# Question 10632: Preserve Original Point-Value Pairing in a Finite Audit [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-0082) 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. R0020-10632 · q-20261013-advance-20261023-r07-multivariable-real-function-0082 ## Original question and givens 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. - G001 — 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. ## 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=[-6, -1, 3, 1, -1, -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 · calculationBefore original point row0 x squared := mul(0, 0) After original point row0 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, S001 S003 · calculationBefore original point row0 x times y := mul(0, 0) After original point row0 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, S002 S004 · calculationBefore original point row0 y squared := mul(0, 0) After original point row0 y 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, S003 S005 · impliesBefore original point row0:x=0;y=0;coefficients=[-6, -1, 3, 1, -1, -1] After fixed monomial values [1,x,y,x^2,xy,y^2]=[1, 0, 0, 0, 0, 0] 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, S004 S006 · calculationBefore original point row0 monomial term0 := mul(-6, 1) After original point row0 monomial term0 = -6 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, S005 S007 · calculationBefore original point row0 monomial term1 := mul(-1, 0) After original point row0 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, S006 S008 · calculationBefore original point row0 monomial term2 := mul(3, 0) After original point row0 monomial term2 = 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, S007 S009 · calculationBefore original point row0 monomial term3 := mul(1, 0) After original point row0 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, S008 S010 · calculationBefore original point row0 monomial term4 := mul(-1, 0) After original point row0 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, S009 S011 · calculationBefore original point row0 monomial term5 := mul(-1, 0) After original point row0 monomial term5 = 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 original point row0 complete polynomial sum := sum(-6, 0, 0, 0, 0, 0) After original point row0 complete polynomial sum = -6 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 · impliesBefore original row=0;point=[0, 0];aligned claim=-6 After actual value=-6;match=Yes Conditions: none Retain original point order,repeated points and every aligned claim.Do not sort points by semantic identity and mispair their values. Step identifiers and dependenciesRule: point_value_original_pairing_not_set_sort · Uses: G001, S012 S014 · calculationBefore original point row1 x squared := mul(1, 1) After original point row1 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 original point row1 x times y := mul(1, 0) After original point row1 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, S014 S016 · calculationBefore original point row1 y squared := mul(0, 0) After original point row1 y 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, S015 S017 · impliesBefore original point row1:x=1;y=0;coefficients=[-6, -1, 3, 1, -1, -1] After fixed monomial values [1,x,y,x^2,xy,y^2]=[1, 1, 0, 1, 0, 0] 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 original point row1 monomial term0 := mul(-6, 1) After original point row1 monomial term0 = -6 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 original point row1 monomial term1 := mul(-1, 1) After original point row1 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 original point row1 monomial term2 := mul(3, 0) After original point row1 monomial term2 = 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, S019 S021 · calculationBefore original point row1 monomial term3 := mul(1, 1) After original point row1 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 original point row1 monomial term4 := mul(-1, 0) After original point row1 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 original point row1 monomial term5 := mul(-1, 0) After original point row1 monomial term5 = 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, S022 S024 · calculationBefore original point row1 complete polynomial sum := sum(-6, -1, 0, 1, 0, 0) After original point row1 complete polynomial sum = -6 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 · impliesBefore original row=1;point=[1, 0];aligned claim=-6 After actual value=-6;match=Yes Conditions: none Retain original point order,repeated points and every aligned claim.Do not sort points by semantic identity and mispair their values. Step identifiers and dependenciesRule: point_value_original_pairing_not_set_sort · Uses: G001, S024 S026 · calculationBefore original point row2 x squared := mul(0, 0) After original point row2 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, S025 S027 · calculationBefore original point row2 x times y := mul(0, 1) After original point row2 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, S026 S028 · calculationBefore original point row2 y squared := mul(1, 1) After original point row2 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, S027 S029 · impliesBefore original point row2:x=0;y=1;coefficients=[-6, -1, 3, 1, -1, -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, S028 S030 · calculationBefore original point row2 monomial term0 := mul(-6, 1) After original point row2 monomial term0 = -6 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 original point row2 monomial term1 := mul(-1, 0) After original point row2 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, S030 S032 · calculationBefore original point row2 monomial term2 := mul(3, 1) After original point row2 monomial term2 = 3 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 original point row2 monomial term3 := mul(1, 0) After original point row2 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, S032 S034 · calculationBefore original point row2 monomial term4 := mul(-1, 0) After original point row2 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, S033 S035 · calculationBefore original point row2 monomial term5 := mul(-1, 1) After original point row2 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, S034 S036 · calculationBefore original point row2 complete polynomial sum := sum(-6, 0, 3, 0, 0, -1) After original point row2 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, S035 S037 · impliesBefore original row=2;point=[0, 1];aligned claim=-4 After actual value=-4;match=Yes Conditions: none Retain original point order,repeated points and every aligned claim.Do not sort points by semantic identity and mispair their values. Step identifiers and dependenciesRule: point_value_original_pairing_not_set_sort · Uses: G001, S036 S038 · impliesBefore all original audit rows=[[0, 0, -6, -6, Yes], [1, 0, -6, -6, Yes], [0, 1, -4, -4, Yes]] After all listed rows match=Yes Conditions: none An empty table's all-match condition is vacuously true.Even all true audits only this finite table,not global identity. Step identifiers and dependenciesRule: finite_point_table_not_global_identity · Uses: G001, S037 ## Reverse checks - S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Recompute original point row2 complete polynomial sum with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute original point row2 monomial term5 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute original point row2 monomial term4 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute original point row2 monomial term3 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute original point row2 monomial term2 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute original point row2 monomial term1 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute original point row2 monomial term0 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S028 · premise_and_result_audit — Recompute original point row2 y squared with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute original point row2 x times y with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute original point row2 x squared with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Recompute original point row1 complete polynomial sum with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute original point row1 monomial term5 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute original point row1 monomial term4 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute original point row1 monomial term3 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute original point row1 monomial term2 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute original point row1 monomial term1 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute original point row1 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 original point row1 y squared with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute original point row1 x times y with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute original point row1 x squared with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Recompute original point row0 complete polynomial sum with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute original point row0 monomial term5 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute original point row0 monomial term4 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute original point row0 monomial term3 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute original point row0 monomial term2 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute original point row0 monomial term1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute original point row0 monomial term0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Recompute original point row0 y squared with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute original point row0 x times y with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute original point row0 x squared with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([-6, -1, 3, 1, -1, -1], [[0, 0, -6, -6, Yes], [1, 0, -6, -6, Yes], [0, 1, -4, -4, Yes]], Yes) 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: 11de3a6bb4be4e2a2a1ce85792adeda17131dba47c6ae5f31911740210b9bbd6 - candidate record SHA-256: 12f30de1073237982b212419696468389afe700c580ad7b750672a7925a41d43 - 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-0082.json)