Question 24483: The Symbolic Gradient Keeps Every Euclidean Coordinate in Order

Original question and givens

Use dimension=3, raw polynomial terms=[[8, [2, 1, 0]], [9, [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. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).

• G001 — Use dimension=3, raw polynomial terms=[[8, [2, 1, 0]], [9, [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. Give (dimension, canonical_terms, coordinate_derivative_polynomials, 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-symbolic-polynomial]

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

Gradient ORIGINAL monomial coefficient sum0 := sum(0, 8)

Gradient ORIGINAL monomial coefficient sum0 = 8

source roles = [0, 3, [2, 1, 0], 8]

derived = [8]

Gradient ORIGINAL monomial coefficient sum1 := sum(0, 9)

Gradient ORIGINAL monomial coefficient sum1 = 9

source roles = [1, 3, [0, 1, 1], 9]

derived = [9]

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 = [[[8, [2, 1, 0]], [9, [0, 1, 1]], [1, [0, 0, 0]]]]

derived = [3, [[1, [0, 0, 0]], [9, [0, 1, 1]], [8, [2, 1, 0]]]]

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

derived = [0]

source roles = [0, 1, [0, 1, 1], 9]

derived = [0]

Gradient axis0 term2 power-rule coefficient := mul(8, 2)

Gradient axis0 term2 power-rule coefficient = 16

Gradient axis0 derivative combine2 := sum(0, 16)

Gradient axis0 derivative combine2 = 16

source roles = [0, 2, [2, 1, 0], 8, 2]

derived = [[1, 1, 0], 16]

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

derived = [0]

Gradient axis1 term1 power-rule coefficient := mul(9, 1)

Gradient axis1 term1 power-rule coefficient = 9

Gradient axis1 derivative combine1 := sum(0, 9)

Gradient axis1 derivative combine1 = 9

source roles = [1, 1, [0, 1, 1], 9, 1]

derived = [[0, 0, 1], 9]

Gradient axis1 term2 power-rule coefficient := mul(8, 1)

Gradient axis1 term2 power-rule coefficient = 8

Gradient axis1 derivative combine2 := sum(0, 8)

Gradient axis1 derivative combine2 = 8

source roles = [1, 2, [2, 1, 0], 8, 1]

derived = [[2, 0, 0], 8]

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

derived = [0]

Gradient axis2 term1 power-rule coefficient := mul(9, 1)

Gradient axis2 term1 power-rule coefficient = 9

Gradient axis2 derivative combine1 := sum(0, 9)

Gradient axis2 derivative combine1 = 9

source roles = [2, 1, [0, 1, 1], 9, 1]

derived = [[0, 1, 0], 9]

source roles = [2, 2, [2, 1, 0], 8]

derived = [0]

source roles = [3, [[1, [0, 0, 0]], [9, [0, 1, 1]], [8, [2, 1, 0]]]]

derived = [[[[16, [1, 1, 0]]], [[9, [0, 0, 1]], [8, [2, 0, 0]]], [[9, [0, 1, 0]]]]]

Reverse checks

• 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]], [9, [0, 1, 1]], [8, [2, 1, 0]]], [[[16, [1, 1, 0]]], [[9, [0, 0, 1]], [8, [2, 0, 0]]], [[9, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)

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.