Question 24443: Evaluate Every Coordinate Derivative at the Same Exact Rational Point

Original question and givens

Use dimension=3, raw polynomial terms=[[18, [2, 1, 0]], [19, [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=[18/1, 19/2, 20/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).

• G001 — Use dimension=3, raw polynomial terms=[[18, [2, 1, 0]], [19, [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=[18/1, 19/2, 20/3]. 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, 18)

Gradient ORIGINAL monomial coefficient sum0 = 18

source roles = [0, 3, [2, 1, 0], 18]

derived = [18]

Gradient ORIGINAL monomial coefficient sum1 := sum(0, 19)

Gradient ORIGINAL monomial coefficient sum1 = 19

source roles = [1, 3, [0, 1, 1], 19]

derived = [19]

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 = [[[18, [2, 1, 0]], [19, [0, 1, 1]], [1, [0, 0, 0]]]]

derived = [3, [[1, [0, 0, 0]], [19, [0, 1, 1]], [18, [2, 1, 0]]]]

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

derived = [0]

source roles = [0, 1, [0, 1, 1], 19]

derived = [0]

Gradient axis0 term2 power-rule coefficient := mul(18, 2)

Gradient axis0 term2 power-rule coefficient = 36

Gradient axis0 derivative combine2 := sum(0, 36)

Gradient axis0 derivative combine2 = 36

source roles = [0, 2, [2, 1, 0], 18, 2]

derived = [[1, 1, 0], 36]

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

derived = [0]

Gradient axis1 term1 power-rule coefficient := mul(19, 1)

Gradient axis1 term1 power-rule coefficient = 19

Gradient axis1 derivative combine1 := sum(0, 19)

Gradient axis1 derivative combine1 = 19

source roles = [1, 1, [0, 1, 1], 19, 1]

derived = [[0, 0, 1], 19]

Gradient axis1 term2 power-rule coefficient := mul(18, 1)

Gradient axis1 term2 power-rule coefficient = 18

Gradient axis1 derivative combine2 := sum(0, 18)

Gradient axis1 derivative combine2 = 18

source roles = [1, 2, [2, 1, 0], 18, 1]

derived = [[2, 0, 0], 18]

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

derived = [0]

Gradient axis2 term1 power-rule coefficient := mul(19, 1)

Gradient axis2 term1 power-rule coefficient = 19

Gradient axis2 derivative combine1 := sum(0, 19)

Gradient axis2 derivative combine1 = 19

source roles = [2, 1, [0, 1, 1], 19, 1]

derived = [[0, 1, 0], 19]

source roles = [2, 2, [2, 1, 0], 18]

derived = [0]

source roles = [3, [[1, [0, 0, 0]], [19, [0, 1, 1]], [18, [2, 1, 0]]]]

derived = [[[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]]]

(18)^1

Gradient axis0 term0 coordinate0 power = 18

Gradient axis0 term0 coordinate0 factor := mul(36, 18)

Gradient axis0 term0 coordinate0 factor = 648

(19/2)^1

Gradient axis0 term0 coordinate1 power = 19/2

Gradient axis0 term0 coordinate1 factor := mul(648, 19/2)

Gradient axis0 term0 coordinate1 factor = 6156

(20/3)^0

Gradient axis0 term0 coordinate2 power = 1

Gradient axis0 term0 coordinate2 factor := mul(6156, 1)

Gradient axis0 term0 coordinate2 factor = 6156

Gradient axis0 cumulative term0 := sum(0, 6156)

Gradient axis0 cumulative term0 = 6156

(18)^0

Gradient axis1 term0 coordinate0 power = 1

Gradient axis1 term0 coordinate0 factor := mul(19, 1)

Gradient axis1 term0 coordinate0 factor = 19

(19/2)^0

Gradient axis1 term0 coordinate1 power = 1

Gradient axis1 term0 coordinate1 factor := mul(19, 1)

Gradient axis1 term0 coordinate1 factor = 19

(20/3)^1

Gradient axis1 term0 coordinate2 power = 20/3

Gradient axis1 term0 coordinate2 factor := mul(19, 20/3)

Gradient axis1 term0 coordinate2 factor = 380/3

(18)^2

Gradient axis1 term1 coordinate0 power = 324

Gradient axis1 term1 coordinate0 factor := mul(18, 324)

Gradient axis1 term1 coordinate0 factor = 5832

(19/2)^0

Gradient axis1 term1 coordinate1 power = 1

Gradient axis1 term1 coordinate1 factor := mul(5832, 1)

Gradient axis1 term1 coordinate1 factor = 5832

(20/3)^0

Gradient axis1 term1 coordinate2 power = 1

Gradient axis1 term1 coordinate2 factor := mul(5832, 1)

Gradient axis1 term1 coordinate2 factor = 5832

Gradient axis1 cumulative term0 := sum(0, 380/3)

Gradient axis1 cumulative term0 = 380/3

Gradient axis1 cumulative term1 := sum(380/3, 5832)

Gradient axis1 cumulative term1 = 17876/3

(18)^0

Gradient axis2 term0 coordinate0 power = 1

Gradient axis2 term0 coordinate0 factor := mul(19, 1)

Gradient axis2 term0 coordinate0 factor = 19

(19/2)^1

Gradient axis2 term0 coordinate1 power = 19/2

Gradient axis2 term0 coordinate1 factor := mul(19, 19/2)

Gradient axis2 term0 coordinate1 factor = 361/2

(20/3)^0

Gradient axis2 term0 coordinate2 power = 1

Gradient axis2 term0 coordinate2 factor := mul(361/2, 1)

Gradient axis2 term0 coordinate2 factor = 361/2

Gradient axis2 cumulative term0 := sum(0, 361/2)

Gradient axis2 cumulative term0 = 361/2

source roles = [3, [18, 19/2, 20/3], [[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]]]

derived = [[6156, 17876/3, 361/2], 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]], [19, [0, 1, 1]], [18, [2, 1, 0]]], [18, 19/2, 20/3], [[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]], [6156, 17876/3, 361/2], 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.