# Question 24420: Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0047-24420 · q-20261015-advance-20261113-r06-gradient-vector-0095



## Original question and givens

Use mode=saddle_stationary, positive scale=20, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary).


- G001 — Use mode=saddle_stationary, positive scale=20, with the source-declared explicit counterexample for that mode. Give (mode, function_description, 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](https://ocme.evemisslab.com/en/mko/mko-gradient-vector/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [gradient-boundary-counterexample]

After

derived = [[mode, function_description, point, gradient_values, stationary, extremum_certificate, 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 dependenciesRule: source_ordered_fields_without_JS · Uses: G001





S002 · impliesBefore

source roles = [saddle_stationary, 20]

After

derived = [saddle_stationary, f(x,y)=20*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate]

Conditions: none

The gradient[2sx,−2sy] vanishes at0,but arbitrarily small(h,0) gives positive values and(0,h) negative values,so neither local extremum exists.

Step identifiers and dependenciesRule: 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

(saddle_stationary, f(x,y)=20*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)

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: 0d7ad6dc3d8871fe7009ad96859a2f0eaedad61878e6df72949f5433eac9298e
- candidate record SHA-256: 10e87f84b82199560d7ecccf317a8dde6f1f556b3a3f6367753ca0bd381af9d4
- semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386
- question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261113-r06-gradient-vector-0095.json)