and explanation
([[4, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[4, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
79. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[5, 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,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.
View answer and explanation
([[5, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[5, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
80. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[6, -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.
View answer and explanation
([[6, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[6, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
81. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[7, 0], [1, 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,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.
View answer and explanation
([[7, 0], [1, 1]], Yes, No, No, No, not_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[7, 0], [1, 1]], Yes, No, No, No, not_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
82. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[8, 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.
View answer and explanation
([[8, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
83. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
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,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.
View answer and explanation
([[8, 0], [0, 0]], Yes, Yes, No, No, not_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 0], [0, 0]], Yes, Yes, No, No, not_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
84. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[10, -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.
View answer and explanation
([[10, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[10, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
85. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[11, 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,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.
View answer and explanation
([[11, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[11, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
86. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[12, 1], [2, 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.
View answer and explanation
([[12, 1], [2, 2]], Yes, No, No, No, not_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[12, 1], [2, 2]], Yes, No, No, No, not_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
87. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[13, 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,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.
View answer and explanation
([[13, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[13, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
88. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
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.
View answer and explanation
([[14, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[14, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
89. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 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,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.
View answer and explanation
([[15, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
90. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 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,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.
View answer and explanation
([[15, 0], [0, 0]], Yes, Yes, No, No, not_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 0]], Yes, Yes, No, No, not_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
91. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[17, 2], [3, 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,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.
View answer and explanation
([[17, 2], [3, 5]], Yes, No, No, No, not_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[17, 2], [3, 5]], Yes, No, No, No, not_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
92. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[18, -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.
View answer and explanation
([[18, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[18, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
93. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
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,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.
View answer and explanation
([[19, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[19, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
94. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[20, 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.
View answer and explanation
([[20, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[20, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
95. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[21, 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,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.
View answer and explanation
([[21, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[21, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
96. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[22, -1], [0, 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.
View answer and explanation
([[22, -1], [0, 2]], Yes, No, No, No, not_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[22, -1], [0, 2]], Yes, No, No, No, not_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
97. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[22, 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,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.
View answer and explanation
([[22, 0], [0, 0]], Yes, Yes, No, No, not_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[22, 0], [0, 0]], Yes, Yes, No, No, not_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
98. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[24, 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.
View answer and explanation
([[24, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[24, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
99. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
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,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.
View answer and explanation
([[25, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[25, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
100. Conjoin Bilinearity, Symmetry and Strict Positive Definiteness
Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[26, -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.
View answer and explanation
([[26, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)
The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[26, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product).
Reviewed step-by-step derivation
Real inner-product space · qf-mod28-inner-qualification · candidate
Canonical question JSONL · English JSONL · Validation report