Question 24427: Evaluate Every Coordinate Derivative at the Same Exact Rational Point

Original question and givens

Use dimension=3, raw polynomial terms=[[2, [2, 1, 0]], [3, [0, 1, 1]], [1, [0, 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=[2/1, 3/2, 4/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).

• G001 — Use dimension=3, raw polynomial terms=[[2, [2, 1, 0]], [3, [0, 1, 1]], [1, [0, 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=[2/1, 3/2, 4/3]. 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, 2)

Gradient ORIGINAL monomial coefficient sum0 = 2

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

derived = [2]

Gradient ORIGINAL monomial coefficient sum1 := sum(0, 3)

Gradient ORIGINAL monomial coefficient sum1 = 3

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

derived = [3]

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

Gradient ORIGINAL monomial coefficient sum2 = 1

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

derived = [1]

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

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

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

derived = [0]

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

derived = [0]

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

Gradient axis0 term2 power-rule coefficient = 4

Gradient axis0 derivative combine2 := sum(0, 4)

Gradient axis0 derivative combine2 = 4

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

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

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

derived = [0]

Gradient axis1 term1 power-rule coefficient := mul(3, 1)

Gradient axis1 term1 power-rule coefficient = 3

Gradient axis1 derivative combine1 := sum(0, 3)

Gradient axis1 derivative combine1 = 3

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

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

Gradient axis1 term2 power-rule coefficient := mul(2, 1)

Gradient axis1 term2 power-rule coefficient = 2

Gradient axis1 derivative combine2 := sum(0, 2)

Gradient axis1 derivative combine2 = 2

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

derived = [[2, 0, 0], 2]

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

derived = [0]

Gradient axis2 term1 power-rule coefficient := mul(3, 1)

Gradient axis2 term1 power-rule coefficient = 3

Gradient axis2 derivative combine1 := sum(0, 3)

Gradient axis2 derivative combine1 = 3

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

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

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

derived = [0]

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

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

(2)^1

Gradient axis0 term0 coordinate0 power = 2

Gradient axis0 term0 coordinate0 factor := mul(4, 2)

Gradient axis0 term0 coordinate0 factor = 8

(3/2)^1

Gradient axis0 term0 coordinate1 power = 3/2

Gradient axis0 term0 coordinate1 factor := mul(8, 3/2)

Gradient axis0 term0 coordinate1 factor = 12

(4/3)^0

Gradient axis0 term0 coordinate2 power = 1

Gradient axis0 term0 coordinate2 factor := mul(12, 1)

Gradient axis0 term0 coordinate2 factor = 12

Gradient axis0 cumulative term0 := sum(0, 12)

Gradient axis0 cumulative term0 = 12

(2)^0

Gradient axis1 term0 coordinate0 power = 1

Gradient axis1 term0 coordinate0 factor := mul(3, 1)

Gradient axis1 term0 coordinate0 factor = 3

(3/2)^0

Gradient axis1 term0 coordinate1 power = 1

Gradient axis1 term0 coordinate1 factor := mul(3, 1)

Gradient axis1 term0 coordinate1 factor = 3

(4/3)^1

Gradient axis1 term0 coordinate2 power = 4/3

Gradient axis1 term0 coordinate2 factor := mul(3, 4/3)

Gradient axis1 term0 coordinate2 factor = 4

(2)^2

Gradient axis1 term1 coordinate0 power = 4

Gradient axis1 term1 coordinate0 factor := mul(2, 4)

Gradient axis1 term1 coordinate0 factor = 8

(3/2)^0

Gradient axis1 term1 coordinate1 power = 1

Gradient axis1 term1 coordinate1 factor := mul(8, 1)

Gradient axis1 term1 coordinate1 factor = 8

(4/3)^0

Gradient axis1 term1 coordinate2 power = 1

Gradient axis1 term1 coordinate2 factor := mul(8, 1)

Gradient axis1 term1 coordinate2 factor = 8

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

Gradient axis1 cumulative term0 = 4

Gradient axis1 cumulative term1 := sum(4, 8)

Gradient axis1 cumulative term1 = 12

(2)^0

Gradient axis2 term0 coordinate0 power = 1

Gradient axis2 term0 coordinate0 factor := mul(3, 1)

Gradient axis2 term0 coordinate0 factor = 3

(3/2)^1

Gradient axis2 term0 coordinate1 power = 3/2

Gradient axis2 term0 coordinate1 factor := mul(3, 3/2)

Gradient axis2 term0 coordinate1 factor = 9/2

(4/3)^0

Gradient axis2 term0 coordinate2 power = 1

Gradient axis2 term0 coordinate2 factor := mul(9/2, 1)

Gradient axis2 term0 coordinate2 factor = 9/2

Gradient axis2 cumulative term0 := sum(0, 9/2)

Gradient axis2 cumulative term0 = 9/2

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

derived = [[12, 12, 9/2], No, No]

Reverse checks

• S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S054 · premise_and_result_audit — Recompute Gradient axis2 cumulative term0 with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Gradient axis2 term0 coordinate2 factor with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S051 · premise_and_result_audit — Recompute Gradient axis2 term0 coordinate1 factor with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S049 · premise_and_result_audit — Recompute Gradient axis2 term0 coordinate0 factor with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S047 · premise_and_result_audit — Recompute Gradient axis1 cumulative term1 with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Gradient axis1 cumulative term0 with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute Gradient axis1 term1 coordinate2 factor with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S043 · premise_and_result_audit — Recompute Gradient axis1 term1 coordinate1 factor with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S041 · premise_and_result_audit — Recompute Gradient axis1 term1 coordinate0 factor with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S039 · premise_and_result_audit — Recompute Gradient axis1 term0 coordinate2 factor with exact rational arithmetic, not rounded decimals.
• 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 — Recompute Gradient axis1 term0 coordinate1 factor with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S035 · premise_and_result_audit — Recompute Gradient axis1 term0 coordinate0 factor with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S033 · premise_and_result_audit — Recompute Gradient axis0 cumulative term0 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute Gradient axis0 term0 coordinate2 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 axis0 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 axis0 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — 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 axis2 derivative combine1 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Gradient axis2 term1 power-rule coefficient with exact rational arithmetic, not rounded decimals.
• 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

(3, [[1, [0, 0, 0]], [3, [0, 1, 1]], [2, [2, 1, 0]]], [2, 3/2, 4/3], [[[4, [1, 1, 0]]], [[3, [0, 0, 1]], [2, [2, 0, 0]]], [[3, [0, 1, 0]]]], [12, 12, 9/2], 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.