Question 24428: Evaluate Every Coordinate Derivative at the Same Exact Rational Point

Original question and givens

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. Point=[3/1, 4/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, 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. Point=[3/1, 4/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).

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-gradient-vector

Step-by-step derivation

source roles = [gradient-rational-point]

derived = [[dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate]]

Gradient ORIGINAL monomial coefficient sum0 := sum(0, 3)

Gradient ORIGINAL monomial coefficient sum0 = 3

source roles = [0, 2, [2, 1], 3]

derived = [3]

Gradient ORIGINAL monomial coefficient sum1 := sum(0, 4)

Gradient ORIGINAL monomial coefficient sum1 = 4

source roles = [1, 2, [0, 1], 4]

derived = [4]

Gradient ORIGINAL monomial coefficient sum2 := sum(0, 1)

Gradient ORIGINAL monomial coefficient sum2 = 1

source roles = [2, 2, [0, 0], 1]

derived = [1]

source roles = [[[3, [2, 1]], [4, [0, 1]], [1, [0, 0]]]]

derived = [2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]]]

source roles = [0, 0, [0, 0], 1]

derived = [0]

source roles = [0, 1, [0, 1], 4]

derived = [0]

Gradient axis0 term2 power-rule coefficient := mul(3, 2)

Gradient axis0 term2 power-rule coefficient = 6

Gradient axis0 derivative combine2 := sum(0, 6)

Gradient axis0 derivative combine2 = 6

source roles = [0, 2, [2, 1], 3, 2]

derived = [[1, 1], 6]

source roles = [1, 0, [0, 0], 1]

derived = [0]

Gradient axis1 term1 power-rule coefficient := mul(4, 1)

Gradient axis1 term1 power-rule coefficient = 4

Gradient axis1 derivative combine1 := sum(0, 4)

Gradient axis1 derivative combine1 = 4

source roles = [1, 1, [0, 1], 4, 1]

derived = [[0, 0], 4]

Gradient axis1 term2 power-rule coefficient := mul(3, 1)

Gradient axis1 term2 power-rule coefficient = 3

Gradient axis1 derivative combine2 := sum(0, 3)

Gradient axis1 derivative combine2 = 3

source roles = [1, 2, [2, 1], 3, 1]

derived = [[2, 0], 3]

source roles = [2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]]]

derived = [[[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]]]

(3)^1

Gradient axis0 term0 coordinate0 power = 3

Gradient axis0 term0 coordinate0 factor := mul(6, 3)

Gradient axis0 term0 coordinate0 factor = 18

(2)^1

Gradient axis0 term0 coordinate1 power = 2

Gradient axis0 term0 coordinate1 factor := mul(18, 2)

Gradient axis0 term0 coordinate1 factor = 36

Gradient axis0 cumulative term0 := sum(0, 36)

Gradient axis0 cumulative term0 = 36

(3)^0

Gradient axis1 term0 coordinate0 power = 1

Gradient axis1 term0 coordinate0 factor := mul(4, 1)

Gradient axis1 term0 coordinate0 factor = 4

(2)^0

Gradient axis1 term0 coordinate1 power = 1

Gradient axis1 term0 coordinate1 factor := mul(4, 1)

Gradient axis1 term0 coordinate1 factor = 4

(3)^2

Gradient axis1 term1 coordinate0 power = 9

Gradient axis1 term1 coordinate0 factor := mul(3, 9)

Gradient axis1 term1 coordinate0 factor = 27

(2)^0

Gradient axis1 term1 coordinate1 power = 1

Gradient axis1 term1 coordinate1 factor := mul(27, 1)

Gradient axis1 term1 coordinate1 factor = 27

Gradient axis1 cumulative term0 := sum(0, 4)

Gradient axis1 cumulative term0 = 4

Gradient axis1 cumulative term1 := sum(4, 27)

Gradient axis1 cumulative term1 = 31

source roles = [2, [3, 2], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]]]

derived = [[36, 31], No, No]

Reverse checks

• 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 Gradient axis1 cumulative term1 with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute Gradient axis1 cumulative term0 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute Gradient axis1 term1 coordinate1 factor with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S032 · premise_and_result_audit — Recompute Gradient axis1 term1 coordinate0 factor with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S030 · premise_and_result_audit — Recompute Gradient axis1 term0 coordinate1 factor 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 Gradient axis1 term0 coordinate0 factor with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Recompute Gradient axis0 cumulative term0 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Gradient axis0 term0 coordinate1 factor with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S023 · premise_and_result_audit — Recompute Gradient axis0 term0 coordinate0 factor with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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.

Result

(2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]], [3, 2], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]], [36, 31], exact_rational_coordinate_evaluation)

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.