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R0047-24416 · q-20261015-advance-20261113-r06-gradient-vector-0091
Use mode=saddle_stationary, positive scale=16, 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=16, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary).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.
Before
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.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [saddle_stationary, 16]
After
derived = [saddle_stationary, f(x,y)=16*(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.
Rule: Gradient_mode_specific_counterexample_with_actual_hypotheses · Uses: G001, S001
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.(saddle_stationary, f(x,y)=16*(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.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 9eecc50f2c7b6c1907222dc05809d469511bebf34ca75fbeb64381381378a3ddcandidate record SHA-256: 32dc5a247d9c1616acd7965a28509babc21bb079cefac93bff2bbc98bae5b069semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50bLean: NOT_RUN · human review: NOT_PERFORMED