# Question 24452: A Zero Gradient Certifies Stationarity, Not an Automatic Extremum [Return to catalogue page 123](https://ocme.evemisslab.com/en/questions/page/123/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261113-r06-gradient-vector-foundations/#q-20261015-advance-20261113-r06-gradient-vector-0052) 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. R0047-24452 · q-20261015-advance-20261113-r06-gradient-vector-0052 ## Original question and givens Use dimension=2, raw polynomial terms=[[2, [2, 0]], [3, [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=[[2, [2, 0]], [3, [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](https://ocme.evemisslab.com/en/mko/mko-gradient-vector/) ## Step-by-step derivation S001 · impliesBefore source roles = [gradient-stationary-audit] After derived = [[dimension, 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 · calculationBefore Gradient ORIGINAL monomial coefficient sum0 := sum(0, 2) After Gradient ORIGINAL monomial coefficient sum0 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore source roles = [0, 2, [2, 0], 2] After derived = [2] Conditions: none Combine only identical ordered exponent vectors;remove zeros after combining and keep fixed Euclidean coordinate order. Step identifiers and dependenciesRule: Gradient_original_term_position_and_exponent_arity · Uses: G001, S002 S004 · calculationBefore Gradient ORIGINAL monomial coefficient sum1 := sum(0, 3) After Gradient ORIGINAL monomial coefficient sum1 = 3 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore source roles = [1, 2, [0, 2], 3] After derived = [3] Conditions: none Combine only identical ordered exponent vectors;remove zeros after combining and keep fixed Euclidean coordinate order. Step identifiers and dependenciesRule: Gradient_original_term_position_and_exponent_arity · Uses: G001, S004 S006 · impliesBefore source roles = [[[2, [2, 0]], [3, [0, 2]]]] After derived = [2, [[3, [0, 2]], [2, [2, 0]]]] Conditions: none Use source English canonical-exponent order,not numeric tuple order.Complete cancellation may yield an empty canonical list for the zero polynomial. Step identifiers and dependenciesRule: Gradient_canonical_nonzero_polynomial_terms_source_order · Uses: G001, S005 S007 · impliesBefore source roles = [0, 0, [0, 2], 3] After derived = [0] Conditions: none Exponent0 means derivative0 in this coordinate,without omitting other coordinates. Step identifiers and dependenciesRule: Gradient_constant_in_coordinate_derivative_zero · Uses: G001, S006 S008 · calculationBefore Gradient axis0 term1 power-rule coefficient := mul(2, 2) After Gradient axis0 term1 power-rule coefficient = 4 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Gradient axis0 derivative combine1 := sum(0, 4) After Gradient axis0 derivative combine1 = 4 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore source roles = [0, 1, [2, 0], 2, 2] After derived = [[1, 0], 4] Conditions: none Hold the other variables fixed:the coefficient becomes ck and only this coordinate exponent decreases by1. Step identifiers and dependenciesRule: Gradient_coordinate_power_rule_ordered_exponents · Uses: G001, S009 S011 · calculationBefore Gradient axis1 term0 power-rule coefficient := mul(3, 2) After Gradient axis1 term0 power-rule coefficient = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Gradient axis1 derivative combine0 := sum(0, 6) After Gradient axis1 derivative combine0 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore source roles = [1, 0, [0, 2], 3, 2] After derived = [[0, 1], 6] Conditions: none Hold the other variables fixed:the coefficient becomes ck and only this coordinate exponent decreases by1. Step identifiers and dependenciesRule: Gradient_coordinate_power_rule_ordered_exponents · Uses: G001, S012 S014 · impliesBefore source roles = [1, 1, [2, 0], 2] After derived = [0] Conditions: none Exponent0 means derivative0 in this coordinate,without omitting other coordinates. Step identifiers and dependenciesRule: Gradient_constant_in_coordinate_derivative_zero · Uses: G001, S013 S015 · impliesBefore source roles = [2, [[3, [0, 2]], [2, [2, 0]]]] After derived = [[[[4, [1, 0]]], [[6, [0, 1]]]]] Conditions: none The gradient contains all d ordered component polynomials.Polynomial differentiability makes this Euclidean vector represent the differential,not a sampled single direction. Step identifiers and dependenciesRule: Gradient_ALL_ordered_coordinate_polynomials · Uses: G001, S014 S016 · impliesBefore (0)^1 After Gradient axis0 term0 coordinate0 power = 0 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S015 S017 · calculationBefore Gradient axis0 term0 coordinate0 factor := mul(4, 0) After Gradient axis0 term0 coordinate0 factor = 0 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore (0)^0 After Gradient axis0 term0 coordinate1 power = 1 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S017 S019 · calculationBefore Gradient axis0 term0 coordinate1 factor := mul(0, 1) After Gradient axis0 term0 coordinate1 factor = 0 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Gradient axis0 cumulative term0 := sum(0, 0) After Gradient axis0 cumulative term0 = 0 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore (0)^0 After Gradient axis1 term0 coordinate0 power = 1 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S020 S022 · calculationBefore Gradient axis1 term0 coordinate0 factor := mul(6, 1) After Gradient axis1 term0 coordinate0 factor = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · impliesBefore (0)^1 After Gradient axis1 term0 coordinate1 power = 0 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S022 S024 · calculationBefore Gradient axis1 term0 coordinate1 factor := mul(6, 0) After Gradient axis1 term0 coordinate1 factor = 0 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Gradient axis1 cumulative term0 := sum(0, 0) After Gradient axis1 cumulative term0 = 0 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · impliesBefore source roles = [2, [0, 0], [[[4, [1, 0]]], [[6, [0, 1]]]]] After derived = [[0, 0], Yes, No] Conditions: none Evaluate every component at the same point before deciding stationarity.Zero gradient supplies no automatic extremum certificate;No does not deny actual extrema of special cases such as constant functions. Step identifiers and dependenciesRule: Gradient_SAME_point_all_coordinates_stationary_not_extremum · Uses: G001, S025 ## 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: a4a6540ef5155cef1d1d67d9b080ee0649190caa53d3de8abe2613643b023445 - candidate record SHA-256: 5d9c4c8104892f74f4ad417b44ea776210cc99bc10310a67cf7031d530caf2d8 - 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-0052.json)