Question 24465: A Zero Gradient Certifies Stationarity, Not an Automatic Extremum

Original question and givens

Use dimension=2, raw polynomial terms=[[15, [2, 0]], [16, [0, 2]]], 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=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).

• G001 — Use dimension=2, raw polynomial terms=[[15, [2, 0]], [16, [0, 2]]], 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=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).

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-stationary-audit]

derived = [[dimension, point, gradient_values, stationary, extremum_certificate, boundary]]

Gradient ORIGINAL monomial coefficient sum0 := sum(0, 15)

Gradient ORIGINAL monomial coefficient sum0 = 15

source roles = [0, 2, [2, 0], 15]

derived = [15]

Gradient ORIGINAL monomial coefficient sum1 := sum(0, 16)

Gradient ORIGINAL monomial coefficient sum1 = 16

source roles = [1, 2, [0, 2], 16]

derived = [16]

source roles = [[[15, [2, 0]], [16, [0, 2]]]]

derived = [2, [[16, [0, 2]], [15, [2, 0]]]]

source roles = [0, 0, [0, 2], 16]

derived = [0]

Gradient axis0 term1 power-rule coefficient := mul(15, 2)

Gradient axis0 term1 power-rule coefficient = 30

Gradient axis0 derivative combine1 := sum(0, 30)

Gradient axis0 derivative combine1 = 30

source roles = [0, 1, [2, 0], 15, 2]

derived = [[1, 0], 30]

Gradient axis1 term0 power-rule coefficient := mul(16, 2)

Gradient axis1 term0 power-rule coefficient = 32

Gradient axis1 derivative combine0 := sum(0, 32)

Gradient axis1 derivative combine0 = 32

source roles = [1, 0, [0, 2], 16, 2]

derived = [[0, 1], 32]

source roles = [1, 1, [2, 0], 15]

derived = [0]

source roles = [2, [[16, [0, 2]], [15, [2, 0]]]]

derived = [[[[30, [1, 0]]], [[32, [0, 1]]]]]

(0)^1

Gradient axis0 term0 coordinate0 power = 0

Gradient axis0 term0 coordinate0 factor := mul(30, 0)

Gradient axis0 term0 coordinate0 factor = 0

(0)^0

Gradient axis0 term0 coordinate1 power = 1

Gradient axis0 term0 coordinate1 factor := mul(0, 1)

Gradient axis0 term0 coordinate1 factor = 0

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

Gradient axis0 cumulative term0 = 0

(0)^0

Gradient axis1 term0 coordinate0 power = 1

Gradient axis1 term0 coordinate0 factor := mul(32, 1)

Gradient axis1 term0 coordinate0 factor = 32

(0)^1

Gradient axis1 term0 coordinate1 power = 0

Gradient axis1 term0 coordinate1 factor := mul(32, 0)

Gradient axis1 term0 coordinate1 factor = 0

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

Gradient axis1 cumulative term0 = 0

source roles = [2, [0, 0], [[[30, [1, 0]]], [[32, [0, 1]]]]]

derived = [[0, 0], Yes, No]

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 — Recompute Gradient axis1 cumulative term0 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Gradient axis1 term0 coordinate1 factor with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Recompute Gradient axis1 term0 coordinate0 factor 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 — Recompute Gradient axis0 cumulative term0 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Gradient axis0 term0 coordinate1 factor with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Recompute Gradient axis0 term0 coordinate0 factor with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 axis1 derivative combine0 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Gradient axis1 term0 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 — Recompute Gradient axis0 derivative combine1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Gradient axis0 term1 power-rule coefficient with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)

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.