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Question 24402: Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0047-24402 · q-20261015-advance-20261113-r06-gradient-vector-0077

Original question and givens

Use mode=nondifferentiable_norm, positive scale=2, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary).

  • G001 — Use mode=nondifferentiable_norm, positive scale=2, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partials_exist, gradient_defined, 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.

Step-by-step derivation

S001 · implies

Before

source roles = [gradient-boundary-counterexample]

After

derived = [[mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary]]

Conditions: none

List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [nondifferentiable_norm, 2]

After

derived = [nondifferentiable_norm, f(x,y)=2*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined]

Conditions: none

Along the x-axis the quotient s*|h|/h has one-sided limits s and−s,unequal because s>0;the y-axis is the same.

Step identifiers and dependencies

Rule: Gradient_mode_specific_counterexample_with_actual_hypotheses · Uses: G001, S001

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

(nondifferentiable_norm, f(x,y)=2*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: da611211d394609e3595be260d936ed80ee1ae9baabac6e0966ede47181c3513
  • candidate record SHA-256: 3495e1ee1d31b84c6653e985d0c0596872f2f6a79b1ac0af70e689bbe318c380
  • semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386
  • question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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