# Question 10677: A Fixed-Second-Coordinate Slice on Listed Inputs Only [Return to catalogue page 54](https://ocme.evemisslab.com/en/questions/page/54/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261023-r07-multivariable-real-function-foundations/#q-20261013-advance-20261023-r07-multivariable-real-function-0052) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0020-10677 · q-20261013-advance-20261023-r07-multivariable-real-function-0052 ## Original question and givens 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. - G001 — 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. ## 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](https://ocme.evemisslab.com/en/mko/mko-multivariable-real-function/) ## Step-by-step derivation S001 · impliesBefore original six coefficients=[-11, -1, -2, 1, 0, -1] After declared scalar R2->R polynomial;finite audits keep coordinate/value roles Conditions: none Evaluate the declared formula exactly.A finite point match is not a new general formal proof of global equivalence,continuity or differentiability. Step identifiers and dependenciesRule: bivariate_explicit_slice_scope · Uses: G001 S002 · impliesBefore fixed second coordinate y=-1;raw first-coordinate list=[-1, 0, -2] After sorted distinct first coordinates=[-2, -1, 0] Conditions: none Fix y and normalize only the supplied x list.Do not invent points for an empty list or claim a finite slice determines the whole function. Step identifiers and dependenciesRule: slice_fixed_second_coordinate_not_global_curve · Uses: G001, S001 S003 · calculationBefore slice x-2 x squared := mul(-2, -2) After slice x-2 x squared = 4 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore slice x-2 x times y := mul(-2, -1) After slice x-2 x times y = 2 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore slice x-2 y squared := mul(-1, -1) After slice x-2 y squared = 1 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore slice x-2:x=-2;y=-1;coefficients=[-11, -1, -2, 1, 0, -1] After fixed monomial values [1,x,y,x^2,xy,y^2]=[1, -2, -1, 4, 2, 1] Conditions: none The six coefficients correspond to[1,x,y,x²,xy,y²] in that order.Do not exchange x and y or reinterpret the scalar function as vector-valued. Step identifiers and dependenciesRule: bivariate_fixed_coordinate_monomial_roles · Uses: G001, S005 S007 · calculationBefore slice x-2 monomial term0 := mul(-11, 1) After slice x-2 monomial term0 = -11 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore slice x-2 monomial term1 := mul(-1, -2) After slice x-2 monomial term1 = 2 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore slice x-2 monomial term2 := mul(-2, -1) After slice x-2 monomial term2 = 2 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore slice x-2 monomial term3 := mul(1, 4) After slice x-2 monomial term3 = 4 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore slice x-2 monomial term4 := mul(0, 2) After slice x-2 monomial term4 = 0 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore slice x-2 monomial term5 := mul(-1, 1) After slice x-2 monomial term5 = -1 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore slice x-2 complete polynomial sum := sum(-11, 2, 2, 4, 0, -1) After slice x-2 complete polynomial sum = -4 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore slice x-1 x squared := mul(-1, -1) After slice x-1 x squared = 1 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore slice x-1 x times y := mul(-1, -1) After slice x-1 x times y = 1 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore slice x-1 y squared := mul(-1, -1) After slice x-1 y squared = 1 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore slice x-1:x=-1;y=-1;coefficients=[-11, -1, -2, 1, 0, -1] After fixed monomial values [1,x,y,x^2,xy,y^2]=[1, -1, -1, 1, 1, 1] Conditions: none The six coefficients correspond to[1,x,y,x²,xy,y²] in that order.Do not exchange x and y or reinterpret the scalar function as vector-valued. Step identifiers and dependenciesRule: bivariate_fixed_coordinate_monomial_roles · Uses: G001, S016 S018 · calculationBefore slice x-1 monomial term0 := mul(-11, 1) After slice x-1 monomial term0 = -11 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore slice x-1 monomial term1 := mul(-1, -1) After slice x-1 monomial term1 = 1 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore slice x-1 monomial term2 := mul(-2, -1) After slice x-1 monomial term2 = 2 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore slice x-1 monomial term3 := mul(1, 1) After slice x-1 monomial term3 = 1 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore slice x-1 monomial term4 := mul(0, 1) After slice x-1 monomial term4 = 0 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore slice x-1 monomial term5 := mul(-1, 1) After slice x-1 monomial term5 = -1 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore slice x-1 complete polynomial sum := sum(-11, 1, 2, 1, 0, -1) After slice x-1 complete polynomial sum = -8 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore slice x0 x squared := mul(0, 0) After slice x0 x squared = 0 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore slice x0 x times y := mul(0, -1) After slice x0 x times y = 0 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore slice x0 y squared := mul(-1, -1) After slice x0 y squared = 1 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · impliesBefore slice x0:x=0;y=-1;coefficients=[-11, -1, -2, 1, 0, -1] After fixed monomial values [1,x,y,x^2,xy,y^2]=[1, 0, -1, 0, 0, 1] Conditions: none The six coefficients correspond to[1,x,y,x²,xy,y²] in that order.Do not exchange x and y or reinterpret the scalar function as vector-valued. Step identifiers and dependenciesRule: bivariate_fixed_coordinate_monomial_roles · Uses: G001, S027 S029 · calculationBefore slice x0 monomial term0 := mul(-11, 1) After slice x0 monomial term0 = -11 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore slice x0 monomial term1 := mul(-1, 0) After slice x0 monomial term1 = 0 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore slice x0 monomial term2 := mul(-2, -1) After slice x0 monomial term2 = 2 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore slice x0 monomial term3 := mul(1, 0) After slice x0 monomial term3 = 0 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore slice x0 monomial term4 := mul(0, 0) After slice x0 monomial term4 = 0 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore slice x0 monomial term5 := mul(-1, 1) After slice x0 monomial term5 = -1 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore slice x0 complete polynomial sum := sum(-11, 0, 2, 0, 0, -1) After slice x0 complete polynomial sum = -10 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 ## Reverse checks - S035 · premise_and_result_audit — Recompute slice x0 complete polynomial sum with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute slice x0 monomial term5 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute slice x0 monomial term4 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute slice x0 monomial term3 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute slice x0 monomial term2 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute slice x0 monomial term1 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute slice x0 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 x0 y squared with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute slice x0 x times y with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute slice x0 x squared with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute slice x-1 complete polynomial sum with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute slice x-1 monomial term5 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute slice x-1 monomial term4 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute slice x-1 monomial term3 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute slice x-1 monomial term2 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute slice x-1 monomial term1 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute slice x-1 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 x-1 y squared with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute slice x-1 x times y with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute slice x-1 x squared with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute slice x-2 complete polynomial sum with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute slice x-2 monomial term5 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute slice x-2 monomial term4 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute slice x-2 monomial term3 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute slice x-2 monomial term2 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute slice x-2 monomial term1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute slice x-2 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-2 y squared with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute slice x-2 x times y with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute slice x-2 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 ([-11, -1, -2, 1, 0, -1], -1, [-2, -1, 0], [[-2, -4], [-1, -8], [0, -10]]) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 776efafc6d1b7789f683afaf126d4aad76d10c5b8a4ca1dc5f4edeace0a9f6de - candidate record SHA-256: 5bbd6c31f16f596c52ebe5d19261455680ece0f312d6fe8869d847dee9b66399 - semantic review receipt SHA-256: b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598f - question-first witness SHA-256: 3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261023-r07-multivariable-real-function-0052.json)