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R0047-24478 · q-20261015-advance-20261113-r06-gradient-vector-0003
Use dimension=2, raw polynomial terms=[[3, [2, 1]], [4, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
G001 — Use dimension=2, raw polynomial terms=[[3, [2, 1]], [4, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).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
source roles = [gradient-symbolic-polynomial]
After
derived = [[dimension, canonical_terms, coordinate_derivative_polynomials, certificate]]
Conditions: none
List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
Gradient ORIGINAL monomial coefficient sum0 := sum(0, 3)
After
Gradient ORIGINAL monomial coefficient sum0 = 3
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
source roles = [0, 2, [2, 1], 3]
After
derived = [3]
Conditions: none
Combine only identical ordered exponent vectors;remove zeros after combining and keep fixed Euclidean coordinate order.
Rule: Gradient_original_term_position_and_exponent_arity · Uses: G001, S002
Before
Gradient ORIGINAL monomial coefficient sum1 := sum(0, 4)
After
Gradient ORIGINAL monomial coefficient sum1 = 4
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
source roles = [1, 2, [0, 1], 4]
After
derived = [4]
Conditions: none
Combine only identical ordered exponent vectors;remove zeros after combining and keep fixed Euclidean coordinate order.
Rule: Gradient_original_term_position_and_exponent_arity · Uses: G001, S004
Before
Gradient ORIGINAL monomial coefficient sum2 := sum(0, 1)
After
Gradient ORIGINAL monomial coefficient sum2 = 1
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
source roles = [2, 2, [0, 0], 1]
After
derived = [1]
Conditions: none
Combine only identical ordered exponent vectors;remove zeros after combining and keep fixed Euclidean coordinate order.
Rule: Gradient_original_term_position_and_exponent_arity · Uses: G001, S006
Before
source roles = [[[3, [2, 1]], [4, [0, 1]], [1, [0, 0]]]]
After
derived = [2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]]]
Conditions: none
Use source English canonical-exponent order,not numeric tuple order.Complete cancellation may yield an empty canonical list for the zero polynomial.
Rule: Gradient_canonical_nonzero_polynomial_terms_source_order · Uses: G001, S007
Before
source roles = [0, 0, [0, 0], 1]
After
derived = [0]
Conditions: none
Exponent0 means derivative0 in this coordinate,without omitting other coordinates.
Rule: Gradient_constant_in_coordinate_derivative_zero · Uses: G001, S008
Before
source roles = [0, 1, [0, 1], 4]
After
derived = [0]
Conditions: none
Exponent0 means derivative0 in this coordinate,without omitting other coordinates.
Rule: Gradient_constant_in_coordinate_derivative_zero · Uses: G001, S009
Before
Gradient axis0 term2 power-rule coefficient := mul(3, 2)
After
Gradient axis0 term2 power-rule coefficient = 6
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
Gradient axis0 derivative combine2 := sum(0, 6)
After
Gradient axis0 derivative combine2 = 6
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
source roles = [0, 2, [2, 1], 3, 2]
After
derived = [[1, 1], 6]
Conditions: none
Hold the other variables fixed:the coefficient becomes ck and only this coordinate exponent decreases by1.
Rule: Gradient_coordinate_power_rule_ordered_exponents · Uses: G001, S012
Before
source roles = [1, 0, [0, 0], 1]
After
derived = [0]
Conditions: none
Exponent0 means derivative0 in this coordinate,without omitting other coordinates.
Rule: Gradient_constant_in_coordinate_derivative_zero · Uses: G001, S013
Before
Gradient axis1 term1 power-rule coefficient := mul(4, 1)
After
Gradient axis1 term1 power-rule coefficient = 4
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
Gradient axis1 derivative combine1 := sum(0, 4)
After
Gradient axis1 derivative combine1 = 4
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
source roles = [1, 1, [0, 1], 4, 1]
After
derived = [[0, 0], 4]
Conditions: none
Hold the other variables fixed:the coefficient becomes ck and only this coordinate exponent decreases by1.
Rule: Gradient_coordinate_power_rule_ordered_exponents · Uses: G001, S016
Before
Gradient axis1 term2 power-rule coefficient := mul(3, 1)
After
Gradient axis1 term2 power-rule coefficient = 3
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
Gradient axis1 derivative combine2 := sum(0, 3)
After
Gradient axis1 derivative combine2 = 3
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
source roles = [1, 2, [2, 1], 3, 1]
After
derived = [[2, 0], 3]
Conditions: none
Hold the other variables fixed:the coefficient becomes ck and only this coordinate exponent decreases by1.
Rule: Gradient_coordinate_power_rule_ordered_exponents · Uses: G001, S019
Before
source roles = [2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]]]
After
derived = [[[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]]]
Conditions: none
The gradient contains all d ordered component polynomials.Polynomial differentiability makes this Euclidean vector represent the differential,not a sampled single direction.
Rule: Gradient_ALL_ordered_coordinate_polynomials · Uses: G001, S020
S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S019 · premise_and_result_audit — Recompute Gradient axis1 derivative combine2 with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute Gradient axis1 term2 power-rule coefficient 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 Gradient axis1 derivative combine1 with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute Gradient axis1 term1 power-rule coefficient with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 Gradient axis0 derivative combine2 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute Gradient axis0 term2 power-rule coefficient with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S006 · premise_and_result_audit — Recompute Gradient ORIGINAL monomial coefficient sum2 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 Gradient ORIGINAL monomial coefficient sum1 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Recompute Gradient ORIGINAL monomial coefficient sum0 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.(2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)
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: a70bcace4119a0af64f36c6802ce11ab0422b9308a749521ac1e5b03416f1310candidate record SHA-256: 8e19334daa355e41ccb5af36e5518746a8c49e9caaa38bbd23eed8c11ece5e66semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50bLean: NOT_RUN · human review: NOT_PERFORMED