Question 24409: Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts

Original question and givens

Use mode=incomplete_coordinate_list, positive scale=9, with the source-declared explicit counterexample for that mode. Give (mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary).

• G001 — Use mode=incomplete_coordinate_list, positive scale=9, with the source-declared explicit counterexample for that mode. Give (mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, 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-boundary-counterexample]

derived = [[mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary]]

source roles = [incomplete_coordinate_list, 9]

derived = [incomplete_coordinate_list, 3, [9*partial_x, 9*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3]

Reverse checks

• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(incomplete_coordinate_list, 3, [9*partial_x, 9*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)

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.