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R0020-10637 · q-20261013-advance-20261023-r07-multivariable-real-function-0087
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.
G001 — 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.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.
Before
original six coefficients=[-1, -1, 1, 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.
Rule: bivariate_explicit_slice_scope · Uses: G001
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
original point row0:x=0;y=0;coefficients=[-1, -1, 1, 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.
Rule: bivariate_fixed_coordinate_monomial_roles · Uses: G001, S004
Before
original point row0 monomial term0 := mul(-1, 1)
After
original point row0 monomial term0 = -1
Conditions: none
Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
original point row0 monomial term2 := mul(1, 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.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
original point row0 complete polynomial sum := sum(-1, 0, 0, 0, 0, 0)
After
original point row0 complete polynomial sum = -1
Conditions: none
Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
original row=0;point=[0, 0];aligned claim=-1
After
actual value=-1;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.
Rule: point_value_original_pairing_not_set_sort · Uses: G001, S012
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
original point row1:x=1;y=0;coefficients=[-1, -1, 1, 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.
Rule: bivariate_fixed_coordinate_monomial_roles · Uses: G001, S016
Before
original point row1 monomial term0 := mul(-1, 1)
After
original point row1 monomial term0 = -1
Conditions: none
Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
original point row1 monomial term2 := mul(1, 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.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
original point row1 complete polynomial sum := sum(-1, -1, 0, 1, 0, 0)
After
original point row1 complete polynomial sum = -1
Conditions: none
Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
original row=1;point=[1, 0];aligned claim=-1
After
actual value=-1;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.
Rule: point_value_original_pairing_not_set_sort · Uses: G001, S024
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
original point row2:x=0;y=1;coefficients=[-1, -1, 1, 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.
Rule: bivariate_fixed_coordinate_monomial_roles · Uses: G001, S028
Before
original point row2 monomial term0 := mul(-1, 1)
After
original point row2 monomial term0 = -1
Conditions: none
Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
original point row2 monomial term2 := mul(1, 1)
After
original point row2 monomial term2 = 1
Conditions: none
Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
original point row2 complete polynomial sum := sum(-1, 0, 1, 0, 0, -1)
After
original point row2 complete polynomial sum = -1
Conditions: none
Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
original row=2;point=[0, 1];aligned claim=-1
After
actual value=-1;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.
Rule: point_value_original_pairing_not_set_sort · Uses: G001, S036
Before
all original audit rows=[[0, 0, -1, -1, Yes], [1, 0, -1, -1, Yes], [0, 1, -1, -1, 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.
Rule: finite_point_table_not_global_identity · Uses: G001, S037
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.([-1, -1, 1, 1, 1, -1], [[0, 0, -1, -1, Yes], [1, 0, -1, -1, Yes], [0, 1, -1, -1, Yes]], Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 6439e37bdbbe1af6ad03d3f5b2b518275f09a3c4a6c1056149b4c6f1c6bc87d7candidate record SHA-256: cd3ea28d36aee72b5230bf9b49a83dbdd39eacbf3e79d1fff595a703596e639csemantic review receipt SHA-256: b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598fquestion-first witness SHA-256: 3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4Lean: NOT_RUN · human review: NOT_PERFORMED