Declared Bivariate Polynomial Coordinates and Domain Slices
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batch-20261013-advance-20261023-r07-multivariable-real-function-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[-12, -2, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-2, y=-3. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([-12, -2, -3, 1, -1, -1], -2, -3, -10)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -1], -2, -3, -10).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
2. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[-11, -1, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-1, y=-2. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([-11, -1, -2, 1, 0, -1], -1, -2, -9)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 1, 0, -1], -1, -2, -9).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
3. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[-10, 0, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=0, y=-1. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([-10, 0, -1, 1, 1, -1], 0, -1, -10)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 1, 1, -1], 0, -1, -10).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
4. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[-9, 1, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=1, y=0. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([-9, 1, 0, 1, -1, -1], 1, 0, -7)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 1, -1, -1], 1, 0, -7).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
5. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[-8, 2, 1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=2, y=1. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([-8, 2, 1, 1, 0, -1], 2, 1, 0)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, -1], 2, 1, 0).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
6. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[-7, -2, 2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-2, y=2. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([-7, -2, 2, 1, 1, -1], -2, 2, -3)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 1, 1, -1], -2, 2, -3).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
7. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[-6, -1, 3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-1, y=3. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([-6, -1, 3, 1, -1, -1], -1, 3, -1)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 1, -1, -1], -1, 3, -1).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
8. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[-5, 0, -3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=0, y=-3. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([-5, 0, -3, 1, 0, -1], 0, -3, -5)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 1, 0, -1], 0, -3, -5).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
9. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[-4, 1, -2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=1, y=-2. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([-4, 1, -2, 1, 1, -1], 1, -2, -4)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, -1], 1, -2, -4).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
10. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[-3, 2, -1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=2, y=-1. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([-3, 2, -1, 1, -1, -1], 2, -1, 7)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 1, -1, -1], 2, -1, 7).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
11. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[-2, -2, 0, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-2, y=0. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([-2, -2, 0, 1, 0, -1], -2, 0, 6)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 1, 0, -1], -2, 0, 6).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
12. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[-1, -1, 1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-1, y=1. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([-1, -1, 1, 1, 1, -1], -1, 1, 0)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 1, 1, -1], -1, 1, 0).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
13. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[0, 0, 2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=0, y=2. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([0, 0, 2, 1, -1, -1], 0, 2, 0)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, -1], 0, 2, 0).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
14. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[1, 1, 3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=1, y=3. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([1, 1, 3, 1, 0, -1], 1, 3, 3)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 1, 0, -1], 1, 3, 3).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
15. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[2, 2, -3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=2, y=-3. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([2, 2, -3, 1, 1, -1], 2, -3, 4)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 1, 1, -1], 2, -3, 4).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
16. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[3, -2, -2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-2, y=-2. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([3, -2, -2, 1, -1, -1], -2, -2, 7)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 1, -1, -1], -2, -2, 7).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
17. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[4, -1, -1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-1, y=-1. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([4, -1, -1, 1, 0, -1], -1, -1, 6)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, -1, 6).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
18. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[5, 0, 0, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=0, y=0. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([5, 0, 0, 1, 1, -1], 0, 0, 5)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 1, 1, -1], 0, 0, 5).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
19. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[6, 1, 1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=1, y=1. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([6, 1, 1, 1, -1, -1], 1, 1, 7)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 1, -1, -1], 1, 1, 7).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
20. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[7, 2, 2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=2, y=2. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([7, 2, 2, 1, 0, -1], 2, 2, 15)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 1, 0, -1], 2, 2, 15).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
21. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[8, -2, 3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-2, y=3. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([8, -2, 3, 1, 1, -1], -2, 3, 10)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -1], -2, 3, 10).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
22. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[9, -1, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-1, y=-3. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([9, -1, -3, 1, -1, -1], -1, -3, 8)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 1, -1, -1], -1, -3, 8).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
23. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[10, 0, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=0, y=-2. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([10, 0, -2, 1, 0, -1], 0, -2, 10)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 1, 0, -1], 0, -2, 10).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
24. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[11, 1, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=1, y=-1. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([11, 1, -1, 1, 1, -1], 1, -1, 12)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 1, 1, -1], 1, -1, 12).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
25. Evaluate with Fixed Bivariate Coefficient Roles
Use the declared real polynomial R²→R with coefficients=[12, 2, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=2, y=0. Give (coefficients,x,y,exact polynomial value).
View answer and explanation
([12, 2, 0, 1, -1, -1], 2, 0, 20)
Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, -1], 2, 0, 20).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-evaluate · candidate
26. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-2, 2, -1, 3] and point [x,y]=[-3, -2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-2, 2, -1, 3], [-3, -2], No)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 2, -1, 3], [-3, -2], No).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
27. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-3, 3, -2, 4] and point [x,y]=[-2, -1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-3, 3, -2, 4], [-2, -1], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 3, -2, 4], [-2, -1], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
28. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-4, 4, -3, 5] and point [x,y]=[-1, 0]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-4, 4, -3, 5], [-1, 0], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 4, -3, 5], [-1, 0], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
29. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-5, 5, -4, 6] and point [x,y]=[0, 1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-5, 5, -4, 6], [0, 1], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 5, -4, 6], [0, 1], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
30. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-6, 6, -5, 7] and point [x,y]=[1, 2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-6, 6, -5, 7], [1, 2], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-6, 6, -5, 7], [1, 2], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
31. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-7, 7, -6, 8] and point [x,y]=[2, -2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-7, 7, -6, 8], [2, -2], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-7, 7, -6, 8], [2, -2], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
32. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-8, 8, -7, 9] and point [x,y]=[3, -1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-8, 8, -7, 9], [3, -1], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 8, -7, 9], [3, -1], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
33. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-9, 9, -8, 10] and point [x,y]=[-3, 0]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-9, 9, -8, 10], [-3, 0], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 9, -8, 10], [-3, 0], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
34. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-10, 10, -9, 11] and point [x,y]=[-2, 1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-10, 10, -9, 11], [-2, 1], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 10, -9, 11], [-2, 1], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
35. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-11, 11, -10, 12] and point [x,y]=[-1, 2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-11, 11, -10, 12], [-1, 2], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-11, 11, -10, 12], [-1, 2], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
36. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-12, 12, -11, 13] and point [x,y]=[0, -2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-12, 12, -11, 13], [0, -2], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-12, 12, -11, 13], [0, -2], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
37. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-13, 13, -12, 14] and point [x,y]=[1, -1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-13, 13, -12, 14], [1, -1], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-13, 13, -12, 14], [1, -1], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
38. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-14, 14, -13, 15] and point [x,y]=[2, 0]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-14, 14, -13, 15], [2, 0], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-14, 14, -13, 15], [2, 0], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
39. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-15, 15, -14, 16] and point [x,y]=[3, 1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-15, 15, -14, 16], [3, 1], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-15, 15, -14, 16], [3, 1], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
40. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-16, 16, -15, 17] and point [x,y]=[-3, 2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-16, 16, -15, 17], [-3, 2], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-16, 16, -15, 17], [-3, 2], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
41. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-17, 17, -16, 18] and point [x,y]=[-2, -2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-17, 17, -16, 18], [-2, -2], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-17, 17, -16, 18], [-2, -2], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
42. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-18, 18, -17, 19] and point [x,y]=[-1, -1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-18, 18, -17, 19], [-1, -1], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-18, 18, -17, 19], [-1, -1], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
43. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-19, 19, -18, 20] and point [x,y]=[0, 0]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-19, 19, -18, 20], [0, 0], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-19, 19, -18, 20], [0, 0], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
44. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-20, 20, -19, 21] and point [x,y]=[1, 1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-20, 20, -19, 21], [1, 1], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-20, 20, -19, 21], [1, 1], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
45. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-21, 21, -20, 22] and point [x,y]=[2, 2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-21, 21, -20, 22], [2, 2], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-21, 21, -20, 22], [2, 2], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
46. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-22, 22, -21, 23] and point [x,y]=[3, -2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-22, 22, -21, 23], [3, -2], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-22, 22, -21, 23], [3, -2], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
47. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-23, 23, -22, 24] and point [x,y]=[-3, -1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-23, 23, -22, 24], [-3, -1], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-23, 23, -22, 24], [-3, -1], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
48. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-24, 24, -23, 25] and point [x,y]=[-2, 0]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-24, 24, -23, 25], [-2, 0], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-24, 24, -23, 25], [-2, 0], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
49. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-25, 25, -24, 26] and point [x,y]=[-1, 1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-25, 25, -24, 26], [-1, 1], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-25, 25, -24, 26], [-1, 1], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
50. Inclusive Membership in a Declared Closed Rectangle
Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-26, 26, -25, 27] and point [x,y]=[0, 2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.
View answer and explanation
([-26, 26, -25, 27], [0, 2], Yes)
Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-26, 26, -25, 27], [0, 2], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-rectangle-domain-audit · candidate
51. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[-12, -2, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-2, x input list=[-1, 0, -3] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([-12, -2, -3, 1, -1, -1], -2, [-3, -1, 0], [[-3, -1], [-1, -9], [0, -10]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -1], -2, [-3, -1, 0], [[-3, -1], [-1, -9], [0, -10]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
52. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[-11, -1, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-1, x input list=[-1, 0, -2] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([-11, -1, -2, 1, 0, -1], -1, [-2, -1, 0], [[-2, -4], [-1, -8], [0, -10]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 1, 0, -1], -1, [-2, -1, 0], [[-2, -4], [-1, -8], [0, -10]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
53. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[-10, 0, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=0, x input list=[-1, 0, -1] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([-10, 0, -1, 1, 1, -1], 0, [-1, 0], [[-1, -9], [0, -10]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 1, 1, -1], 0, [-1, 0], [[-1, -9], [0, -10]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
54. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[-9, 1, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=1, x input list=[-1, 0, 0] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([-9, 1, 0, 1, -1, -1], 1, [-1, 0], [[-1, -9], [0, -10]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 1, -1, -1], 1, [-1, 0], [[-1, -9], [0, -10]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
55. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[-8, 2, 1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=2, x input list=[-1, 0, 1] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([-8, 2, 1, 1, 0, -1], 2, [-1, 0, 1], [[-1, -11], [0, -10], [1, -7]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, -1], 2, [-1, 0, 1], [[-1, -11], [0, -10], [1, -7]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
56. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[-7, -2, 2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-2, x input list=[-1, 0, 2] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([-7, -2, 2, 1, 1, -1], -2, [-1, 0, 2], [[-1, -10], [0, -15], [2, -19]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 1, 1, -1], -2, [-1, 0, 2], [[-1, -10], [0, -15], [2, -19]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
57. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[-6, -1, 3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-1, x input list=[-1, 0, 3] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([-6, -1, 3, 1, -1, -1], -1, [-1, 0, 3], [[-1, -9], [0, -10], [3, -1]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 1, -1, -1], -1, [-1, 0, 3], [[-1, -9], [0, -10], [3, -1]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
58. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[-5, 0, -3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=0, x input list=[-1, 0, -3] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([-5, 0, -3, 1, 0, -1], 0, [-3, -1, 0], [[-3, 4], [-1, -4], [0, -5]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 1, 0, -1], 0, [-3, -1, 0], [[-3, 4], [-1, -4], [0, -5]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
59. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[-4, 1, -2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=1, x input list=[-1, 0, -2] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([-4, 1, -2, 1, 1, -1], 1, [-2, -1, 0], [[-2, -7], [-1, -8], [0, -7]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, -1], 1, [-2, -1, 0], [[-2, -7], [-1, -8], [0, -7]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
60. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[-3, 2, -1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=2, x input list=[-1, 0, -1] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([-3, 2, -1, 1, -1, -1], 2, [-1, 0], [[-1, -8], [0, -9]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 1, -1, -1], 2, [-1, 0], [[-1, -8], [0, -9]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
61. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[-2, -2, 0, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-2, x input list=[-1, 0, 0] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([-2, -2, 0, 1, 0, -1], -2, [-1, 0], [[-1, -3], [0, -6]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 1, 0, -1], -2, [-1, 0], [[-1, -3], [0, -6]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
62. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[-1, -1, 1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-1, x input list=[-1, 0, 1] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([-1, -1, 1, 1, 1, -1], -1, [-1, 0, 1], [[-1, 0], [0, -3], [1, -4]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 1, 1, -1], -1, [-1, 0, 1], [[-1, 0], [0, -3], [1, -4]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
63. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[0, 0, 2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=0, x input list=[-1, 0, 2] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([0, 0, 2, 1, -1, -1], 0, [-1, 0, 2], [[-1, 1], [0, 0], [2, 4]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, -1], 0, [-1, 0, 2], [[-1, 1], [0, 0], [2, 4]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
64. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[1, 1, 3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=1, x input list=[-1, 0, 3] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([1, 1, 3, 1, 0, -1], 1, [-1, 0, 3], [[-1, 3], [0, 3], [3, 15]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 1, 0, -1], 1, [-1, 0, 3], [[-1, 3], [0, 3], [3, 15]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
65. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[2, 2, -3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=2, x input list=[-1, 0, -3] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([2, 2, -3, 1, 1, -1], 2, [-3, -1, 0], [[-3, -11], [-1, -11], [0, -8]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 1, 1, -1], 2, [-3, -1, 0], [[-3, -11], [-1, -11], [0, -8]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
66. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[3, -2, -2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-2, x input list=[-1, 0, -2] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([3, -2, -2, 1, -1, -1], -2, [-2, -1, 0], [[-2, 7], [-1, 4], [0, 3]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 1, -1, -1], -2, [-2, -1, 0], [[-2, 7], [-1, 4], [0, 3]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
67. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[4, -1, -1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-1, x input list=[-1, 0, -1] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([4, -1, -1, 1, 0, -1], -1, [-1, 0], [[-1, 6], [0, 4]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, [-1, 0], [[-1, 6], [0, 4]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
68. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[5, 0, 0, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=0, x input list=[-1, 0, 0] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([5, 0, 0, 1, 1, -1], 0, [-1, 0], [[-1, 6], [0, 5]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 1, 1, -1], 0, [-1, 0], [[-1, 6], [0, 5]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
69. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[6, 1, 1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=1, x input list=[-1, 0, 1] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([6, 1, 1, 1, -1, -1], 1, [-1, 0, 1], [[-1, 7], [0, 6], [1, 7]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 1, -1, -1], 1, [-1, 0, 1], [[-1, 7], [0, 6], [1, 7]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
70. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[7, 2, 2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=2, x input list=[-1, 0, 2] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([7, 2, 2, 1, 0, -1], 2, [-1, 0, 2], [[-1, 6], [0, 7], [2, 15]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 1, 0, -1], 2, [-1, 0, 2], [[-1, 6], [0, 7], [2, 15]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
71. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[8, -2, 3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-2, x input list=[-1, 0, 3] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([8, -2, 3, 1, 1, -1], -2, [-1, 0, 3], [[-1, 3], [0, -2], [3, -5]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -1], -2, [-1, 0, 3], [[-1, 3], [0, -2], [3, -5]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
72. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[9, -1, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-1, x input list=[-1, 0, -3] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([9, -1, -3, 1, -1, -1], -1, [-3, -1, 0], [[-3, 20], [-1, 12], [0, 11]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 1, -1, -1], -1, [-3, -1, 0], [[-3, 20], [-1, 12], [0, 11]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
73. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[10, 0, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=0, x input list=[-1, 0, -2] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([10, 0, -2, 1, 0, -1], 0, [-2, -1, 0], [[-2, 14], [-1, 11], [0, 10]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 1, 0, -1], 0, [-2, -1, 0], [[-2, 14], [-1, 11], [0, 10]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
74. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[11, 1, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=1, x input list=[-1, 0, -1] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([11, 1, -1, 1, 1, -1], 1, [-1, 0], [[-1, 8], [0, 9]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 1, 1, -1], 1, [-1, 0], [[-1, 8], [0, 9]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
75. A Fixed-Second-Coordinate Slice on Listed Inputs Only
Use the declared real polynomial R²→R with coefficients=[12, 2, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=2, x input list=[-1, 0, 0] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.
View answer and explanation
([12, 2, 0, 1, -1, -1], 2, [-1, 0], [[-1, 9], [0, 8]])
Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, -1], 2, [-1, 0], [[-1, 9], [0, 8]]).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-x-slice-table · candidate
76. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[-12, -2, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-12, -13, -15]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([-12, -2, -3, 1, -1, -1], [[0, 0, -12, -12, Yes], [1, 0, -13, -13, Yes], [0, 1, -15, -16, No]], No)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -1], [[0, 0, -12, -12, Yes], [1, 0, -13, -13, Yes], [0, 1, -15, -16, No]], No).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
77. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[-11, -1, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-11, -11, -14]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([-11, -1, -2, 1, 0, -1], [[0, 0, -11, -11, Yes], [1, 0, -11, -11, Yes], [0, 1, -14, -14, Yes]], Yes)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 1, 0, -1], [[0, 0, -11, -11, Yes], [1, 0, -11, -11, Yes], [0, 1, -14, -14, Yes]], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
78. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[-10, 0, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-10, -9, -12]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([-10, 0, -1, 1, 1, -1], [[0, 0, -10, -10, Yes], [1, 0, -9, -9, Yes], [0, 1, -12, -12, Yes]], Yes)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 1, 1, -1], [[0, 0, -10, -10, Yes], [1, 0, -9, -9, Yes], [0, 1, -12, -12, Yes]], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
79. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[-9, 1, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-9, -7, -10]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([-9, 1, 0, 1, -1, -1], [[0, 0, -9, -9, Yes], [1, 0, -7, -7, Yes], [0, 1, -10, -10, Yes]], Yes)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 1, -1, -1], [[0, 0, -9, -9, Yes], [1, 0, -7, -7, Yes], [0, 1, -10, -10, Yes]], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
80. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[-8, 2, 1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-8, -5, -7]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([-8, 2, 1, 1, 0, -1], [[0, 0, -8, -8, Yes], [1, 0, -5, -5, Yes], [0, 1, -7, -8, No]], No)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, -1], [[0, 0, -8, -8, Yes], [1, 0, -5, -5, Yes], [0, 1, -7, -8, No]], No).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
81. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[-7, -2, 2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-7, -8, -6]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([-7, -2, 2, 1, 1, -1], [[0, 0, -7, -7, Yes], [1, 0, -8, -8, Yes], [0, 1, -6, -6, Yes]], Yes)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 1, 1, -1], [[0, 0, -7, -7, Yes], [1, 0, -8, -8, Yes], [0, 1, -6, -6, Yes]], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
82. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[-6, -1, 3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-6, -6, -4]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([-6, -1, 3, 1, -1, -1], [[0, 0, -6, -6, Yes], [1, 0, -6, -6, Yes], [0, 1, -4, -4, Yes]], Yes)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 1, -1, -1], [[0, 0, -6, -6, Yes], [1, 0, -6, -6, Yes], [0, 1, -4, -4, Yes]], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
83. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[-5, 0, -3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-5, -4, -9]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([-5, 0, -3, 1, 0, -1], [[0, 0, -5, -5, Yes], [1, 0, -4, -4, Yes], [0, 1, -9, -9, Yes]], Yes)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 1, 0, -1], [[0, 0, -5, -5, Yes], [1, 0, -4, -4, Yes], [0, 1, -9, -9, Yes]], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
84. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[-4, 1, -2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-4, -2, -6]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([-4, 1, -2, 1, 1, -1], [[0, 0, -4, -4, Yes], [1, 0, -2, -2, Yes], [0, 1, -6, -7, No]], No)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, -1], [[0, 0, -4, -4, Yes], [1, 0, -2, -2, Yes], [0, 1, -6, -7, No]], No).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
85. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[-3, 2, -1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-3, 0, -5]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([-3, 2, -1, 1, -1, -1], [[0, 0, -3, -3, Yes], [1, 0, 0, 0, Yes], [0, 1, -5, -5, Yes]], Yes)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 1, -1, -1], [[0, 0, -3, -3, Yes], [1, 0, 0, 0, Yes], [0, 1, -5, -5, Yes]], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
86. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[-2, -2, 0, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-2, -3, -3]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([-2, -2, 0, 1, 0, -1], [[0, 0, -2, -2, Yes], [1, 0, -3, -3, Yes], [0, 1, -3, -3, Yes]], Yes)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 1, 0, -1], [[0, 0, -2, -2, Yes], [1, 0, -3, -3, Yes], [0, 1, -3, -3, Yes]], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
87. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[-1, -1, 1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-1, -1, -1]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([-1, -1, 1, 1, 1, -1], [[0, 0, -1, -1, Yes], [1, 0, -1, -1, Yes], [0, 1, -1, -1, Yes]], Yes)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 1, 1, -1], [[0, 0, -1, -1, Yes], [1, 0, -1, -1, Yes], [0, 1, -1, -1, Yes]], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
88. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[0, 0, 2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[0, 1, 2]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([0, 0, 2, 1, -1, -1], [[0, 0, 0, 0, Yes], [1, 0, 1, 1, Yes], [0, 1, 2, 1, No]], No)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, -1], [[0, 0, 0, 0, Yes], [1, 0, 1, 1, Yes], [0, 1, 2, 1, No]], No).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
89. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[1, 1, 3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[1, 3, 3]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([1, 1, 3, 1, 0, -1], [[0, 0, 1, 1, Yes], [1, 0, 3, 3, Yes], [0, 1, 3, 3, Yes]], Yes)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 1, 0, -1], [[0, 0, 1, 1, Yes], [1, 0, 3, 3, Yes], [0, 1, 3, 3, Yes]], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
90. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[2, 2, -3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[2, 5, -2]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([2, 2, -3, 1, 1, -1], [[0, 0, 2, 2, Yes], [1, 0, 5, 5, Yes], [0, 1, -2, -2, Yes]], Yes)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 1, 1, -1], [[0, 0, 2, 2, Yes], [1, 0, 5, 5, Yes], [0, 1, -2, -2, Yes]], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
91. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[3, -2, -2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[3, 2, 0]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([3, -2, -2, 1, -1, -1], [[0, 0, 3, 3, Yes], [1, 0, 2, 2, Yes], [0, 1, 0, 0, Yes]], Yes)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 1, -1, -1], [[0, 0, 3, 3, Yes], [1, 0, 2, 2, Yes], [0, 1, 0, 0, Yes]], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
92. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[4, -1, -1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[4, 4, 3]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([4, -1, -1, 1, 0, -1], [[0, 0, 4, 4, Yes], [1, 0, 4, 4, Yes], [0, 1, 3, 2, No]], No)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], [[0, 0, 4, 4, Yes], [1, 0, 4, 4, Yes], [0, 1, 3, 2, No]], No).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
93. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[5, 0, 0, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[5, 6, 4]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([5, 0, 0, 1, 1, -1], [[0, 0, 5, 5, Yes], [1, 0, 6, 6, Yes], [0, 1, 4, 4, Yes]], Yes)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 1, 1, -1], [[0, 0, 5, 5, Yes], [1, 0, 6, 6, Yes], [0, 1, 4, 4, Yes]], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
94. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[6, 1, 1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[6, 8, 6]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([6, 1, 1, 1, -1, -1], [[0, 0, 6, 6, Yes], [1, 0, 8, 8, Yes], [0, 1, 6, 6, Yes]], Yes)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 1, -1, -1], [[0, 0, 6, 6, Yes], [1, 0, 8, 8, Yes], [0, 1, 6, 6, Yes]], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
95. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[7, 2, 2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[7, 10, 8]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([7, 2, 2, 1, 0, -1], [[0, 0, 7, 7, Yes], [1, 0, 10, 10, Yes], [0, 1, 8, 8, Yes]], Yes)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 1, 0, -1], [[0, 0, 7, 7, Yes], [1, 0, 10, 10, Yes], [0, 1, 8, 8, Yes]], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
96. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[8, -2, 3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[8, 7, 11]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([8, -2, 3, 1, 1, -1], [[0, 0, 8, 8, Yes], [1, 0, 7, 7, Yes], [0, 1, 11, 10, No]], No)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -1], [[0, 0, 8, 8, Yes], [1, 0, 7, 7, Yes], [0, 1, 11, 10, No]], No).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
97. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[9, -1, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[9, 9, 5]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([9, -1, -3, 1, -1, -1], [[0, 0, 9, 9, Yes], [1, 0, 9, 9, Yes], [0, 1, 5, 5, Yes]], Yes)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 1, -1, -1], [[0, 0, 9, 9, Yes], [1, 0, 9, 9, Yes], [0, 1, 5, 5, Yes]], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
98. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[10, 0, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[10, 11, 7]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([10, 0, -2, 1, 0, -1], [[0, 0, 10, 10, Yes], [1, 0, 11, 11, Yes], [0, 1, 7, 7, Yes]], Yes)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 1, 0, -1], [[0, 0, 10, 10, Yes], [1, 0, 11, 11, Yes], [0, 1, 7, 7, Yes]], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
99. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[11, 1, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[11, 13, 9]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([11, 1, -1, 1, 1, -1], [[0, 0, 11, 11, Yes], [1, 0, 13, 13, Yes], [0, 1, 9, 9, Yes]], Yes)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 1, 1, -1], [[0, 0, 11, 11, Yes], [1, 0, 13, 13, Yes], [0, 1, 9, 9, Yes]], Yes).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
100. Preserve Original Point-Value Pairing in a Finite Audit
Use the declared real polynomial R²→R with coefficients=[12, 2, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[12, 15, 12]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.
View answer and explanation
([12, 2, 0, 1, -1, -1], [[0, 0, 12, 12, Yes], [1, 0, 15, 15, Yes], [0, 1, 12, 11, No]], No)
Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, -1], [[0, 0, 12, 12, Yes], [1, 0, 15, 15, Yes], [0, 1, 12, 11, No]], No).
Reviewed step-by-step derivation
Multivariable real-valued function · qf-mod23-multivar-point-table-audit · candidate
Canonical question JSONL · English JSONL · Validation report