Question 10696: A Fixed-Second-Coordinate Slice on Listed Inputs Only

Original question and givens

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.

• G001 — 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.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-multivariable-real-function

Step-by-step derivation

original six coefficients=[8, -2, 3, 1, 1, -1]

declared scalar R2->R polynomial;finite audits keep coordinate/value roles

fixed second coordinate y=-2;raw first-coordinate list=[-1, 0, 3]

sorted distinct first coordinates=[-1, 0, 3]

slice x-1 x squared := mul(-1, -1)

slice x-1 x squared = 1

slice x-1 x times y := mul(-1, -2)

slice x-1 x times y = 2

slice x-1 y squared := mul(-2, -2)

slice x-1 y squared = 4

slice x-1:x=-1;y=-2;coefficients=[8, -2, 3, 1, 1, -1]

fixed monomial values [1,x,y,x^2,xy,y^2]=[1, -1, -2, 1, 2, 4]

slice x-1 monomial term0 := mul(8, 1)

slice x-1 monomial term0 = 8

slice x-1 monomial term1 := mul(-2, -1)

slice x-1 monomial term1 = 2

slice x-1 monomial term2 := mul(3, -2)

slice x-1 monomial term2 = -6

slice x-1 monomial term3 := mul(1, 1)

slice x-1 monomial term3 = 1

slice x-1 monomial term4 := mul(1, 2)

slice x-1 monomial term4 = 2

slice x-1 monomial term5 := mul(-1, 4)

slice x-1 monomial term5 = -4

slice x-1 complete polynomial sum := sum(8, 2, -6, 1, 2, -4)

slice x-1 complete polynomial sum = 3

slice x0 x squared := mul(0, 0)

slice x0 x squared = 0

slice x0 x times y := mul(0, -2)

slice x0 x times y = 0

slice x0 y squared := mul(-2, -2)

slice x0 y squared = 4

slice x0:x=0;y=-2;coefficients=[8, -2, 3, 1, 1, -1]

fixed monomial values [1,x,y,x^2,xy,y^2]=[1, 0, -2, 0, 0, 4]

slice x0 monomial term0 := mul(8, 1)

slice x0 monomial term0 = 8

slice x0 monomial term1 := mul(-2, 0)

slice x0 monomial term1 = 0

slice x0 monomial term2 := mul(3, -2)

slice x0 monomial term2 = -6

slice x0 monomial term3 := mul(1, 0)

slice x0 monomial term3 = 0

slice x0 monomial term4 := mul(1, 0)

slice x0 monomial term4 = 0

slice x0 monomial term5 := mul(-1, 4)

slice x0 monomial term5 = -4

slice x0 complete polynomial sum := sum(8, 0, -6, 0, 0, -4)

slice x0 complete polynomial sum = -2

slice x3 x squared := mul(3, 3)

slice x3 x squared = 9

slice x3 x times y := mul(3, -2)

slice x3 x times y = -6

slice x3 y squared := mul(-2, -2)

slice x3 y squared = 4

slice x3:x=3;y=-2;coefficients=[8, -2, 3, 1, 1, -1]

fixed monomial values [1,x,y,x^2,xy,y^2]=[1, 3, -2, 9, -6, 4]

slice x3 monomial term0 := mul(8, 1)

slice x3 monomial term0 = 8

slice x3 monomial term1 := mul(-2, 3)

slice x3 monomial term1 = -6

slice x3 monomial term2 := mul(3, -2)

slice x3 monomial term2 = -6

slice x3 monomial term3 := mul(1, 9)

slice x3 monomial term3 = 9

slice x3 monomial term4 := mul(1, -6)

slice x3 monomial term4 = -6

slice x3 monomial term5 := mul(-1, 4)

slice x3 monomial term5 = -4

slice x3 complete polynomial sum := sum(8, -6, -6, 9, -6, -4)

slice x3 complete polynomial sum = -5

Reverse checks

• S035 · premise_and_result_audit — Recompute slice x3 complete polynomial sum with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute slice x3 monomial term5 with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute slice x3 monomial term4 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute slice x3 monomial term3 with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute slice x3 monomial term2 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute slice x3 monomial term1 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute slice x3 monomial term0 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Recompute slice x3 y squared with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute slice x3 x times y with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute slice x3 x squared with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute slice x0 complete polynomial sum with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute slice x0 monomial term5 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute slice x0 monomial term4 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute slice x0 monomial term3 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute slice x0 monomial term2 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute slice x0 monomial term1 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute slice x0 monomial term0 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Recompute slice x0 y squared with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute slice x0 x times y with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute slice x0 x squared with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute slice x-1 complete polynomial sum with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute slice x-1 monomial term5 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute slice x-1 monomial term4 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute slice x-1 monomial term3 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute slice x-1 monomial term2 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute slice x-1 monomial term1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute slice x-1 monomial term0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute slice x-1 y squared with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute slice x-1 x times y with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute slice x-1 x squared with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([8, -2, 3, 1, 1, -1], -2, [-1, 0, 3], [[-1, 3], [0, -2], [3, -5]])

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

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.